Recent zbMATH articles in MSC 46https://www.zbmath.org/atom/cc/462021-02-27T13:50:00+00:00WerkzeugOn the duality of variable Triebel-Lizorkin spaces.https://www.zbmath.org/1453.460262021-02-27T13:50:00+00:00"Drihem, Douadi"https://www.zbmath.org/authors/?q=ai:drihem.douadiLet \(1<q<\infty\), \(\frac{1}{q} + \frac{1}{q'} =1\) and \(s\in \mathbb R\). The duality
\[
F^s_{1,q} (\mathbb R^n)' = F^{-s}_{\infty, q'} (\mathbb R^n)
\]
is one of the cornerstones of the theory of the spaces \(F^s_{\infty,q} (\mathbb R^n)\), including \(bmo^s (\mathbb R^n) = F^s_{\infty,2} (\mathbb R^n)\). It is the
main aim of this paper to extend this assertion to the spaces \(F^{s(\cdot)}_{1, q(\cdot)} (\mathbb R^n)\) with variable \(s(x)\) and \(q(x)\).
Reviewer: Hans Triebel (Jena)Corrigendum to: ``Sobolev spaces with variable exponents on complete manifolds''.https://www.zbmath.org/1453.460272021-02-27T13:50:00+00:00"Gaczkowski, Michał"https://www.zbmath.org/authors/?q=ai:gaczkowski.michal"Górka, Przemysław"https://www.zbmath.org/authors/?q=ai:gorka.przemyslaw"Pons, Daniel J."https://www.zbmath.org/authors/?q=ai:pons.daniel-jFrom the text: In this note, we correct some misprints in our paper [J. Funct. Anal. 270, No. 4, 1379--1415 (2016; Zbl 1346.46026)]. In particular, we give the correct formulation of Theorem 6.1, and for the reader's convenience, we provide some elements of the proof.Analysis of Legendre polynomial kernel in support vector machines.https://www.zbmath.org/1453.681492021-02-27T13:50:00+00:00"Djelloul, Naima"https://www.zbmath.org/authors/?q=ai:djelloul.naima"Amir, Abdessamad"https://www.zbmath.org/authors/?q=ai:amir.abdessamadSummary: For several types of machines learning problems, the support vector machine is a method of choice. The kernel functions are a basic ingredient in support vector machine theory. Kernels based on the concepts of orthogonal polynomials gave the great satisfaction in practice. In this paper we identify the reproducing kernel Hilbert space of Legendre polynomial kernel which allows us to understand its ability to extract more discriminative features. We also show that without being a universal kernel, Legendre kernel possesses the same separation properties. The Legendre, Gaussian and polynomial kernel performance has been first evaluated on two dimensional illustrative examples in order to give a graphical comparison, then on real world data sets from UCI repository. For nonlinearly separable data, Legendre kernel always gives satisfaction regarding classification accuracy and reduction in the number of support vectors.A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics.https://www.zbmath.org/1453.681632021-02-27T13:50:00+00:00"E, Weinan"https://www.zbmath.org/authors/?q=ai:e.weinan"Ma, Chao"https://www.zbmath.org/authors/?q=ai:ma.chao"Wu, Lei"https://www.zbmath.org/authors/?q=ai:wu.lei.2|wu.lei.1|wu.lei.3|wu.lei.4Summary: A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert space.The John-Nirenberg inequality for the regularized BLO space on non-homogeneous metric measure spaces.https://www.zbmath.org/1453.460392021-02-27T13:50:00+00:00"Lin, Haibo"https://www.zbmath.org/authors/?q=ai:lin.haibo"Liu, Zhen"https://www.zbmath.org/authors/?q=ai:liu.zhen|liu.zhen.1"Wang, Chenyan"https://www.zbmath.org/authors/?q=ai:wang.chenyanIn the classical Euclidean space (the Euclidean space equipped with the Lebesgue measure), the John-Nirenberg inequality for the space BMO established by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)] examines the rate of logarithmic growth of functions in BMO. After this, a variety of John-Nirenberg type inequalities have been established, but not for the regularized BLO. In this paper, such an inequality is established for the regularized BLO in a metric measure space satisfying the geometrically doubling condition and the upper doubling condition.
Reviewer: Elijah Liflyand (Ramat-Gan)Regular variation and free regular infinitely divisible laws.https://www.zbmath.org/1453.601022021-02-27T13:50:00+00:00"Chakrabarty, Arijit"https://www.zbmath.org/authors/?q=ai:chakrabarty.arijit"Chakraborty, Sukrit"https://www.zbmath.org/authors/?q=ai:chakraborty.sukrit"Hazra, Rajat Subhra"https://www.zbmath.org/authors/?q=ai:hazra.rajat-subhraSummary: In this article the relation between the tail behaviours of a free regular infinitely divisible probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of \textit{P. Embrechts} et al. [Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 335--347 (1979; Zbl 0397.60024)] and our result using the Bercovici-Pata bijection.Adaptive Haar type wavelets on manifolds.https://www.zbmath.org/1453.420332021-02-27T13:50:00+00:00"Dem'yanovich, Yu. K."https://www.zbmath.org/authors/?q=ai:demyanovich.yuri-kSummary: We consider embedded Haar type spaces associated with cell subdivisions of a smooth manifold. We use an adaptivity criterion connected with a nonnegative set function possessing certain monotonicity properties. We propose an algorithm for constructing embedded spaces satisfying the adaptivity criterion. To construct the wavelet decomposition, we apply the nonclassical approach and obtain the adaptive wavelet decomposition of the Haar type space on the manifold. Some model examples are given.Fundamental formulas for modified generalized integral transforms.https://www.zbmath.org/1453.601392021-02-27T13:50:00+00:00"Chung, Hyun Soo"https://www.zbmath.org/authors/?q=ai:chung.hyun-sooSummary: Various fundamental formulas and results for integral transforms on a function space have been studied in many papers. However, there are many limitations with regard to obtaining the fundamental formulas and results with respect to integral transforms on the function space, because generalized Brownian motion has the nonzero mean function. Despite recent attempts address this problem, it has yet to be resolved. In this paper, based on the definitions of the modified generalized integral transform and (generalized) convolution product on function space, we establish the fundamental formulas relating the two. The fundamental formulas obtained via the translation theorem are applied in several examples to demonstrate the usefulness of our fundamental approach.Quantum decomposition associated with the \(q\)-deformed Lévy-Meixner white noise.https://www.zbmath.org/1453.810332021-02-27T13:50:00+00:00"Riahi, Anis"https://www.zbmath.org/authors/?q=ai:riahi.anis"Ettaieb, Amine"https://www.zbmath.org/authors/?q=ai:ettaieb.amine``Small step`` remodeling and counterexamples for weighted estimates with arbitrarily ``smooth'' weights.https://www.zbmath.org/1453.420112021-02-27T13:50:00+00:00"Kakaroumpas, S."https://www.zbmath.org/authors/?q=ai:kakaroumpas.spyridon"Treil, S."https://www.zbmath.org/authors/?q=ai:treil.sergeiSummary: For an \(A_p\) weight \(w\) the norm of the Hilbert transform in \(L^p(w)\), \(1 < p < \infty\) is estimated by \([ w ]_{A_p}^s\), where \([ w ]_{A_p}\) is the \(A_p\) characteristic of the weight \(w\) and \(s = \max(1, 1 /(p - 1))\); as simple examples with power weights show, these estimates are sharp.
A natural question to ask, is whether it is possible to improve the exponent \(s\) in the above estimate if one replaces the \(A_p\) characteristic by its ``fattened'' version, where the averages are replaced by Poisson-like averages. For power weights (for example with \(p = 2\) and Poisson averages) one can see that there is indeed an improvement in the exponent: but is it true for general weights?
In this paper we show that the optimal exponent \(s\) remains the same by constructing counterexamples for arbitrarily ``smooth'' weights (in the sense that the doubling constant is arbitrarily close to 2), so the ``fattened'' \( A_p\) characteristic is equivalent to the classical one, and such that \(\| T \|_{L^p ( w )} \sim [ w ]_{A_p}^s\).
We use the ideas from the manuscript by \textit{F. Nazarov} [``A counterexample to Sarason's conjecture'', unpublished] disproving Sarason's conjecture. We start from simple classical counterexamples for dyadic models, and then by using what we call ``small step construction'' we transform them into examples with weights that are arbitrarily dyadically smooth. F. Nazarov had used Bellman function method to prove the existence of such examples, but our construction gives a way to get such examples from the standard dyadic ones. We then use a modification of ``remodeling'', introduced by \textit{J. Bourgain} [Ark. Mat. 21, 163--168 (1983; Zbl 0533.46008)] and developed by F. Nazarov, to get from examples for dyadic models to examples for the Hilbert transform. As an added bonus, we present a proof that the \(L^p\) analog of Sarason's conjecture is false for all \(p\), \(1 < p < \infty \).Few remarks on quasi quantum quadratic operators on \(\mathbb{M}_2(\mathbb{C})\).https://www.zbmath.org/1453.810282021-02-27T13:50:00+00:00"Mukhamedov, Farrukh"https://www.zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Syam, Sondos M."https://www.zbmath.org/authors/?q=ai:syam.sondos-m"Almazrouei, Shamma A. Y."https://www.zbmath.org/authors/?q=ai:almazrouei.shamma-a-yA non-exponential extension of Sanov's theorem via convex duality.https://www.zbmath.org/1453.600742021-02-27T13:50:00+00:00"Lacker, Daniel"https://www.zbmath.org/authors/?q=ai:lacker.danielSummary: This work is devoted to a vast extension of Sanov's theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.On the uniqueness of invariant states.https://www.zbmath.org/1453.810352021-02-27T13:50:00+00:00"Bambozzi, Federico"https://www.zbmath.org/authors/?q=ai:bambozzi.federico"Murro, Simone"https://www.zbmath.org/authors/?q=ai:murro.simoneIn many cases the quantum physical system is endowed with an action by a group of \(*\)-automorphisms which is not compact nor abelian, even if it acts ergodically. An example is the abelian Chern- Simons theory. In this case the general criterion for the existence and uniqueness of invariant states is still missing. However in [\textit{F. Bambozzi} et al., ``Invariant states on noncommutative tori'', Preprint, \url{arXiv:1802.02487}] ``invariant states by the symplectic group of automorphisms on group algebras with involutions that define irrational non-commutative tori have been classified using elementary, and mostly algebraic, methods.More precisely, it was shown that for irrational rotational algebras the only state invariant under the action of the symplectic group is the canonical trace state''.
The present paper continues the mentioned paper. The main result is the following one. ``Given an abelian group \(G\) endowed with a \(T=\mathbb{R}/\mathbb{Z}\)-pre-symplectic form, the authors assign to it a symplectic twisted group \(*\)-algebra \(W_G\) and then provide criteria for the uniqueness of states invariant under the ergodic action of the symplectic group of automorphism''
Reviewer: Dmitry Artamonov (Moskva)Extended Banach \(\overrightarrow{G}\)-flow spaces on differential equations with applications.https://www.zbmath.org/1453.460172021-02-27T13:50:00+00:00"Mao, Linfan"https://www.zbmath.org/authors/?q=ai:mao.linfanSummary: Let \(\mathcal V\) be a Banach space over a field \(\mathcal F\). A \(\overrightarrow{G}\)-flow is a graph \(\overrightarrow{G}\) embedded in a topological space \(\mathcal S\) associated with an injective mappings \(L:u^v\to L(u^v)\in\mathcal V\) such that \(L(u^v)=-L(v^u)\) for all \((u, v)\in X (\overrightarrow{G})\) holding with conservation laws
\[
\sum_{u\in N_{G}(v)}L(v^u) = 0\ \text{for all } v\in V (\overset{\rightarrow}{G}),
\]
where \(u^v\) denotes the semi-arc of \((u, v)\in X(\overrightarrow{G})\), which is a mathematical object for things embedded in a topological space. The main purpose of this paper is to extend Banach spaces on topological graphs with operator actions and show all of these extensions are also Banach space with a bijection with a bijection between linear continuous functionals and elements, which enables to solve linear functional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, find multi-space solutions on equations, for instance, the Einstein's gravitational equations. A few well-known results in classical mathematics are generalized such as those of the fundamental theorem in algebra, Hilbert and Schmidt's result on integral equations, and the stability of such \(\overrightarrow{G}\)-flow solutions with applications to ecologically industrial systems are also discussed in this paper.Some remarks on Schauder bases in Lipschitz free spaces.https://www.zbmath.org/1453.460092021-02-27T13:50:00+00:00"Novotný, Matěj"https://www.zbmath.org/authors/?q=ai:novotny.matejThe paper deals with the existence of a Schauder basis in ``Lipschitz-free Banach spaces'' \(\mathcal{F}(M)\) (known also as ``Arens-Eells spaces'' or ``transportation cost spaces'').
The author provides an example of a discrete metric space \(M\) such that in \(\mathcal{F}(M)\) there exists an ``extensional'' Schauder basis, but there does not exist any ``retractional'' Schauder basis. Moreover, he shows that on \(\mathcal{F}(\mathbb{Z}^d)\) (\(d\geq 2\)) there does not exist a ``retractional'' unconditional Schauder basis (while it is known that a ``retractional'' Schauder basis does exist). The notions of an ``extensional'' and ``retractional'' Schauder basis are summarized below.
It is well known that any Schauder basis in a Banach space is given by a collection of projections \((P_n)\). We say the Schauder basis given by the projections \((P_n)\) in a Lipschitz-free space \(\mathcal{F}(M)\) is \textit{extensional} if there are finite subsets \(M_n\subset M\) such that the image of each \(P_n\) is naturally isometric to \(\mathcal{F}(M_n)\) and the adjoints \(P_n^*:\operatorname{Lip}_0(M_n)\to \operatorname{Lip}_0(M)\) provide extensions of Lipschitz functions from \(M_n\) to the space \(M\). This notion is very natural, because as far as the reviewer knows, all the Schauder bases constructed in Lipschitz-free spaces are extensional. Moreover, the basis is \textit{retractional} if the adjoints \(P_n^*\) satisfy \(P_n^*(f) = f\circ r_n\), where \(r_n:M\to M_n\) are certain Lipschitz retractions. Again, this notion is natural as it is often the case that a Schauder basis constructed in \(\mathcal{F}(M)\), where \(M\) is discrete, is retractional.
Reviewer: Marek Cúth (Praha)Non-asymptotic \(\ell_1\) spaces with unique \(\ell_1\) asymptotic model.https://www.zbmath.org/1453.460112021-02-27T13:50:00+00:00"Argyros, Spiros A."https://www.zbmath.org/authors/?q=ai:argyros.spiros-a"Georgiou, Alexandros"https://www.zbmath.org/authors/?q=ai:georgiou.alexandros"Motakis, Pavlos"https://www.zbmath.org/authors/?q=ai:motakis.pavlosSummary: A recent result of Freeman, Odell, Sari, and Zheng [\textit{D.\,Freeman} et al., Trans. Am. Math. Soc. 370, No.\,10, 6933--6953 (2018; 1427.46006)] states that whenever a separable Banach space not containing \(\ell_1\) has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$, then the space is asymptotic \(c_0\). We show that, if we replace \(c_0\) with \(\ell_1\), then this result is no longer true. Moreover, a~stronger result of Maurey-Rosenthal type [\textit{B.\,Maurey} and \textit{H.\,P.\thinspace Rosenthal}, Stud. Math. 61, 77--98 (1977; Zbl 0357.46025)] is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting \(\ell_1\) as a unique asymptotic model, whereas any subsequence of the basis generates a non-asymptotic \(\ell_1\) subspace.Operator inequalities of Morrey spaces associated with Karamata regular variation.https://www.zbmath.org/1453.420172021-02-27T13:50:00+00:00"Wang, Jiajia"https://www.zbmath.org/authors/?q=ai:wang.jiajia"Fu, Zunwei"https://www.zbmath.org/authors/?q=ai:fu.zunwei"Shi, Shaoguang"https://www.zbmath.org/authors/?q=ai:shi.shaoguang"Mi, Ling"https://www.zbmath.org/authors/?q=ai:mi.lingLet \(\omega: \mathbb{R}\times \mathbb{R}^{+}\to \mathbb{R}\) be a positive measurable function satisfying that for any \( 0 < C_{1} < C_{2}\), there exists a constant \(C>0\) such that
\[C^{-1}\leq \frac{\omega(x_0,\xi h)}{\omega(x_0,h)}\leq C,\]
where \(\xi\in[C_1, C_2]\) and
\[\int_{h}^{\infty}\frac{\omega(x_0,t)}{t^{s+1}}dt\leq C\frac{\omega(x_0,h)}{h^s}\]
for any \(x_0\in \mathbb{R}\), \(h>0\) and \(s>0\). The Morrey space associated with Karamata's regular variation is given by
\[
M_{K}^{p,\omega}(\mathbb{R})=\left\{f\in L_{\mathrm{loc}}^{p}(\mathbb{R}):\Vert f\Vert_{M_{K}^{p,\omega} (\mathbb{R})}=:\sup_{I}\frac{1}{\omega(I)}\int_{I}|f(x)|^{p}dx<\infty\right\},\ 1\leq p<\infty.
\]
The authors establish that the one-sided Hardy-Littlewood maximal operator \(M^{+}\) given by
\[M^{+}f(x)=\sup_{h>0}\frac{1}{h}\int_{x}^{x+h}|f(y)|dy\]
is bounded on the Morrey space \(M_{K}^{p,\omega}(\mathbb{R})\) for \( 1 < p<\infty\) and bounded from the Morrey space \(M_{K}^{1,\omega}(\mathbb{R})\) to the weak \(L^1(\mathbb{R})\) space. Similar results for the one-sided Calderón-Zygmund singular integral operator, the Riemann-Liouville fractional integral and the Weyl fractional integral are also obtained. In the last section, the authors also extend the results in the one-dimensional case to the \(n\)-dimensional case.
Reviewer: Fayou Zhao (Shanghai)Quantum anomalies via differential properties of Lebesgue-Feynman generalized measures.https://www.zbmath.org/1453.810312021-02-27T13:50:00+00:00"Gough, John E."https://www.zbmath.org/authors/?q=ai:gough.john-e"Ratiu, Tudor S."https://www.zbmath.org/authors/?q=ai:ratiu.tudor-stefan"Smolyanov, Oleg G."https://www.zbmath.org/authors/?q=ai:smolyanov.oleg-georgievichSummary: We address the problem concerning the origin of quantum anomalies, which has been the source of disagreement in the literature. Our approach is novel as it is based on the differentiability properties of families of generalized measures. To this end, we introduce a space of test functions over a locally convex topological vector space, and define the concept of logarithmic derivatives of the corresponding generalized measures. In particular, we show that quantum anomalies are readily understood in terms of the differential properties of the Lebesgue-Feynman generalized measures (equivalently, of the Feynman path integrals). We formulate a precise definition for quantum anomalies in these terms.Choquet order and Jordan morphisms of operator algebras.https://www.zbmath.org/1453.460532021-02-27T13:50:00+00:00"Turilova, E. A."https://www.zbmath.org/authors/?q=ai:turilova.e-a"Hamhalter, J."https://www.zbmath.org/authors/?q=ai:hamhalter.janThis paper continues the investigation started by the same authors in [Int. J. Theor. Phys. 53, No. 10, 3333--3345 (2014; Zbl 1302.81046)] of describing the ordinal isomorphisms of the partially ordered set of orthogonal measures on the state-space of operator algebras. It is shown that, for \(\sigma\)-finite von Neumann algebras, every such ordinal isomorphism (w.r.t.\ the Choquet partial order) is generated by a~Jordan \(\ast\)-isomorphism.
Reviewer: Emanuel Chetcuti (Msida)Monic representations of finite higher-rank graphs.https://www.zbmath.org/1453.460502021-02-27T13:50:00+00:00"Farsi, Carla"https://www.zbmath.org/authors/?q=ai:farsi.carla"Gillaspy, Elizabeth"https://www.zbmath.org/authors/?q=ai:gillaspy.elizabeth"Jorgensen, Palle"https://www.zbmath.org/authors/?q=ai:jorgensen.palle-e-t"Kang, Sooran"https://www.zbmath.org/authors/?q=ai:kang.sooran"Packer, Judith"https://www.zbmath.org/authors/?q=ai:packer.judith-aSummary: In this paper, we define the notion of monic representation for the \(C^{\ast}\)-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative \(C^{\ast}\)-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the \(\Lambda\)-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer
[\textit{C.~Farsi} et al., J. Math. Anal. Appl. 434, No.~1, 241--270 (2016; Zbl 1352.46049)]
and also provide a universal representation model for non-negative monic representations.\(C^*\)-algebra-valued modular spaces and fixed point theorems.https://www.zbmath.org/1453.470092021-02-27T13:50:00+00:00"Shateri, Tayebe Lal"https://www.zbmath.org/authors/?q=ai:shateri.tayebeh-lalSummary: In the present paper a concept of \(C^*\)-algebra-valued modular space is introduced which is a generalization of a modular space. Next, some fixed point theorems for self-maps with contractive or expansive conditions on such spaces are proved.Convolution operators on weighted spaces of continuous functions and supremal convolution.https://www.zbmath.org/1453.440062021-02-27T13:50:00+00:00"Kleiner, T."https://www.zbmath.org/authors/?q=ai:kleiner.tillmann"Hilfer, R."https://www.zbmath.org/authors/?q=ai:hilfer.rudolfSummary: The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.Uniqueness of the approximative trace.https://www.zbmath.org/1453.460342021-02-27T13:50:00+00:00"Sauter, Manfred"https://www.zbmath.org/authors/?q=ai:sauter.manfredSummary: We study the \emph{approximative trace} for individual elements in the Sobolev space \(W^{1,p}(\Omega)\) for \(1\le p\le\infty \). This notion of a trace was introduced for \(p=2\) in
[\textit{W.~Arendt} and \textit{A.~F.~M. ter Elst}, J. Differ. Equations 251, No.~8, 2100--2124 (2011; Zbl 1241.47036)] in the setting of general open sets \(\Omega\subset\mathbb{R}^d\). The approximative trace exhibits a curious nonuniqueness phenomenon. We provide a detailed analysis of this phenomenon based on methods of geometric measure theory and are able to give very weak geometric conditions that are sufficient for the uniqueness of the approximative trace. In particular, we prove that the approximative trace is unique on open sets with continuous boundary and on arbitrary connected domains in \(\mathbb{R}^2\). Furthermore, we provide an example which shows that the uniqueness of the approximative trace depends on \(p\). These results answer several open questions.Some sufficient conditions for fixed points of multivalued nonexpansive mappings in Banach spaces.https://www.zbmath.org/1453.460122021-02-27T13:50:00+00:00"Wang, Xi"https://www.zbmath.org/authors/?q=ai:wang.xi"Zhang, Chiping"https://www.zbmath.org/authors/?q=ai:zhang.chiping"Cui, Yunan"https://www.zbmath.org/authors/?q=ai:cui.yunanSummary: In this paper, we show some sufficient conditions on a Banach space \(X\) concerning the generalized von Neumann-Jordan constant, the coefficient \(R(1,X)\), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings.A quantitative version of Helly's selection principle in Banach spaces and its applications.https://www.zbmath.org/1453.460072021-02-27T13:50:00+00:00"García, Gonzalo"https://www.zbmath.org/authors/?q=ai:garcia.gonzalo-aHelly's selection principle states that every sequence of real valued functions, defined on a compact interval, of uniformly bounded variation, has a pointwise converging subsequence. In other words, a set of real and uniformly bounded variation functions defined on a compact interval is sequentially compact in the pointwise topology. The main result of this paper is a quantitative version of this principle. The proofs use the so-called degree of nondensifiability, which measures (in a specified sense) the distance of a given convex subset of a Banach space to the class of its Peano continua. As application of the general results, the author analyzes the solvability of certain Volterra integral equations, the integral being considered in the sense of Bochner.
Reviewer: Ioan Raşa (Cluj-Napoca)Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure.https://www.zbmath.org/1453.470032021-02-27T13:50:00+00:00"Ablet, Ehmet"https://www.zbmath.org/authors/?q=ai:ablet.ehmet"Cheng, Lixin"https://www.zbmath.org/authors/?q=ai:cheng.lixin"Cheng, Qingjin"https://www.zbmath.org/authors/?q=ai:cheng.qingjin"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.2Summary: In this paper, we show that every infinite-dimensional Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Therefore, it resolves a long-standing question.Dirichlet forms and convergence of Besov norms on self-similar sets.https://www.zbmath.org/1453.460282021-02-27T13:50:00+00:00"Gu, Qingsong"https://www.zbmath.org/authors/?q=ai:gu.qingsong"Lau, Ka-Sing"https://www.zbmath.org/authors/?q=ai:lau.kasingLet \(K\) be a compact \(d\)-set in \(\mathbb{R}^n\), \(0<d<n\), furnished with the related Hausdorff measure. Let \(B^\sigma_{2,2} (K)\) and \(B^\sigma_{2,\infty} (K)\) be the usual Besov spaces on \(K\) defined in terms of first differences, \(\sigma >0\). Further, let
\[
\sigma^* = \sup \big\{ \sigma: \ B^\sigma_{2,\infty} \cap C(K) \ \text{is dense in} \ C(K) \big\}.
\]
It is the main aim of the paper to study the convergence of the \(B^\sigma_{2,2} (K)\)-norms to the \(B^{\sigma^*}_{2,\infty} (K)\)-norm if \(\sigma <\sigma^*\) and \(\sigma \to \sigma^*\).
Reviewer: Hans Triebel (Jena)Boundedness of operators on certain weighted Morrey spaces beyond the Muckenhoupt range.https://www.zbmath.org/1453.420192021-02-27T13:50:00+00:00"Duoandikoetxea, Javier"https://www.zbmath.org/authors/?q=ai:duoandikoetxea.javier"Rosenthal, Marcel"https://www.zbmath.org/authors/?q=ai:rosenthal.marcelThe authors develop a certain unified approach and prove that for operators satisfying weighted inequalities with \(A_p\) weights the boundedness on a certain class of Morrey spaces holds with weights of the form \(|x|^{\alpha}w(x)\) for
\(w\in A_p\), where \(A_p\) is the Muckenhoupt class of weights. More exactly the main result is the following.
The Morrey space \(\mathcal{L}^{p,\lambda}(\omega)\) is defined by the norm
\[
||f||_{\mathcal{L}^{p,\lambda}(\omega)}=\sup\limits_{x\in\mathbb R^n, r>0}\left(\frac{1}{r^{\lambda}}\int\limits_{B(x,r)}|f|^pw\right)^{1/p},
\]
and the norm in the standard weighted \(L^p\)-space is the following
\[
||f||_{{ L}^{p}(\omega)}=\left(\int\limits_{\mathbb R^n}|f|^pw\right)^{1/p}.
\]
Moreover, the authors use so called reverse Hölder class \(RH_{\sigma}\) and consider a collection \(\mathcal F\) of nonnegative measurable pairs of functions. If we assume that for every pair \((f,g)\in\mathcal F\) and for every \(\omega\in A_{p_0}\), \(1\leq p_0<\infty\) we have
\[
||g||_{{ L}^{p_0}(\omega)}\leq C||f||_{{ L}^{p_0}(\omega)}
\]
with \(C\) non-depending on the pair \((f,g)\) then we can obtain the following inequality
\[
||g||_{\mathcal L^{p,\lambda}(|x|^{\alpha}\omega)}\leq C||f||_{\mathcal L^{p,\lambda}(|x|^{\alpha}\omega)}
\]
for \(0\leq\lambda<n/\sigma^\prime\) and \(0\leq\alpha<\lambda\).
This is the main result of the paper. This article includes a lot of particular results which were considered earlier.
Reviewer: Vladimir Vasilyev (Belgorod)Universal regularity of higher-dimensional disorder and density of states under non-local interactions. I: Infinite smoothness and localization.https://www.zbmath.org/1453.820382021-02-27T13:50:00+00:00"Chulaevsky, Victor"https://www.zbmath.org/authors/?q=ai:chulaevsky.victorSummary: It is shown that in a large class of disordered systems with singular alloy-type disorder and non-local media-particle interactions, the marginal measures of the induced random potential and the finite-volume integrated density of states (IDS) are infinitely differentiable in higher dimensions. The proposed approach complements the classical Wegner estimate which says that the IDS in the short-range models is at least as regular as the marginal distribution of the disorder. In the models with non-local interaction the finite-volume IDS is much more regular than the underlying disorder. In turn, smoothness of the finite-volume IDS is responsible for a mechanism complementing the Lifshitz tails phenomenon.
The new eigenvalue concentration estimates give rise to relatively simple proofs of Anderson localization in several classes of discrete and continuous long-range models with arbitrarily singular disorder. The present paper addresses the model with power-law decay of the potential.Closed ideals with bounded approximate identities in some Banach algebras.https://www.zbmath.org/1453.460482021-02-27T13:50:00+00:00"Mohamadi, Issa"https://www.zbmath.org/authors/?q=ai:mohamadi.issaThe existence of bounded approximate identity for Banach algebra is a ``nice'' property for them. For example, any group algebra over a locally compact group possesses a bounded approximate identity. Also, the Fourier algebra over a locally compact group \(G\) has a bounded approximate identity iff \(G\) is amenable. This paper is about characterizing the existence of a bounded approximate identity for a~certain closed ideal of a Banach algebra. In fact, the author shows for which \(m\in A^{**}\) the closed ideal \(I_{m}=\{a\in A:am=0\}\) possesses a bounded approximate identity. Thereby a~question raised by \textit{A. T. M. Lau} and \textit{A. Ülger} [Trans. Am. Math. Soc. 366, No. 8, 4151--4171 (2014; Zbl 1312.46050)] is partially solved in this paper.
Reviewer: Amir Sahami (Tehran)Almost convergence and Euler totient matrix.https://www.zbmath.org/1453.460042021-02-27T13:50:00+00:00"Demiriz, Serkan"https://www.zbmath.org/authors/?q=ai:demiriz.serkan"İlkhan, Merve"https://www.zbmath.org/authors/?q=ai:ilkhan.merve"Kara, Emrah Evren"https://www.zbmath.org/authors/?q=ai:kara.emrah-evrenThe \(\Phi\)-transform of complex-valued sequences is defined by
\[
v_{n}=\sum_{k\mid n}\varphi( k) u_{k},\ n\in\mathbb{N},
\]
where \(\varphi\) denotes the Euler totient function. Let
\[
\widehat{c}\left(\Phi\right) :=\left\{(u_{k})\biggm\vert\exists\alpha\in\mathbb{C}:\lim_{m\rightarrow\infty}\sum_{j=0}^{m}\frac{v_{n+j}}{m+1}=\alpha\text{ uniformly in }n\right\}.
\]
From the authors' abstract: ``It is proved that the space \(\widehat{c}\left(\Phi\right)\) and the space of all almost convergent sequences are linearly isomorphic. Further, the \(\beta\)-dual of the space \(\widehat{c}\left(\Phi\right)\) is determined and Euler totient core of a complex-valued sequence has been defined. Finally, inclusion theorems related to this new
type of core are obtained.''
Reviewer: Toivo Leiger (Tartu)Gabor duality theory for Morita equivalent \(C^\ast\)-algebras.https://www.zbmath.org/1453.420252021-02-27T13:50:00+00:00"Austad, Are"https://www.zbmath.org/authors/?q=ai:austad.are"Jakobsen, Mads S."https://www.zbmath.org/authors/?q=ai:jakobsen.mads-sielemann"Luef, Franz"https://www.zbmath.org/authors/?q=ai:luef.franzThe authors establish a far-reaching generalization to Morita equivalent \(C^\ast\)-algebras where the equivalence bimodule is a finitely generated projective Hilbert \(C^\ast\)-module. They formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group \(C^\ast\)-algebras of a lattice in the phase space. All these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule are obtained. The authors also introduced \((n,d)\)-matrix frames, which generalize superframes and multi-window frames. Density theorems for \((n,d)\)-matrix frames are established.
Reviewer: Ashok Kumar Sah (New Delhi)Sharp \(L^2\)-Caffarelli-Kohn-Nirenberg inequalities for Grushin vector fields.https://www.zbmath.org/1453.350062021-02-27T13:50:00+00:00"Flynn, Joshua"https://www.zbmath.org/authors/?q=ai:flynn.joshuaOn \(\mathbb{R}^{N+1}\) with coordinates \(z \in \mathbb{R}^{N}\) and \(t\in \mathbb{R}\) define the Grushin gradient
\(\nabla_G=(\nabla_z,|z|\partial_t)\) and the Grushin operator \(\Delta_G= \Delta_z+|z|^2\partial_t^2\) associated to the vector fields \(X_j = \partial_{z_j}\), \(j=1, \dots, N\), and \(T=|z|\partial_t\).
In this paper are characterized the best constants and minimizers for inequalities
\[C \int_{\mathbb{R}^{N+1}} |u|^2 \frac{\psi}{\rho^{a+b+1}}\ dz\,dt \le \left( \int_{\mathbb{R}^{N+1}} \frac{|\nabla_G u|^2 }{\rho^{2b}}\ dz\,dt \right)^\frac12 \left( \int_{\mathbb{R}^{N+1}} |u|^2 \frac{\psi}{\rho^{2a}}\ dz\,dt \right)^\frac12,
\]
where \((a,b) \in \mathbb{R}^2\), \(\psi = \frac{|z|^2}{\rho^2}\), and \(\rho= \left( |z|^{4}+4t^{2}\right)\).
This analysis is done by a clever use of special polar coordinates and by taking advantage of the commutation relations between \(\Delta_{G}\) and an anisotropic Kelvin transform.
Reviewer: Florin Catrina (New York)A note on geodesics of projections in the Calkin algebra.https://www.zbmath.org/1453.530492021-02-27T13:50:00+00:00"Andruchow, Esteban"https://www.zbmath.org/authors/?q=ai:andruchow.estebanThe author considers the Calkin algebra on a Hilbert space, i.e., the algebra of bounded operators modulo compact operators.
Inside the Calkin algebra, he considers the image \(\mathcal{P}\) of the self-adjoint projection operators, a kind of infinite-dimensional Grassmannian. The operator norm on each tangent space of \(\mathcal{P}\) induces a length metric on \(\mathcal{P}\). The author proves that any two points of \(\mathcal{P}\) are connected by a minimal geodesic in \(\mathcal{P}\) if and only if they arise from bounded operators \(P,Q\) so that the null spaces of \(P-Q\pm 1\) are both finite-dimensional or both infinite-dimensional. The minimal geodesic is unique if \(P+Q-1\) has quotient in the Calkin algebra with trivial annihilator.
Reviewer: Benjamin McKay (Cork)Spaces of strongly lacunary invariant summable sequences.https://www.zbmath.org/1453.460052021-02-27T13:50:00+00:00"Savaş, E."https://www.zbmath.org/authors/?q=ai:savas.ekremSummary: In this paper, we introduce and examine some properties of three sequence spaces defined using lacunary sequence and invariant mean which generalize several known sequence spaces.On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type \(\text{III}_1\).https://www.zbmath.org/1453.460562021-02-27T13:50:00+00:00"Masuda, Toshihiko"https://www.zbmath.org/authors/?q=ai:masuda.toshihikoSummary: We show that the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type \(\text{III}_1\) with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type \(\text{III}_1\).On Banach space projective tensor product of \(C^*\)-algebras.https://www.zbmath.org/1453.460522021-02-27T13:50:00+00:00"Gupta, Ved Prakash"https://www.zbmath.org/authors/?q=ai:gupta.ved-prakash"Jain, Ranjana"https://www.zbmath.org/authors/?q=ai:jain.ranjanaSummary: For \(C^*\)-algebras \(A\) and \(B\), we show that there is a natural homeomorphism from the product of spaces of maximal (resp., maximal modular) ideals of \(A\) and \(B\) onto the space of maximal (resp., maximal modular) ideals of \(A \otimes^\gamma B\), with respect to the hull-kernel topology. This is based on and preceded by an analysis of the structure of closed ideals of \(A \otimes^\gamma B\) in terms of those of \(A\) and \(B\). During the process, we also identify the center of \(A \otimes^\gamma B\) with \(\mathcal{Z}(A) \otimes^\gamma\mathcal{Z}(B)\).Parts formulas involving the Fourier-Feynman transform associated with Gaussian paths on Wiener space.https://www.zbmath.org/1453.460422021-02-27T13:50:00+00:00"Chang, Seung Jun"https://www.zbmath.org/authors/?q=ai:chang.seung-jun"Choi, Jae Gil"https://www.zbmath.org/authors/?q=ai:choi.jae-gilSummary: \textit{C. Park} and \textit{D. Skoug} [Panam. Math. J. 8, No. 4, 1--11 (1998; Zbl 0958.46042)] established several integration by parts formulas involving analytic Feynman integrals, analytic Fourier-Feynman transforms, and the first variation of cylinder-type functionals of standard Brownian motion paths in Wiener space \(C_0[0,T]\). In this paper, using a very general Cameron-Storvick theorem on the Wiener space \(C_0[0,T]\), we establish various integration by parts formulas involving generalized analytic Feynman integrals, generalized analytic Fourier-Feynman transforms, and the first variation (associated with Gaussian processes) of functionals \(F\) on \(C_0[0,T]\) having the form
\[ F(x)=f(\langle{\alpha_1,x}\rangle,\dots,\langle{\alpha_n,x}\rangle) \]
for scale-invariant almost every \(x\in C_0[0,T]\), where \(\langle{\alpha ,x}\rangle\) denotes the Paley-Wiener-Zygmund stochastic integral \(\int_0^T \alpha (t)\,dx(t)\), and \(\{\alpha_1,\dots ,\alpha_n\}\) is an orthogonal set of nonzero functions in \(L_2[0,T]\). The Gaussian processes used in this paper are not stationary.On the constants in a Kato inequality for the Euler and Navier-Stokes equations.https://www.zbmath.org/1453.351382021-02-27T13:50:00+00:00"Morosi, Carlo"https://www.zbmath.org/authors/?q=ai:morosi.carlo"Pizzocchero, Livio"https://www.zbmath.org/authors/?q=ai:pizzocchero.livioSummary: We continue an analysis, started in our work [Appl. Math. Lett. 26, No. 2, 277--284 (2013; Zbl 1426.76092)], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a \(d\)-dimensional torus \(T^d\). More specifically, we consider the quadratic term in these equations; this arises from the bilinear map \((v, w) \mapsto v \cdot \partial w\), where \(v, w : T^d \to R^d\) are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants \(G_{nd} \equiv G_n\) in the Kato inequality \(|\langle v \cdot \partial w|w \rangle_n| \leqslant G_n||v||_n||w||^2_n\), where \(n \in (d/2 + 1, +\infty)\) and \(v, w\) are in the Sobolev spaces \(\mathbb{H}^n_{\Sigma_0}, \mathbb{H}^{n+1}_{\Sigma_0}\) of zero mean, divergence free vector fields of orders \(n\) and \(n + 1\), respectively. As examples, the numerical values of our upper and lower bounds are reported for \(d = 3\) and some values of \(n\). When combined with the results of [loc. cit.] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.Core inverse in Banach algebras.https://www.zbmath.org/1453.460452021-02-27T13:50:00+00:00"Mosić, Dijana"https://www.zbmath.org/authors/?q=ai:mosic.dijana"Li, Tingting"https://www.zbmath.org/authors/?q=ai:li.tingting"Chen, Jianlong"https://www.zbmath.org/authors/?q=ai:chen.jianlongSummary: We define and characterize the core inverse in the context of Banach algebras. The Banach space operator case is also considered. Using the core inverse, we present new characterizations of EP Banach space operators and EP Banach algebra elements. The dual core inverse for Banach algebra elements is presented too. Some new characterizations of co-EP Banach algebra elements are given by means the core inverse and dual core inverse.Associate spaces of logarithmic interpolation spaces and generalized Lorentz-Zygmund spaces.https://www.zbmath.org/1453.460152021-02-27T13:50:00+00:00"Besoy, Blanca F."https://www.zbmath.org/authors/?q=ai:besoy.blanca-f"Cobos, Fernando"https://www.zbmath.org/authors/?q=ai:cobos.fernando"Fernández-Cabrera, Luz M."https://www.zbmath.org/authors/?q=ai:fernandez-cabrera.luz-mSummary: We determine the associate space of the logarithmic interpolation space \((X_0,X_1)_{1,q,\mathbf{A}}\) where \(X_0\) and \(X_1\) are Banach function spaces over a \(\sigma\)-finite measure space \(( \Omega,\mu)\). Particularizing the results for the case of the couple \((L_1,L_\infty)\) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces \(L_{( \infty ,q;\mathbf{A})}\). We also establish the corresponding results for sequence spaces.On octahedrality and Müntz spaces.https://www.zbmath.org/1453.460102021-02-27T13:50:00+00:00"Martiny, André"https://www.zbmath.org/authors/?q=ai:martiny.andreLet \(\Lambda =(\lambda_n)_{n\ge0}\) be a strictly increasing sequence of real numbers with \(\lambda_0=0\) and \(\lambda_n\to\infty\). The closed linear hull of the functions \(t\mapsto t^\lambda\), \(\lambda \in \Lambda\), in \(C[0,1]\) is called a Müntz space, denoted by \(M(\Lambda)\), if \(\sum_{k=1}^\infty 1/\lambda_k <\infty\); the latter condition is equivalent to \(M(\Lambda)\) being a proper subspace of \(C[0,1]\), by the Müntz-Szász theorem.
Combining known results the author proves that all Müntz spaces embed isomorphically into~\(c_0\); previously this was known to hold only under additional assumptions on \(\Lambda\). More precisely, he proves that, for \(\lambda_1\ge1\), \(M(\Lambda)\) embeds almost isometrically into~\(c\).
In Section~3 it is proved that no Müntz space is locally octahedral and no Müntz space is almost square (ASQ). This is of interest because ASQ implies a certain diameter-\(2\) property, and this property is known to hold for Müntz spaces.
Reviewer: Dirk Werner (Berlin)Just interpolate: kernel ``ridgeless'' regression can generalize.https://www.zbmath.org/1453.681552021-02-27T13:50:00+00:00"Liang, Tengyuan"https://www.zbmath.org/authors/?q=ai:liang.tengyuan"Rakhlin, Alexander"https://www.zbmath.org/authors/?q=ai:rakhlin.alexanderSummary: In the absence of explicit regularization, Kernel ``Ridgeless'' Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.Common best proximity pairs in strictly convex Banach spaces.https://www.zbmath.org/1453.470072021-02-27T13:50:00+00:00"Gabeleh, Moosa"https://www.zbmath.org/authors/?q=ai:gabeleh.moosaSummary: A mapping \(T: A\cup B\to A\cup B\) such that \(T(A)\subseteq A\) and \(T(B)\subseteq B\) is called a noncyclic mapping, where \(A\) and \(B\) are two nonempty subsets of a Banach space \(X\). A~best proximity pair \((p,q)\in A\times B\) for such a mapping \(T\) is a~point such that \(p=Tp\), \(q=Tq\) and \(\|p-q\|=\operatorname{dist}(A,B)\). In the present paper, we establish some existence results of best proximity pairs in strictly convex Banach spaces. The presented theorems improve and extend some recent results in the literature. We also obtain a generalized version of Markov-Kakutani's theorem for best proximity pairs in a strictly convex Banach space setting.A second look of Sobolev spaces on metrizable groups.https://www.zbmath.org/1453.460382021-02-27T13:50:00+00:00"Górka, Przemyslaw"https://www.zbmath.org/authors/?q=ai:gorka.przemyslaw"Kostrzewa, Tomasz"https://www.zbmath.org/authors/?q=ai:kostrzewa.tomaszSummary: We continue our study of Sobolev spaces on locally compact abelian groups. In this paper we mainly focus on the case of metrizable groups. We show the density of the Bruhat-Schwartz space in Sobolev space. We prove the trace theorem on the cartesian product of topological groups. The comparison of Sobolev and fractional Sobolev spaces are given. In particular, it is proved that in the case of any abelian connected Lie group Sobolev and fractional Sobolev spaces coincide. Most of the theorems are illustrated by \(p\)-adic groups.Anti M-Weierstrass function sequences.https://www.zbmath.org/1453.400032021-02-27T13:50:00+00:00"Calderón-Moreno, María del Carmen"https://www.zbmath.org/authors/?q=ai:calderon-moreno.maria-del-carmen"Gerlach-Mena, Pablo José"https://www.zbmath.org/authors/?q=ai:gerlach-mena.pablo-jose"Prado-Bassas, José Antonio"https://www.zbmath.org/authors/?q=ai:prado-bassas.jose-aSummary: Large algebraic structures are found inside the space of sequences of continuous functions on a compact interval having the property that, the series defined by each sequence converges absolutely and uniformly on the interval but the series of the upper bounds diverges. So showing that there exist many examples satisfying the conclusion but not the hypothesis of the Weierstrass M-test.On the structure of ideals and multipliers: a unified approach.https://www.zbmath.org/1453.460462021-02-27T13:50:00+00:00"Mbekhta, Mostafa"https://www.zbmath.org/authors/?q=ai:mbekhta.mostafa"Neufang, Matthias"https://www.zbmath.org/authors/?q=ai:neufang.matthiasThe aim of the paper under review is to present a general setting in which the structure of one- or two-sided closed ideals in a Banach algebra \(\mathcal{A}\) can be determined. Special attention is given to the case when the Banach algebra \(\mathcal{A}\) is a one-sided ideal in its bidual.
Among other results, it is proved that left (right) closed ideals with a bounded right (left) approximate identity in a Banach algebra \(\mathcal{A}\) are the image of an idempotent right (left) multiplier on \(\mathcal{A}\), whenever \(\mathcal{A}\) is a right (left) ideal in its bidual. The same is true for one-sided closed ideals arising from the action of power bounded right (left) multipliers on \(\mathcal{A}\).
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Weighted \(p\)-regular kernels for reproducing kernel Hilbert spaces and Mercer theorem.https://www.zbmath.org/1453.460182021-02-27T13:50:00+00:00"Agud, L."https://www.zbmath.org/authors/?q=ai:agud.lucia"Calabuig, J. M."https://www.zbmath.org/authors/?q=ai:calabuig.jose-m"Sánchez Pérez, E. A."https://www.zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonsoFractional order Sobolev spaces for the Neumann Laplacian and the vector Laplacian.https://www.zbmath.org/1453.460302021-02-27T13:50:00+00:00"Kim, Seungil"https://www.zbmath.org/authors/?q=ai:kim.seungilSummary: In this paper we study fractional Sobolev spaces characterized by a norm based on eigenfunction expansions. The goal of this paper is twofold. The first one is to define fractional Sobolev spaces of order \(-1\le s\le 2\) equipped with a norm defined in terms of Neumann eigenfunction expansions. Due to the zero Neumann trace of Neumann eigenfunctions on a boundary, fractional Sobolev spaces of order \(3/2\le s\le 2\) characterized by the norm are the spaces of functions with zero Neumann trace on a boundary. The spaces equipped with the norm are useful for studying cross-sectional traces of solutions to the Helmholtz equation in waveguides with a homogeneous Neumann boundary condition. The second one is to define fractional Sobolev spaces of order \(-1\le s\le 1\) for vector-valued functions in a simply-connected, bounded and smooth domain in \(\mathbb{R}^2\). These spaces are defined by a norm based on series expansions in terms of eigenfunctions of the vector Laplacian with boundary conditions of zero tangential component or zero normal component. The spaces defined by the norm are important for analyzing cross-sectional traces of time-harmonic electromagnetic fields in perfectly conducting waveguides.Compactness result and its applications in integral equations.https://www.zbmath.org/1453.540052021-02-27T13:50:00+00:00"Krukowski, Mateusz"https://www.zbmath.org/authors/?q=ai:krukowski.mateusz"Przeradzki, Bogdan"https://www.zbmath.org/authors/?q=ai:przeradzki.bogdanSummary: A version of the Arzelà-Ascoli theorem for \(X\) being a \(\sigma\)-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operators, are derived on the basis of Schauder fixed point theorem.Hilbert spaces of symmetric analytical functions on \(\ell_{1}\).https://www.zbmath.org/1453.460472021-02-27T13:50:00+00:00"Holubchak, O. M."https://www.zbmath.org/authors/?q=ai:holubchak.o-mSummary: We consider completions of the space of symmetric polynomials on \(\ell_{1}\) with respect to some Hilbert norm and investigate conditions under which the obtained spaces consist of analytic functions with domains in \(\ell_{1}\). Some connections with abstract Fock spaces are established.Convex linear metric spaces are normable.https://www.zbmath.org/1453.460032021-02-27T13:50:00+00:00"Singh, Jitender"https://www.zbmath.org/authors/?q=ai:singh.jitender"Narang, T. D."https://www.zbmath.org/authors/?q=ai:narang.tulsi-dasThe authors call a linear metric space \((X,d)\) over the field \(\mathbb R\) convex if \(d(\lambda x+(1-\lambda)y,0)\le\lambda d(x,0)+(1-\lambda)d(y,0)\) holds for all \(x,y\in X\) and each \(\lambda\), \(0\le\lambda\le1\). They prove that each (real) convex linear metric space is normable (with the norm given by \(\|x\|=d(x,0)\). The same is true in the case of complex scalars if the metric \(d\) is rotation invariant (i.e., \(d(\alpha x,0)=d(|\alpha|x,0)\) for all complex scalars \(\alpha\) and all \(x\in X\)), but not in general.
Reviewer: Zoran Kadelburg (Beograd)A note on vanishing Morrey \(\rightarrow\) VMO result for fractional integrals of variable order.https://www.zbmath.org/1453.460232021-02-27T13:50:00+00:00"Rafeiro, Humberto"https://www.zbmath.org/authors/?q=ai:rafeiro.humberto"Samko, Stefan"https://www.zbmath.org/authors/?q=ai:samko.stefan-gSummary: In the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.Directed-completeness of quantum statistical experiments in the randomization order.https://www.zbmath.org/1453.810042021-02-27T13:50:00+00:00"Kuramochi, Yui"https://www.zbmath.org/authors/?q=ai:kuramochi.yuiPure states of maximum uncertainty with respect to a given POVM.https://www.zbmath.org/1453.810032021-02-27T13:50:00+00:00"Szymusiak, Anna"https://www.zbmath.org/authors/?q=ai:szymusiak.annaGeneralized Fourier-Feynman transforms and generalized convolution products on Wiener space. II.https://www.zbmath.org/1453.460432021-02-27T13:50:00+00:00"Shim, Sang Kil"https://www.zbmath.org/authors/?q=ai:shim.sang-kil"Choi, Jae Gil"https://www.zbmath.org/authors/?q=ai:choi.jae-gilSummary: The purpose of this article is to present the second type fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product on Wiener space. The relationships in this article are also natural extensions (to the case on an infinite dimensional Banach space) of the structure which exists between the Fourier transform and the convolution of functions on Euclidean spaces.
For Part~I, see [\textit{S.\,J.\thinspace Chang} et al., Indag. Math., New Ser. 28, No.~2, 566--579 (2017; Zbl 1378.42006)].Stability of a pair of Banach spaces for coarse Lipschitz embeddings.https://www.zbmath.org/1453.460162021-02-27T13:50:00+00:00"Dai, Duanxu"https://www.zbmath.org/authors/?q=ai:dai.duanxuSummary: A pair of Banach spaces \((X,Y)\) is said to be coarsely stable if for every coarse Lipschitz embedding \(f:X\rightarrow Y\), there exist \(\alpha,\gamma>0\) and a Lipschitz mapping \(T:L(f)\rightarrow X\) with its Lipschitz constant \(\Vert T\Vert_{\mathrm{Lip}}\le \alpha\) such that \(\Vert Tf(x)-x\Vert \le \gamma\) for all \(x\in X\), where \(L(f)\) is the closed linear span of \(f(X)\). In this paper, we study the coarse stability of a pair of Banach spaces \((X, Y)\) when \(X\) is an absolute Lipschitz retract (resp. \(X\) is an arbitrary Banach space; \(L_2)\) and \(Y\) is an arbitrary Banach space (resp. \(Y\) is a Hilbert space; \(L_p\) for \(2<p<+\infty)\).Extension of Campanato-Sobolev type spaces associated with Schrödinger operators.https://www.zbmath.org/1453.460292021-02-27T13:50:00+00:00"Huang, Jizheng"https://www.zbmath.org/authors/?q=ai:huang.jizheng"Li, Pengtao"https://www.zbmath.org/authors/?q=ai:li.pengtao"Liu, Yu"https://www.zbmath.org/authors/?q=ai:liu.yuSummary: Let \(L=-\varDelta+V\) be a Schrödinger operator acting on \(L^2(\mathbb{R}^d)\), where \(V\) belongs to the reverse Hölder class \(B_q\) for some \(q\geq d\). For \(\alpha,\beta\in [0,1)\), let \(\varLambda_{\alpha,\beta}^L(\mathbb{R}^d)\) be the Campanato-Sobolev space associated with \(L\). Via the Poisson semigroup \(\{e^{-t\sqrt{L}}\}_{t\geq 0}\), we extend \(\varLambda_{\alpha,\beta}^L(\mathbb{R}^d)\) to \(\mathcal{T}^{\alpha,\beta}_L(\mathbb{R}^{d+1}_+)\) which is defined as the set of all distributional solutions \(u\) of \(-u_{tt}+Lu=0\) on the upper half space \(\mathbb{R}_+^{d+1}\) satisfying
\[
\sup_{(x_0,r)\in\mathbb{R}_+^{d+1}}r^{-(2\alpha+d)}\int_{B(x_0,r)}\int_0^r|\nabla_{x,t}u(x,t)|^2t^{1-2\beta}\,dt\,dx<\infty.
\]Property \(T\ C^\ast\)-algebras with amenable tracial states.https://www.zbmath.org/1453.460492021-02-27T13:50:00+00:00"Chen, Yuzhang"https://www.zbmath.org/authors/?q=ai:chen.yuzhang.1"Ng, Chi-Keung"https://www.zbmath.org/authors/?q=ai:ng.chi-keungSummary: As a non-unital analogue of the main result in [\textit{N.~P. Brown}, J. Funct. Anal. 240, No.~1, 290--296 (2006; Zbl 1114.46042)], we show in this article that if \(A\) is a separable quasi-central \(C^\ast\)-algebra with property \(T\) and is nuclear, then there is a sequence of positive integers \(\{n_k\}_{k \in \mathbb{N}}\) such that the \(c_0\)-direct sum of the family \(\{\mathbb{M}_{n_k}\}_{k \in \mathbb{N}}\) of matrix algebras is an ideal \(J_A\) of \(A\) and that \(A / J_A\) has no tracial state. In particular, a separable \(C^\ast\)-algebra \(A\) is a \(c_0\)-direct sum of matrix algebras if and only if \(A\) is quasi-central, is nuclear, has property \(T\) and every unital simple quotient of \(A\) has a tracial state. We also show that a separable property \(T\ C^\ast\)-algebra \(A\) is residually finite dimensional if and only if \(A\) admits an amenable faithful tracial state and every ideal of \(A\) has non-zero center.Property \(T\) and strong property \(T\) for unital \(*\)-homomorphisms.https://www.zbmath.org/1453.460512021-02-27T13:50:00+00:00"Meng, Qing"https://www.zbmath.org/authors/?q=ai:meng.qingSummary: We introduce and study property \(T\) and strong property \(T\) for unital \(*\)-homomorphisms between two unital \(C^*\)-algebras. We also consider the relations between property \(T\) and invariant subspaces for some canonical unital \(*\)-representations. As a corollary, we show that when \(G\) is a discrete group, \(G\) is finite if and only if \(G\) is amenable and the inclusion map \(i : C^*_r(G) \rightarrow \mathscr{B} (\ell^2(G))\) has property \(T\). We also give some new equivalent forms of property \(T\) for countable discrete groups and strong property \(T\) for unital \(C^*\)-algebras.The base change in the Atiyah and the Lück approximation conjectures.https://www.zbmath.org/1453.200072021-02-27T13:50:00+00:00"Jaikin-Zapirain, Andrei"https://www.zbmath.org/authors/?q=ai:jaikin-zapirain.andreiSummary: Let \(F\) be a free finitely generated group and \({A \in \mathrm{Mat}_{n \times m}(\mathbb{C}[F])}\). For each quotient \(G = F/N\) of \(F\) we can define a von Neumann rank function \(\mathrm{rk}_{G}(A)\) associated with the \(l^2\)-operator \(l^2(G)^n \rightarrow l^2(G)^m\) induced by right multiplication by \(A\). For example, in the case where \(G\) is finite, \(\mathrm{rk}_G(A)=\frac{\mathrm{rk}_{\mathbb{C}}(\bar{A})}{|G|}\) is the normalized rank of the matrix \(\bar{A} \in \mathrm{Mat}_{n \times m}(\mathbb{C}[G])\) obtained by reducing the coefficients of \(A\) modulo \(N\). One of the variations of the Lück approximation conjecture claims that the function \({N\mapsto \mathrm{rk}_{F/N}(A)}\) is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of \(G\) is finite, then \({\mathrm{rk}_G(A) \in \frac{1}{\mathrm{lcm} (G)}\mathbb{Z}}\). In our first result we prove the sofic Lück approximation conjecture. In particular, we show that the function \({N \mapsto \mathrm{rk}_{F/N}(A)}\) is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Lück approximation and we show that the strong Atiyah conjecture holds for groups from a class \({{\mathcal{D}}}\), virtually compact special groups, Artin's braid groups and torsion-free \(p\)-adic analytic pro-\(p\) groups.The representation and continuity of a generalized metric projection onto half-spaces in Banach spaces.https://www.zbmath.org/1453.410072021-02-27T13:50:00+00:00"Liu, Ruijuan"https://www.zbmath.org/authors/?q=ai:liu.ruijuan"Liu, Chunyan"https://www.zbmath.org/authors/?q=ai:liu.chunyan"Zhang, Zihou"https://www.zbmath.org/authors/?q=ai:zhang.zihouLet \(X\) be a real Banach space and let \(C\) be a closed bounded convex subset of \(X\) having \(0\) in its interior. Denote by \(p_C\) the Minkowski functional of \(C\). Let \(G\) be a nonempty subset of \(X\) and let \(x\in X\). If there exists \(g_0 \in G\) such that \(p_C(g_0-x) = \inf \{p_C(g-x): g\in G\}\), then \(g_0\) is called the generalized best approximation to \(x\) from \(G\). The set of all generalized best approximations to \(x\) from \(G\) is called the \textit{generalized metric projection} onto \(G\).
The aim of the paper under review is to establish a formula for the generalized metric projection onto a half-space in \(X\) and to investigate the continuity of this generalized metric projection. The classical case is obtained when \(C\) is the unit ball of \(X\).
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Weighted sub-Bergman Hilbert spaces in the unit ball of \(\mathbb{C}^n\).https://www.zbmath.org/1453.320052021-02-27T13:50:00+00:00"Rososzczuk, Renata"https://www.zbmath.org/authors/?q=ai:rososzczuk.renata"Symesak, Frédéric"https://www.zbmath.org/authors/?q=ai:symesak.fredericSummary: In this note, we study defect operators in the case of holomorphic functions of the unit ball of \(\mathbb{C}^n\). These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.Polynomial tempered distributions.https://www.zbmath.org/1453.460402021-02-27T13:50:00+00:00"Sharyn, S. V."https://www.zbmath.org/authors/?q=ai:sharyn.sergiiSummary: In the article, polynomial (nonlinear) analogue of tempered Schwartz distributions is constructed. Generalized operation of differentiation in the space of polynomial generalized functions as well as Fourier-Laplace transformation of such distributions are considered. Some examples are given.Algebra and geometry of Sobolev embeddings.https://www.zbmath.org/1453.460372021-02-27T13:50:00+00:00"Visintin, Augusto"https://www.zbmath.org/authors/?q=ai:visintin.augustoSummary: We recall the definition of Hölder, Sobolev and Slobodeckii spaces, review some of the classical embeddings, and disprove other statements via scaling arguments. We then provide an algebraic and geometric representation of these results.Fractional Sobolev inequalities revisited: the maximal function approach.https://www.zbmath.org/1453.460322021-02-27T13:50:00+00:00"Nguyen Anh Dao"https://www.zbmath.org/authors/?q=ai:nguyen-anh-dao."Díaz, Jesús Ildefonso"https://www.zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonso"Nguyen, Quoc-Hung"https://www.zbmath.org/authors/?q=ai:nguyen.quoc-hung.1|nguyen.quoc-hungSummary: We revisit Sobolev-Gagliardo-Nirenberg type inequalities involving fractional norms. We prove general embedding \(W^{s,p} (\mathbb{R}^d)\) results by using the Hardy-Littlewood maximal functions as technique instead of the usual interpolation methods.Some interpolation formulae for grand and small Lorentz spaces.https://www.zbmath.org/1453.460192021-02-27T13:50:00+00:00"Ahmed, Irshaad"https://www.zbmath.org/authors/?q=ai:ahmed.irshaad"Fiorenza, Alberto"https://www.zbmath.org/authors/?q=ai:fiorenza.alberto"Hafeez, Aneesa"https://www.zbmath.org/authors/?q=ai:hafeez.aneesaSummary: We consider grand and small Lorentz spaces involving slowly varying functions, for which we establish some interpolation formulae. Our approach is based on certain limiting reiteration formulae as well as on computation of certain \(K\)-functionals.Smoothness parameter of power of Euclidean norm.https://www.zbmath.org/1453.460412021-02-27T13:50:00+00:00"Rodomanov, Anton"https://www.zbmath.org/authors/?q=ai:rodomanov.anton"Nesterov, Yurii"https://www.zbmath.org/authors/?q=ai:nesterov.yuriiSummary: In this paper, we study derivatives of powers of Euclidean norm. We prove their Hölder continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the Hölder continuity transforms into the Lipschitz continuity, we improve this result and obtain the optimal constants.On bornologicalness in locally convex algebras.https://www.zbmath.org/1453.460442021-02-27T13:50:00+00:00"Haralampidou, Marina"https://www.zbmath.org/authors/?q=ai:haralampidou.marina"Oudadess, Mohamed"https://www.zbmath.org/authors/?q=ai:oudadess.mohamed"Palacios, Lourdes"https://www.zbmath.org/authors/?q=ai:palacios.lourdes"Signoret, Carlos"https://www.zbmath.org/authors/?q=ai:signoret.carlos-j-eThe authors define six subclasses of bornological algebras within the class of locally convex algebras and study the properties of these subclasses of topological algebras. They show that some subclasses are more general than others and provide characterizations of four subclasses of bornological algebras within the class of Hausdorff locally convex algebras. The characterizations use some properties of continuity of vector space seminorms and linear mappings to any locally convex space.
Quite a long part of the paper is dedicated to examples of topological algebras which belong to some of the subclasses of bornological algebras. The authors provide examples showing that these subclasses of bornological algebras are really different from each other.
In the last two chapters, the case of locally \(A\)-convex algebras and the case of pseudo-Banach algebras are considered and some bornology-related results for these cases are obtained.
Reviewer: Mart Abel (Tartu)Embeddings and associated spaces of Copson-Lorentz spaces.https://www.zbmath.org/1453.460202021-02-27T13:50:00+00:00"Křepela, Martin"https://www.zbmath.org/authors/?q=ai:krepela.martinSummary: Let \(m, p, q \in (0, \infty )\) and let \(u, v, w\) be nonnegative weights. We characterize the validity of the inequality
\[ \left( \int_0^\infty w(t) (f^*(t))^q\,\mathrm{d}t\right)^{\frac{1}{q}} \leq C \left( \int_0^\infty v(t) \left( \int_t^\infty u(s)(f^*(s))^m\,\mathrm{d}s \right)^{\frac{p}{m}}\mathrm{d}t \right)^{\frac{1}{p}} \] for all measurable functions \(f\) defined on \(\mathbb{R}^n\) and provide equivalent estimates of the optimal constant \(C > 0\) in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson-Lorentz space \(CL^{m,p} (u,v)\), generated by the functional \[\|f\|_{CL^{m, p}(u, v)}:= \left( \int_0^\infty v(t) \left( \int_t^\infty u(s)(f^*(s))^m\,\mathrm{d}s \right)^{\frac{p}{m}}\mathrm{d}t \right)^{\frac{1}{p}},\] into the Lorentz space \(\Lambda^q(w)\). Moreover, the results are applied to describe the associated space of the Copson-Lorentz space \(CL^{ m,p }(u,v)\) for the full range of exponents \(m, p \in (0, \infty )\).A separation theorem for nonconvex sets and its applications.https://www.zbmath.org/1453.460632021-02-27T13:50:00+00:00"Ivanov, G. E."https://www.zbmath.org/authors/?q=ai:ivanov.grigorii-e"Lopushanski, M. S."https://www.zbmath.org/authors/?q=ai:lopushanski.mariana-sThe paper is concerned with the separation of closed subsets of a Banach space \(E\) by spheres or by boundaries of quasi-balls (a quasi-ball is a ball corresponding to an asymmetric norm). Denote by \(\mathfrak{B}_r(x)\) the closed ball in \(E\) of center \(x\) and radius \(r>0\) and by \(\mathfrak{B}_r\) the ball with center 0. A set \(C\) is called a summand of \(\mathfrak{B}_r\) if \(\mathfrak{B}_r=C+D\) for some \(D\subset E\). As a sample we mention Theorem~2.1: Let \(E\) be a uniformly convex and uniformly smooth Banach space, \(0< r < R\), \(A\subset E\) closed and uniformly prox-regular, \(C\subset E\) a convex closed summand of \(\mathfrak{B}_r \) with nonempty interior and such that \(A\cap\mathrm{int}\, C=\emptyset\). Then there exist \(a,c\in E\) such that
\[
\mathrm{int}\, C\subset\mathrm{int}\,\mathfrak{B}_r(c)\subset\mathrm{int}\,\mathfrak{B}_R(a)\subset E\setminus A\,.
\]
The separation by quasi-balls of weakly convex sets (see, for instance, [\textit{G. E. Ivanov}, Weakly convex sets and functions. Theory and applications. (Russian). Moskva: Fizmatlit (2006; Zbl 1171.26006)]) is treated in Section~3.
From the authors' abstract: ``These separation theorems are applied for proving some theorems on the continuity (with respect to Hausdorff metric) of the intersection of two multifunctions, the values of one of them being prox-regular or weakly convex (nonconvex, in general), and the values of the other being convex and summands of a ball or quasi-ball. As a corollary, a theorem on the continuity of a multifunction with values bounded by the graphs of two functions is obtained (in Section~4).''
Reviewer: Stefan Cobzaş (Cluj-Napoca)Logarithmic interpolation methods and measure of non-compactness.https://www.zbmath.org/1453.460142021-02-27T13:50:00+00:00"Besoy, Blanca F."https://www.zbmath.org/authors/?q=ai:besoy.blanca-f"Cobos, Fernando"https://www.zbmath.org/authors/?q=ai:cobos.fernandoSummary: We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with \(\theta=0,1\) between quasi-Banach spaces. Applications are given to operators between Lorentz-Zygmund spaces.Optimal embedding for Calderon type spaces and \(J\)-method spaces.https://www.zbmath.org/1453.460622021-02-27T13:50:00+00:00"Gogatishvili, A."https://www.zbmath.org/authors/?q=ai:gogatishvili.amiran"Ovchinnikov, V. I."https://www.zbmath.org/authors/?q=ai:ovchinnikov.vladimir-iSummary: New description of the optimal target rearrangement invariant space for embedding of the Calderon spaces \(\Lambda(F,E)\) in terms of the \(J\)-method interpolation spaces is found. We show that the corresponding Lorentz space \(\Lambda_E\) is involved rather than the rearrangement invariant space \(E\) itself.Correction to: The standard model in noncommutative geometry: fundamental fermions as internal forms.https://www.zbmath.org/1453.580032021-02-27T13:50:00+00:00"Dąbrowski, Ludwik"https://www.zbmath.org/authors/?q=ai:dabrowski.ludwik"D'Andrea, Francesco"https://www.zbmath.org/authors/?q=ai:dandrea.francesco"Sitarz, Andrzej"https://www.zbmath.org/authors/?q=ai:sitarz.andrzejSummary: Corrects a grant number in the authors's paper [ibid. 108, No. 5, 1323--1340 (2018; Zbl 1395.58007)].Torsion and \(K\)-theory for some free wreath products.https://www.zbmath.org/1453.460612021-02-27T13:50:00+00:00"Freslon, Amaury"https://www.zbmath.org/authors/?q=ai:freslon.amaury"Martos, Rubén"https://www.zbmath.org/authors/?q=ai:martos.rubenSummary: We classify torsion actions of free wreath products of arbitrary compact quantum groups by \(S_N^+\) and use this to prove that if \(\mathbb{G}\) is a torsion-free compact quantum group satisfying the strong Baum-Connes property then \(\mathbb{G}\wr_{\ast }S_N^+\) also satisfies the strong Baum-Connes property. We then compute the $K$-theory of free wreath products of classical and quantum free groups by \(SO_q(3)\).Connections between optimal constants in some norm inequalities for differential forms.https://www.zbmath.org/1453.460242021-02-27T13:50:00+00:00"Zsuppán, Sándor"https://www.zbmath.org/authors/?q=ai:zsuppan.sandorSummary: We derive an improved Poincaré inequality in connection with the Babuška-Aziz and Friedrichs-Velte inequalities for differential forms by estimating the domain specific optimal constants in the respective inequalities with each other provided the domain supports the Hardy inequality. We also derive upper estimates for the constants of a star-shaped domain by generalizing the known Horgan-Payne type estimates for planar and spatial domains to higher dimensional ones.Direct limits of adèle rings and their completions.https://www.zbmath.org/1453.111512021-02-27T13:50:00+00:00"Kelly, James P."https://www.zbmath.org/authors/?q=ai:kelly.james-pierre"Samuels, Charles L."https://www.zbmath.org/authors/?q=ai:samuels.charles-lFor a Galois extension \(E/F\), with F a global field, the paper defines a topological ring, denoted by \(\overline{\mathbb{V}}_E\), called the generalized adèle ring of \(E\).
Denote by \(\mathcal{J}_E\) the set \(\{K \subseteq E : K/F\text{ finite Galois}\}\), by \(\mathbb{A}_K\) the adèle ring of \(K\) and by \(\overline{\mathbb{A}}_K\) its completion with respect to some (any) invariant metric on \(\mathbb{A}_K\).
Main theorems are now stated in a short form.
Theorem 1. If \(E/F\) is a Galois extension, then the following hold:
\begin{itemize}
\item[i)] \(\overline{\mathbb{V}}_E\) is a metrizable topological ring which is complete with respect to any invariant metric on \(\overline{\mathbb{V}}_E\).
\item[ii)] If \(\mathbb{V}_E = \bigcup_{K\in\mathcal J_E} \mathbb{V}_K\), then \(\overline{\mathbb{V}}_E\) equals the closure of \(\mathbb{V}_E\) in \(\overline{\mathbb{V}}_E\).
\item[iii)] There exists a topological ring isomorphism \(\phi: \overline{\mathbb{A}}_E \to \overline{\mathbb{V}}_E\) such that \(\phi(\mathbb{A}_E) = \mathbb{V}_E\).
\end{itemize}
Theorem 2.
If \(E/F\) is an infinite Galois extension, then \(\mathbb{A}_E\) has empty interior in \(\overline{\mathbb{A}}_E\).
Reviewer: Stelian Mihalas (Timişoara)Approaching the UCT problem via crossed products of the Razak-Jacelon algebra.https://www.zbmath.org/1453.460542021-02-27T13:50:00+00:00"Barlak, Selçuk"https://www.zbmath.org/authors/?q=ai:barlak.selcuk"Szabó, Gábor"https://www.zbmath.org/authors/?q=ai:szabo.gabor-jSummary: We show that the UCT problem for separable, nuclear C*-algebras relies only on whether the UCT holds for crossed products of certain finite cyclic group actions on the Razak-Jacelon algebra. This observation is analogous to and in fact recovers a characterization of the UCT problem in terms of finite group actions on the Cuntz algebra \(\mathcal{O}_2\) established in previous work by the authors. Although based on a similar approach, the new conceptual ingredients in the finite context are the recent advances in the classification of stably projectionless C*-algebras, as well as a known characterization of the UCT problem in terms of certain tracially AF C*-algebras due to \textit{M. Dadarlat} [in: Operator algebras and mathematical physics. Proceedings of the conference, Constanţa, Romania, July 2--7, 2001. Bucharest: Theta. 65--74 (2003; Zbl 1284.19008)].Furstenberg boundary of minimal actions.https://www.zbmath.org/1453.460592021-02-27T13:50:00+00:00"Naghavi, Zahra"https://www.zbmath.org/authors/?q=ai:naghavi.zahraSummary: For a countable discrete group \(\Gamma\) and a minimal \(\Gamma\)-space \(X\), we study the notion of \((\Gamma, X)\)-boundary, which is a natural generalization of the notion of topological \(\Gamma\)-boundary in the sense of \textit{H. Furstenberg} [Ann. Math. (2) 77, 335--386 (1963; Zbl 0192.12704)]. We give characterizations of the \((\Gamma, X)\)-boundary in terms of essential or proximal extensions. The characterization is used to answer a problem of \textit{D. Hadwin} and \textit{V. I. Paulsen} [Sci. China, Math. 54, No. 11, 2347--2359 (2011; Zbl 1247.46065)] in negative. As an application, we find necessary and sufficient condition for the corresponding reduced crossed product to be exact.On higher-order proto-differentiability of perturbation maps.https://www.zbmath.org/1453.901712021-02-27T13:50:00+00:00"Tung, L. T."https://www.zbmath.org/authors/?q=ai:tung.le-thanhThe author considers a class of parametrized vector optimization problems. Based on several kinds of derivatives for related set-valued mappings, the paper presents results on higher-order sensitivity analysis for this problem class. In particular, under certain qualification conditions, the so-called higher-order proto-differentiability for some corresponding perturbation maps is shown.
Reviewer: Jan-Joachim Rückmann (Bergen)Measurable and continuous units of an \(E_0\)-semigroup.https://www.zbmath.org/1453.460582021-02-27T13:50:00+00:00"Murugan, S. P."https://www.zbmath.org/authors/?q=ai:murugan.s-pazhani-bala"Sundar, S."https://www.zbmath.org/authors/?q=ai:sundar.subbiah|sundar.sobers|sundar.s-babu|sundar.shyamSummary: Let \(P\) be a closed convex cone in \(\mathbb{R}^d\) which is spanning, i.e., \(P-P=\mathbb{R}^d\) and pointed, i.e., \(P\,\cap -P=\{0\}\). Let \(\alpha :=\{\alpha_x\}_{x\in P}\) be an \(E_0\)-semigroup over \(P\) and let \(E\) be the product system associated to \(\alpha\). We show that there exists a bijective correspondence between the units of \(\alpha\) and the units of \(E\).K-theory and index theory for some boundary groupoids.https://www.zbmath.org/1453.580062021-02-27T13:50:00+00:00"Carrillo Rouse, Paulo"https://www.zbmath.org/authors/?q=ai:carrillo-rouse.paulo"So, Bing Kwan"https://www.zbmath.org/authors/?q=ai:so.bing-kwanSpecifying the exact relation between ellipticity and the Fredholm property is an open problem, specially for operators associated with singular spaces. The paper under review indicates that the answer is related with the Atiyah-Singer index theorem.
The authors examine the problem in a groupoid framework, making use of the Fredholm conditions given by Carvalho, Nistor and Qiao in [\textit{C. Carvalho} et al., Oper. Theory: Adv. Appl. 267, 79--122 (2018; Zbl 1446.58011)]. In particular, the authors focus on a large class of Lie groupoids, namely the holonomy groupoids of almost regular foliations (in the sense of Debord, see [\textit{C. Debord}, J. Differ. Geom. 58, No. 3, 467--500 (2001; Zbl 1034.58017); \textit{C. Debord} et al., J. Reine Angew. Math. 628, 1--35 (2009; Zbl 1169.58005)]) on a compact manifold \(M\) whose leaves are the connected components of a compact submanifold \(M_1\) and manifold \(M_0 = M \setminus M_1\). Further, the isotropy Lie group over \(M_1\) has dimension equal to the codimension of \(M_1\) in \(M\). The main result is the following: When the codimension is odd, every elliptic operator admits a compact perturbation to a Fredholm one. However, when the codimension is even, it is shown that the Atiyah-Singer theorem imposes an obstruction to such a perturbation. These results follow from a careful computation of the \(K\)-theory in both cases.
Reviewer: Iakovos Androulidakis (Athína)On modulated topological vector spaces and applications.https://www.zbmath.org/1453.460082021-02-27T13:50:00+00:00"Kozlowski, Wojciech M."https://www.zbmath.org/authors/?q=ai:kozlowski.wojciech-mIf \(X\) is a real vector space, an even, convex function \(\rho:X\to [0,\infty]\) vanishing at \(0\) is a \textit{convex modular} and the set \(X_\rho = \{x\in X: \rho(\lambda x)\to 0\ \text{as}\ \lambda\to 0\}\) is a \textit{modular space}. In [Fixed point theory in modular function spaces. Cham: Birkhäuser/Springer (2015; Zbl 1318.47002)], \textit{M.\,A.\thinspace Khamsi} and the author published a treatise on fixed point theory in modular function spaces.
In the article under review, the author introduces a class of vector spaces that includes Banach spaces, modular function spaces, and many others, and obtains fixed point results in this larger class of vector spaces. With \(\rho\) and \(X_\rho\) as above and \(\tau\) a linear Hausdorff topology on \(X\), the triple \((X_\rho,\rho,\tau)\) is a \textit{modulated topological vector space} if \(\rho\) is \(\tau\)-lower semicontinuous on \(X\) and, if whenever the sequence \((x_n)\) in \(X\) \(\rho\)-converges to \(x\) in \(X\), there exists a subsequence of \((x_n)\) that is \(\tau\)-convergent to \(x\). The author uses a modular analogue of normal structure to obtain an analogue of Kirk's well-known fixed point theorem in the setting of modulated topological vector spaces. In particular, let \((X_\rho, \rho, \tau)\) be a \(\rho\)-complete modulated topological vector space and let \(C\) be a nonempty, \(\rho\)-closed, \(\rho\)-bounded subset of \(X_\rho\). Then, if \(C\) is \(\tau\)-sequentially compact and satisfies a \(\rho\)-analogue of normal structure, then every \(\rho\)-nonexpansive mapping \(T:C\to C\) has a fixed point. The author notes that analogous results can be obtained if \(\rho\) is an \(s\)-convex modular for any \(s>0\).
Reviewer: Barry Turett (Rochester)Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure.https://www.zbmath.org/1453.470042021-02-27T13:50:00+00:00"Ablet, Ehmet"https://www.zbmath.org/authors/?q=ai:ablet.ehmet"Cheng, Lixin"https://www.zbmath.org/authors/?q=ai:cheng.lixin"Cheng, Qingjin"https://www.zbmath.org/authors/?q=ai:cheng.qingjin"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.2From the text: In the authors' paper [ibid. 62, No. 1, 147--156 (2019; Zbl 1453.47003)], Theorem 4.4 states that every infinite-dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. However, there is a gap in the proof. In fact, we found that [loc. cit., Lemma 4.3] is not true. In this erratum, we give a corrected proof of [loc. cit., Theorem 4.4].Physicist's approach to public transportation networks: between data processing and statistical physics.https://www.zbmath.org/1453.820792021-02-27T13:50:00+00:00"Korduba, Yaryna"https://www.zbmath.org/authors/?q=ai:korduba.yaryna"Holovatch, Yurij"https://www.zbmath.org/authors/?q=ai:holovatch.yurij"De Regt, Robin"https://www.zbmath.org/authors/?q=ai:de-regt.robinSummary: In this paper we aim to demonstrate how physical perspective enriches statistical analysis when dealing with a complex system of many interacting agents of non-physical origin. To this end, we discuss analysis of urban public transportation networks viewed as complex systems. In such studies, a multi-disciplinary approach is applied by integrating methods in both data processing and statistical physics to investigate the correlation between public transportation network topological features and their operational stability. These studies incorporate concepts of coarse graining and clusterization, universality and scaling, stability and percolation behavior, diffusion and fractal analysis.A decomposition by non-negative functions in the Sobolev space \(W^{k,1}\).https://www.zbmath.org/1453.460332021-02-27T13:50:00+00:00"Ponce, Augusto"https://www.zbmath.org/authors/?q=ai:ponce.augusto-c"Spector, Daniel"https://www.zbmath.org/authors/?q=ai:spector.daniel-eSummary: We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space \(W^{k,1}(\mathbb{R}^d)\) as the difference of two non-negative functions in the same space with control of their norms.Jordan-von Neumann type constant and fixed points for multivalued nonexpansive mappings.https://www.zbmath.org/1453.460132021-02-27T13:50:00+00:00"Zuo, Zhanfei"https://www.zbmath.org/authors/?q=ai:zuo.zhanfei"Tang, Chunlei"https://www.zbmath.org/authors/?q=ai:tang.chunlei|tang.chun-lei"Chen, Xiaochun"https://www.zbmath.org/authors/?q=ai:chen.xiaochun"Wang, Liangwei"https://www.zbmath.org/authors/?q=ai:wang.liangweiSummary: We give some sufficient conditions for the Domínguez-Lorenzo condition in terms of the Jordan-von Neumann type constant, and the coefficient of weak orthogonality. As a consequence, we obtain some sufficient conditions for normal structure and fixed point theorems for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in recent papers.Product of extension domains is still an extension domain.https://www.zbmath.org/1453.460312021-02-27T13:50:00+00:00"Koskela, Pekka"https://www.zbmath.org/authors/?q=ai:koskela.pekka"Zhu, Zheng"https://www.zbmath.org/authors/?q=ai:zhu.zhengSummary: Our main result gives a functional property of the class of \(W^{1,p}\)-extension domains. Let \(\Omega_1\subset\mathbb{R}^n\) and \(\Omega_2\subset\mathbb{R}^m\) both be \(W^{1,p} \)-extension domains for some \(1<p\leq\infty \). We prove that \(\Omega_1\times\Omega_2\subset\mathbb{R}^{n+m}\) is also a \(W^{1,p}\)-extension domain. We also establish the converse statement.A noninequality for the fractional gradient.https://www.zbmath.org/1453.460352021-02-27T13:50:00+00:00"Spector, Daniel"https://www.zbmath.org/authors/?q=ai:spector.daniel-eSummary: In this paper we give a streamlined proof of an inequality recently obtained by the author: For every \(\alpha \in (0, 1)\) there exists a constant \(C=C(\alpha, d) > 0\) such that \[\|u\|_{L^{d/(d-\alpha), 1}(\mathbb{R}^d)} \leq C\|D^\alpha u\|_{L^1(\mathbb{R}^d; \mathbb{R}^d)}\] for all \(u \in L^q(\mathbb{R}^d)\) for some \(1 \leq q < d/(1-\alpha)\) such that \(D^\alpha u:=\nabla I_{1-\alpha} u \in L^1(\mathbb{R}^d; \mathbb{R}^d)\). We also give a counterexample which shows that in contrast to the case \(\alpha =1\), the fractional gradient does not admit an \(L^1\) trace inequality, i.e., \(\| D^\alpha u\|_{L^1(\mathbb{R}^d; \mathbb{R}^d)}\) cannot control the integral of \(u\) with respect to the Hausdorff content \(\mathscr{H}^{d-\alpha}_\infty\). The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space \(L^1(\mathscr{H}^{d-\beta}_\infty), \beta \in [1, d)\). It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to \(\beta \in (0, 1)\).Tensor products of \(C^*\)-algebras and operator spaces. The Connes-Kirchberg problem.https://www.zbmath.org/1453.460022021-02-27T13:50:00+00:00"Pisier, Gilles"https://www.zbmath.org/authors/?q=ai:pisier.gilles\textit{E.~Kirchberg}, in his seminal work [Invent. Math. 112, No.~3, 449--489 (1993; Zbl 0803.46071)], provided an in-depth investigation of operator algebras based on tensor products of \(C^*\)-algebras, on Lance's extension property WEP (``weak expectation property''), and on the ``local lifting property'' LLP, introduced in [loc. cit.]. At the end of his paper he offered several conjectures grouped into two sections A and~B, the conjectures in A and B being equivalent to the other conjectures in the same section.
The conjectures in A were refuted by \textit{M.~Junge} and the author [Geom. Funct. Anal. 5, No.~2, 329--363 (1995; Zbl 0832.46052)]; a representative of this set of conjectures is the implication WEP \(\Rightarrow\) LLP. It is the conjectures in B, represented by the implication LLP \(\Rightarrow\) WEP, that are at the centre of interest in the monograph under review; the above is referred to as Kirchberg's conjecture. According to the author, this is considered as one of the most important, if not the most important open problem in operator algebra theory. The fact, already proved by Kirchberg, that ``LLP \(\Rightarrow\) WEP'' is equivalent to Connes's embedding problem (every II\(_1\)-factor on a separable Hilbert space embeds into an ultrapower of the hyperfinite II\(_1\)-factor) lends itself to support this assessment.
The key notion of the monograph is that of a nuclear pair. A~pair of \(C^*\)-algebras \((A,B)\) is a nuclear pair if there is only one \(C^*\)-algebra norm on \(A\otimes B\); recall that \(A\) is nuclear if this is so for every \(C^*\)-algebra~\(B\). A fundamental result is that, for \(\mathscr{B}= B(\ell_2)\) and \(\mathscr{C}= C^*(\mathbb{F}_\infty)\) with \(\mathbb{F}_\infty\), the free group on countably many generators, the pair \((\mathscr{C}, \mathscr{B})\) is nuclear, and one incarnation of Kirchberg's conjecture is that \((\mathscr{C}, \mathscr{C})\) is a nuclear pair. Also, ``WEP \(\Rightarrow\) LLP'' is equivalent to \((\mathscr{B}, \mathscr{B})\) being a nuclear pair, which was disproved by Junge and the autho [loc.\,cit.]; a simpler argument is given in Chapter~18.
One feature of the author's presentation is the use of operator space techniques whenever feasible. This monograph thus studies fundamental properties of operator spaces and operator systems in addition to tensor norms, operators between tensor products, and group \(C^*\)-algebras in the first few chapters. Then, ramifications of WEP, LLP, and other properties are investigated, and various equivalences of Kirchberg's conjecture are given. Apart from Connes's embedding problem, there is Tsirelson's problem (from quantum information theory) and a problem from Banach space theory (whether the dual of every \(C^*\)-algebra is finitely representable in the space of trace class operators) which are discussed. A~chapter on open problems concludes the text, and an appendix provides necessary prerequisites.
This is a very rich and detailed monograph on an enormously important subject. It is written in the crystal clear and elegant style that is the hallmark of its author, and it offers a lot of information to specialists and novices alike. The book will certainly become an authoritative guide.
Reviewer's remark. At the beginning of 2020, a paper was uploaded to arXiv claiming to give a negative solution to Tsirelson's problem and thus to the Connes-Kirchberg problem; however, a revised version of that paper admits gaps in the original version, and claims towards the Tsirelson problem are no longer included. To the best of the reviewer's knowledge, the Connes-Kirchberg problem remains open; see, however, the author's recent paper [\textit{G.~Pisier}, Invent. Math. 222, No.~2, 513--544 (2020; Zbl 07269002)].
Reviewer: Dirk Werner (Berlin)New examples on Lavrentiev gap using fractals.https://www.zbmath.org/1453.350822021-02-27T13:50:00+00:00"Balci, Anna Kh."https://www.zbmath.org/authors/?q=ai:balci.anna-kh"Diening, Lars"https://www.zbmath.org/authors/?q=ai:diening.lars"Surnachev, Mikhail"https://www.zbmath.org/authors/?q=ai:surnachev.mikail-dmitrievichThe paper deals with the Lavrentiev gap, i.e., the phenomenon which occurs when the minimum of an integral functional \(\mathcal{G}\) taken over smooth functions differs from the one taken over the associated energy space. The Lavrentiev gap is clearly closely related to the (non-)density of smooth functions, i.e. to the fact that \(H^{1,p(\cdot)}(\Omega)\neq W^{1,p(\cdot)}(\Omega)\), where \(p(\cdot)\) is variable exponent. In [Izv. Akad. Nauk SSSR Ser. Mat., 50, No. 4, 675--710 (1986)], \textit{VV. Zhikov} presented a two-dimensional checkboard example with a Lavrentiev gap. In such an example and in others, the dimension played a critical role, since the exponent presents a saddle point where it crossed the dimension \(d\). In the present paper, the authors provide new examples of variable exponents, such that the Lavrentiev gap occurs but which do not need to cross the dimensional threshold. They also show that \(H^{1,p(\cdot)}(\Omega)\neq W^{1,p(\cdot)}(\Omega)\) and the ambiguity of the notion of \(p(\cdot)\)-harmonicity.
Reviewer: Paolo Musolino (Padova)A course on topological vector spaces.https://www.zbmath.org/1453.460012021-02-27T13:50:00+00:00"Voigt, Jürgen"https://www.zbmath.org/authors/?q=ai:voigt.jurgenThis booklet of just about 150 pages provides a concise yet very readable treatment of the basics of the theory of topological vector spaces which was developed from the 1940's till the 1970's. Many classical treatises of that period such as the books of Bourbaki or Köthe were written in a strict Bourbaki style, always presenting the most general and structural viewpoint, and are not easy to read. Compared to this style, the present author's course provides an introduction in a streamlined and reader-friendly way -- the somewhat baroque typography of the classics is cleaned up and the typesetting, e.g., of the theorems in coloured boxes makes orientation in this book very comfortable.
The main topics covered by the book are the principles of topological vector spaces which serve as tools, e.g., for partial differential equations or semigroup theory on Banach spaces. The course treats
\begin{itemize}
\item basic facts and constructions for topological vector spaces like initial and final locally convex topologies,
\item implications of the Hahn-Banach theorem for duality theory (presented for general dual pairs) with fortunately only a few of the huge zoo of locally convex properties (such as barrelled or bornological spaces),
\item completeness properties of topological vector spaces including Grothendieck's construction of the completion, and
\item some basic facts about Fréchet spaces and the dual concept of DF-spaces.
\end{itemize}
A relatively large part of the book is devoted to (weak) compactness, including standard characterizations of reflexivity and compactness criteria of Eberlein, Grothendieck, Krein, and Šmulian.
Almost entirely avoided are systematic results about the \emph{morphisms} of the category of topological vector spaces, i.e., continuous linear operators. Therefore, important topics such as closed graph theorems, adjoints, or tensor products are not covered -- even the uniform boundedness principle of Banach and Steinhaus is only stated for families of functionals instead of operators.
The book is thus not a panorama of the theory, but confines itself strictly to the tools serving for other disciplines of functional analysis.
Within this goal, the author presents a compact, precise, and very well written introduction to the theory of topological vector spaces, including many examples and, due to the restricted length, only a few applications (such as Bernstein's characterization of completely monotone functions via the Krein-Milman theorem). The text is practically free of typos and errors (the only exceptions the referee noticed are the definition of the absolutely convex hull, to make the theorem of bipolars correct the hull of \(\emptyset\) should not be empty but \(\{0\}\), and a missing reference that Theorem~7.11 is V.\,Klee's solution to a problem posed by Banach).
The book may be highly recommended to all students and researchers with some knowledge of Banach or Hilbert space oriented functional analysis who want to learn its general abstract foundations.
Reviewer: Jochen Wengenroth (Trier)Besov and Triebel-Lizorkin spaces on Lie groups.https://www.zbmath.org/1453.460252021-02-27T13:50:00+00:00"Bruno, Tommaso"https://www.zbmath.org/authors/?q=ai:bruno.tommaso"Peloso, Marco M."https://www.zbmath.org/authors/?q=ai:peloso.marco-maria"Vallarino, Maria"https://www.zbmath.org/authors/?q=ai:vallarino.mariaThe well-known spaces \(B^s_{p,q}(\mathbb R^n)\) and \(F^s_{p,q} (\mathbb R^n)\) with \(s \ge 0\) and \(1<p,q<\infty\) on \(\mathbb R^n\) can be characterized in terms of the Gauss-Weierstrass semi-group \(e^{-t \Delta}\), where \(\Delta\) is the Laplacian on \(\mathbb R^n\), as the collection of all distributions \(f\in S'(\mathbb R^n)\) such that
\[
\big\| e^{-t_0 \Delta} f \, |L_p (\mathbb R^n) \big\| + \Big( \int^1_0 t^{-sq/2} \big\| (t \Delta)^m e^{-t \Delta} f \, | L_p (\mathbb R^n) \big\|^q
\frac{dt}{t} \Big)^{1/q}
\]
and
\[
\big\| e^{-t_0 \Delta} f \, |L_p (\mathbb R^n) \big\| + \Big\| \Big(\int^1_0 t^{-sq/2} \big| \big(t \Delta)^m e^{-t \Delta} f (\cdot)
\big|^q \frac{dt}{t} \Big)^{1/q} \, | L_p (\mathbb R^n) \Big\|
\]
are finite. Here, \(t_0 \in (0,1)\) and \(s/2 <m\) is a natural number (equivalent norms). It is the main aim of the paper to develop a corresponding theory for the spaces \(B^s_{p,q} (G)\) and \(F^s_{p,q} (G)\) where \(G\) is a general non-compact Lie group endowed with a sub-Riemannian structure and a suitable substitute of the Euclidean Laplacian and the Gauss-Weierstrass semi-group.
Reviewer: Hans Triebel (Jena)Refinement of Novikov-Betti numbers and of Novikov homology provided by an angle valued map.https://www.zbmath.org/1453.550072021-02-27T13:50:00+00:00"Burghelea, D."https://www.zbmath.org/authors/?q=ai:burghelea.danGiven a compact ANR \(X\) and a map \(f\colon X\to S^1\), a field \(\kappa\) and a non-negative integer, the author assigns to this a finite configuration of complex numbers with multiplicities, and a finite configuration of free \(\kappa[t,t^{-1}]\)-modules indexed over the same complex numbers. The finite configuration of complex numbers refine the Novikov-Betti numbers, while the configurations of free modules refine Novikov homology of \(X\) correspoding to \(f\). In the case \(\kappa = \mathbb{C}\) the author shows that the configuration of free modules can be completed to mutually orthogonal closed Hilbert submodules of the \(L^2\)-homology of the infinite cyclic covering determined by \(f\).
Reviewer: Dirk Schütz (Durham)Emergence of \(\delta^\prime \)-waves in the zero pressure gas dynamic system.https://www.zbmath.org/1453.351282021-02-27T13:50:00+00:00"Sarrico, C. O. R."https://www.zbmath.org/authors/?q=ai:sarrico.carlos-orlando-rThe author studies singular solutions to the system of pressureless gas dynamics with initial data containing \(\delta\)-measures in both components. It is demonstrated that the solution may develop more singular \(\delta'\)-waves in positive time. This research is conducted strictly within the framework of distribution theory and is based on a suitable approach to multiplication of distributions.
Reviewer: Evgeniy Panov (Novgorod)Weakly Radon-Nikodým Boolean algebras and independent sequences.https://www.zbmath.org/1453.060122021-02-27T13:50:00+00:00"Avilés, Antonio"https://www.zbmath.org/authors/?q=ai:aviles.antonio"Martínez-Cervantes, Gonzalo"https://www.zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Plebanek, Grzegorz"https://www.zbmath.org/authors/?q=ai:plebanek.grzegorzSummary: A compact space is said to be weakly Radon-Nikodým (WRN) if it can be weak\(^*\)-embedded into the dual of a Banach space not containing \(\ell_1\). We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated non-WRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodríguez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses.Spectral pairs and positive-definite-tempered distributions.https://www.zbmath.org/1453.420312021-02-27T13:50:00+00:00"Jorgensen, Palle E. T."https://www.zbmath.org/authors/?q=ai:jorgensen.palle-e-t"Tian, Feng"https://www.zbmath.org/authors/?q=ai:tian.fengSimple equivariant \(\mathrm{C}^\ast\)-algebras whose full and reduced crossed products coincide.https://www.zbmath.org/1453.460602021-02-27T13:50:00+00:00"Suzuki, Yuhei"https://www.zbmath.org/authors/?q=ai:suzuki.yuheiSummary: For any second countable locally compact group \(G\), we construct a simple \(G\)-\(\mathrm{C}^\ast\)- algebra whose full and reduced crossed product norms coincide. We then construct its \(G\)-equivariant representation on another simple \(G\)-\(\mathrm{C}^\ast\)-algebra without the coincidence condition. This settles two problems posed by \textit{C. Anantharaman-Delaroche} [Trans. Am. Math. Soc. 354, No. 10, 4153--4178 (2002; Zbl 1035.46039)]. Some constructions involve the Baire category theorem.Factors generated by \textit{XY}-model with competing Ising interactions on the Cayley tree.https://www.zbmath.org/1453.460572021-02-27T13:50:00+00:00"Mukhamedov, Farrukh"https://www.zbmath.org/authors/?q=ai:mukhamedov.farruh-m"El Gheteb, Soueidy"https://www.zbmath.org/authors/?q=ai:el-gheteb.soueidyAuthors' abstract: In the present paper, we consider a quantum Markov chain corresponding to the \textit{XY}-model with competing Ising interactions on the Cayley tree of order two. Earlier, it was proved that this state does exist and is unique. Moreover, it has clustering property. This means that the von Neumann algebra generated by this state is a factor. In the present paper, we establish that the factor generated by this state may have type \(\text{III}_{\lambda }$, $ \lambda \in (0,1)\), which is unusual for states associated with models with nontrivial interactions.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Sobolev's theorem for double phase functionals.https://www.zbmath.org/1453.460212021-02-27T13:50:00+00:00"Mizuta, Yoshihiro"https://www.zbmath.org/authors/?q=ai:mizuta.yoshihiro"Ohno, Takao"https://www.zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://www.zbmath.org/authors/?q=ai:shimomura.tetsuSummary: Our aim in this paper is to establish generalizations of Sobolev's theorem for double phase functionals \(\Phi(x,t) =t^p+\{b(x)t(\log(e+t))^\tau\}^q\), where \(1<p\leq q< \infty\), \(\tau >0\) and \(b\) is a nonnegative bounded function satisfying \(|b(x)- b(y)|\leq C|x - y|^\theta (\log(e+|x - y|^{-1}))^{- \tau}\) for \(0\leq \theta <1\).On a family of representations of residually finite groups.https://www.zbmath.org/1453.220042021-02-27T13:50:00+00:00"Manuilov, V."https://www.zbmath.org/authors/?q=ai:manuilov.v-mLet \(G\) be a residually finite group, and fix a chain of finite-index normal subgroups \(G\supset G_1\supset G_2\supset \cdots\) satisfying \(\bigcap G_n=\{1\}\). In this paper the author considers a family of \(G\)-invariant subspaces \(H_\gamma\) of the ultraproduct (for some free ultrafilter on \(\mathbb{N}\)) of the quasi-regular representations \(\ell^2(G/G_n)\). The subspaces are defined by imposing restrictions on the rate of growth of the supports of functions on the sequence of finite sets \(G/G_n\). When no such restrictions are imposed, the resulting representation of \(G\) is weakly equivalent to the direct sum \(\bigoplus_n \ell^2(G/G_n)\). At the other extreme, the author proves that for all sufficiently slow rates of growth \(\gamma\), the resulting \(G\)-representation \(H_\gamma\) is weakly equivalent to the regular representation of \(G\). The author also analyses two contrasting cases of intermediate growth rates. On the one hand it is shown that if \(G\) has property \((\tau)\) [\textit{A. Lubotzky}, Discrete groups, expanding graphs and invariant measures. Appendix by Jonathan D. Rogawski. Basel: Birkhäuser (1994; Zbl 0826.22012)], then for all growth rates \(\gamma\) slower than the maximal one the representation \(H_\gamma\) does not weakly contain the trivial representation. On the other hand the author proves that for a certain sequence of finite-index normal subgroups of the free group \(\mathbb{F}_2\) (studied previously in [\textit{G. Arzhantseva} and \textit{E. Guentner}, Math. Ann. 354, No. 3, 863--870 (2012; Zbl 1270.20041)]), and for a certain intermediate growth rate \(\gamma\), the representation \(H_\gamma\) does weakly contain the trivial representation of \(\mathbb{F}_2\).
The author states in the introduction that his original motivation was to study the \(C^*\)-algebra norms on \(\mathbb{C}[G]\) induced by the family of representations \(H_\gamma\), in the hope of finding norms intermediate between the reduced \(C^*\)-algebra norm, and the norm induced by the sum of the quasi-regular representations. This hope has not, as yet, been realised; but, as the author notes, even if every norm in the family turned out to be equal to one of those two extreme norms, one would still be left with the interesting question of determining the critical rate of growth separating the two cases.
Reviewer: Tyrone Crisp (Orono)Some Hardy and Carleson measure spaces estimates for Bochner-Riesz means.https://www.zbmath.org/1453.420162021-02-27T13:50:00+00:00"Tan, Jian"https://www.zbmath.org/authors/?q=ai:tan.jian.1|tan.jian.2|tan.jianSummary: In this paper, we show that the Bochner-Riesz means are bounded on weighted and variable Hardy spaces by using the finite atomic decomposition theories. The boundedness of Bochner-Riesz means on weighted and variable Carleson measure spaces is also obtained. Moreover, we also prove that the maximal Bochner-Riesz means are bounded from weighted or variable Hardy spaces to weighted or variable Lebesgue spaces.On some double Cesàro sequence spaces.https://www.zbmath.org/1453.460062021-02-27T13:50:00+00:00"Sever, Yurdal"https://www.zbmath.org/authors/?q=ai:sever.yurdal"Altay, Bilal"https://www.zbmath.org/authors/?q=ai:altay.bilalSummary: In this study, we define the double Cesàro sequence spaces Ces$_p$, Ces$_{bp}$ and Ces$_{bp0}$ and examine some properties of those sequence spaces. Furthermore, we determine the $\beta(bp)$-duals of the spaces Ces$_{bp}$ and Ces$_p$.On various functional representations of the space of Schwartz operators.https://www.zbmath.org/1453.810052021-02-27T13:50:00+00:00"Amosov, G. G."https://www.zbmath.org/authors/?q=ai:amosov.grigori-gSummary: In this paper, we discuss various representations in which the space \(S\) of Schwartz operators turns into the space of test functions, whereas the dual space \(S'\) turns into the space of generalized functions.Variance function of Boolean additive convolution.https://www.zbmath.org/1453.600442021-02-27T13:50:00+00:00"Fakhfakh, Raouf"https://www.zbmath.org/authors/?q=ai:fakhfakh.raoufSummary: Suppose \(\mathbb{V}_\nu\) is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family \(\mathcal{K}_+ ( \nu )\) generated by a non degenerate probability measure \(\nu\) with support bounded from above. We determine the formula for pseudo-variance function (or variance function \(V_\nu\) in case of existence) under boolean additive convolution power. This formula is used to identify the relation between variance functions under Boolean Bercovici-Pata Bijection between probability measures. We also give the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers.On embeddings of grand grand Sobolev-Morrey spaces with dominant mixed derivatives.https://www.zbmath.org/1453.460222021-02-27T13:50:00+00:00"Najafov, Alik M."https://www.zbmath.org/authors/?q=ai:najafov.alik-m"Babayev, Rovshan F."https://www.zbmath.org/authors/?q=ai:babayev.rovshan-fSummary: In this paper it is constructed a new grand grand Sobolev-Morrey \(S_{p),\varkappa),a,\alpha}^lW(G)\) spaces with dominant mixed derivatives. With help integral representation of generalized mixed derivatives of functions, defined on \(n\)-dimensional domains satisfying flexible horn condition, an embedding theorem is proved. In other works, the embedding theorem is proved in these spaces and belonging of the generalized mixed derivatives of functions from these spaces to the Hölder class, was studied.Set transformations, symmetrizations and isoperimetric inequalities.https://www.zbmath.org/1453.260342021-02-27T13:50:00+00:00"Van Schaftingen, J."https://www.zbmath.org/authors/?q=ai:van-schaftingen.jean"Willem, M."https://www.zbmath.org/authors/?q=ai:willem.michelFor the entire collection see [Zbl 1066.35001].Simultaneous approximations in Banach space-valued Bochner-Lebesgue spaces with variable exponent.https://www.zbmath.org/1453.410092021-02-27T13:50:00+00:00"Wei, Haihua"https://www.zbmath.org/authors/?q=ai:wei.haihua"Xu, Jingshi"https://www.zbmath.org/authors/?q=ai:xu.jingshiThe \textit{variable} Lebesgue and Lebesgue-Bochner spaces, as their name implies, are a generalization of the classical Lebesgue and Lebesgue-Bochner spaces \(L^{p}\) and \(L^{p}(X)\), replacing the constant exponent \(p\) with a variable exponent function \(p(\cdot)\). The resulting Banach function spaces \(L^{p(\cdot)}\) and \(L^{p(\cdot)}(X)\) have many properties similar to their classical counterparts, but they also differ in surprising and subtle ways. For this reason the variable Lebesgue-type spaces have an intrinsic interest, but they are also important in some applications.
In the manuscript under review the authors prove some \(N\)-simultaneous proximinality results of \(L^{p(\cdot)}(A,\mathcal{A}_0,Y)\) in \(L^{p(\cdot)}(A,\mathcal{A},X)\), where \((A,\mathcal{A},\mu)\) is a \(\sigma\)-finite positive measure space, \(\mathcal{A}_0\) is a sub-algebra of \(\mathcal{A}\), \(X\) is a Banach space and \(Y\) is a nonempty locally weakly compact convex subset of \(X\).
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Ranks of operators in simple \(C^*\)-algebras with stable rank one.https://www.zbmath.org/1453.460552021-02-27T13:50:00+00:00"Thiel, Hannes"https://www.zbmath.org/authors/?q=ai:thiel.hannesIt is obtained that the TW (Toms-Winter) conjecture holds for AsH (approximately subhomogeneous) \(C^*\)-algebras with SR (stable rank) 1. In particular, it is obtained that the JS (Jiang-Su algebra) stability and the SC (strict comparison) property of positive elements are equivalent for SU (simple, separable, unital, non-elementary) \(C^*\)-algebras with SR 1 and ND (nuclear dimension) locally finite. It is shown that for such \(C^*\)-algebras \(A\), any PCA (strictly positive, lower semi-continuous, affine) function on the simplex of normalized quasi-traces of \(A\) is equal to the PCA rank function for some positive element of \(A\) tensored with the \(C^*\)-algebra \(\mathbb K\) of compact operators. Historical motivation for considering comparison and rank theory is given in the lengthy Introduction.
Reviewer: Takahiro Sudo (Nishihara)Hardy's inequality on Hardy-Morrey spaces.https://www.zbmath.org/1453.260142021-02-27T13:50:00+00:00"Ho, Kwok-Pun"https://www.zbmath.org/authors/?q=ai:ho.kwok-punSummary: We generalize the Hardy inequality to Hardy-Morrey spaces.On absolute Riesz summability factors of infinite series and their application to Fourier series.https://www.zbmath.org/1453.260192021-02-27T13:50:00+00:00"Bor, Hüseyin"https://www.zbmath.org/authors/?q=ai:bor.huseyinSummary: In this paper, some known results on the absolute Riesz summability factors of infinite series and trigonometric Fourier series have been generalized for the \(|\bar{N}, p_n; \theta_n|_k\) summability method. Some new and known results are also obtained.Yet another note on the arithmetic-geometric mean inequality.https://www.zbmath.org/1453.600712021-02-27T13:50:00+00:00"Kabluchko, Zakhar"https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Prochno, Joscha"https://www.zbmath.org/authors/?q=ai:prochno.joscha"Vysotsky, Vladislav"https://www.zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: It was shown by \textit{E. Gluskin} and \textit{V. Milman} [Lect. Notes Math. 1807, 131--135 (2003; Zbl 1035.26020)] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional \(\ell_p\)-sphere, and also show that sampling the point according to either the cone probability measure on \(\ell_p\) or the uniform distribution on the ball enclosed by \(\ell_p\) yields the same results. First, we prove a central limit theorem, which allows us to identify the precise constants in the reverse inequality. Second, we prove the large deviations counterpart to the central limit theorem, thereby describing the asymptotic behaviour beyond the Gaussian scale, and identify the rate function.On the continuity of the trace operator in GSBV\((\Omega)\) and GSBD\((\Omega)\).https://www.zbmath.org/1453.460362021-02-27T13:50:00+00:00"Tasso, Emanuele"https://www.zbmath.org/authors/?q=ai:tasso.emanueleSummary: In this paper, we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed \((n - 1)\)-dimensional set \(\Gamma \), with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems.https://www.zbmath.org/1453.651692021-02-27T13:50:00+00:00"Arqub, Omar Abu"https://www.zbmath.org/authors/?q=ai:arqub.omar-abu"Al-Smadi, Mohammed"https://www.zbmath.org/authors/?q=ai:al-smadi.mohammed-h"Momani, Shaher"https://www.zbmath.org/authors/?q=ai:momani.shaher-m"Hayat, Tasawar"https://www.zbmath.org/authors/?q=ai:hayat.tasawarThe paper is concerned with the analytic and numerical solution of the two-point fuzzy boundary value problem
\[
y''(x) + f \left(x, y(x), y'(x) \right)= g \left(x, y(x), y'(x) \right) \text{ on } (a,b)
\]
subject to boundary conditions
\[
y(a)=\alpha, \quad y(b)=\beta,
\]
where \(a\) and \(b\) are real numbers, \(\alpha\) and \(\beta\) are fuzzy numbers, and \(f,g,y\) are fuzzy-valued functions. First, the concept of fuzzy analysis is briefly reviewed. Then the problem is reformulated using \(r\)-cut representations of functions under the assumption of strongly generalized differentiability and appropriate Hilbert spaces with reproducing kernels are introduced. An orthonormal system of functions is constructed and it is used for the representation of the analytic solution as an infinite sum of functions and an approximate solution as a finite sum of functions. These solutions can be computed directly in linear cases. For nonlinear equations the authors propose an iterative method and prove its convergence. Numerical experiments and a numerical comparison with the results computed by the homotopy analysis method and the Adomian decomposition method are presented.
Reviewer: Dana Černá (Liberec)The \(C^*\)-algebra of the semi-direct product \(K \ltimes A\).https://www.zbmath.org/1453.220052021-02-27T13:50:00+00:00"Regeiba, Hedi"https://www.zbmath.org/authors/?q=ai:regeiba.hedi"Ludwig, Jean"https://www.zbmath.org/authors/?q=ai:ludwig.jeanEvery \(C^*\)-algebra can be realised as an algebra of operator-valued functions on its spectrum. This realisation becomes most useful when one has an explicit description of the spectrum as a topological space, and an explicit characterisation of those operator-valued functions that correspond to elements of the \(C^*\)-algebra.
In this paper the authors consider the case of \(C^*(K\ltimes A)\), the group \(C^*\)-algebra of the semi-direct product group associated to an action of a compact group \(K\) on a locally compact abelian group \(A\). The spectrum of this \(C^*\)-algebra -- or in other words, the unitary dual \(\widehat{K\ltimes A}\) -- has been computed by \textit{G. W. Mackey} [Ann. Math. (2) 55, 101--139 (1952; Zbl 0046.11601)], and its topology as been described in detail by \textit{L. Baggett} [Trans. Am. Math. Soc. 132, 175--215 (1968; Zbl 0162.18802)]. The main result of this paper is a characterisation of the image of \(C^*(K\ltimes A)\) in the algebra of operator-valued functions on \(\widehat{K\ltimes A}\) in terms of three conditions: a fibrewise-compactness condition; a vanishing-at-infinity condition; and a continuity condition. The continuity condition requires a comparison of operators acting on a priori different Hilbert spaces, and much of the paper is devoted to getting around this difficulty by realising certain families of representations of \(K\ltimes A\) as subspaces of a single Hilbert space.
The main result is illustrated through the discussion of two examples: the Cartan motion group \(SO(n)\ltimes \mathbb{R}^n\), whose \(C^*\)-algebra had previously been computed in [\textit{F. Abdelmoula} et al., Bull. Sci. Math. 135, No. 2, 166--177 (2011; Zbl 1216.46050)]; and the semi-direct product for the action of the direct product \(\{ \pm 1\}^{\infty}\) by multiplication on the direct sum \(\mathbb{Z}^{\oplus\infty}\).
Reviewer: Tyrone Crisp (Orono)A measurable selector in Kadison's Carpenter's theorem.https://www.zbmath.org/1453.420262021-02-27T13:50:00+00:00"Bownik, Marcin"https://www.zbmath.org/authors/?q=ai:bownik.marcin"Szyszkowski, Marcin"https://www.zbmath.org/authors/?q=ai:szyszkowski.marcinSummary: We show the existence of a measurable selector in Carpenter's theorem due to \textit{R. V. Kadison} [Proc. Natl. Acad. Sci. USA 99, No. 7, 4178--4184 (2002; Zbl 1013.46049); ibid. 99, No. 8, 5217--5222 (2002; Zbl 1013.46050)]. This solves a problem posed by \textit{M. Bownik} and \textit{J. Jasper} [Can. Math. Bull. 57, No. 3, 463--476 (2014; Zbl 1314.42032)]. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of \(L^2(\mathbb{R}^d)\) and Carpenter's theorem for type \(\text{I}_{\infty }\) von Neumann algebras.