Recent zbMATH articles in MSC 43https://www.zbmath.org/atom/cc/432022-05-16T20:40:13.078697ZWerkzeugA fresh look at the notion of normalityhttps://www.zbmath.org/1483.111542022-05-16T20:40:13.078697Z"Bergelson, Vitaly"https://www.zbmath.org/authors/?q=ai:bergelson.vitaly"Downarowicz, Tomasz"https://www.zbmath.org/authors/?q=ai:downarowicz.tomasz"Misiurewicz, Michał"https://www.zbmath.org/authors/?q=ai:misiurewicz.michalThe point of departure of this paper is the classical notion of normality of a \(0\)-\(1\) sequence \((x_n)_{n\in\mathbb N}\in \{0,1\}^{\mathbb N}\): for any positive integer \(k\) and any word \(w\in\{0,1\}^k\), the asymptotic density of the set of indices \(n\in\mathbb N\) such that \((x_n,x_{n+1},\ldots,x_{n+k-1})=w\) has to be equal to \(2^{-k}\).
This definition is extended to Følner sequences \((F_n)_{n\in \mathbb N}\), which are sequences of finite subsets \(F_n\subset\mathbb N\) satisfying the Følner condition
\[
(\text{ for all } k\in\mathbb N) \lim_{n\rightarrow\infty}\frac{\lvert F_n\cap(F_n-k)\rvert}{\lvert F_n\rvert}=1.
\]
A sequence \(x\in\{0,1\}^{\mathbb N}\) is \((F_n)\)-normal if for all \(k\in\mathbb N\) and any \(w\in\{0,1\}^k\) we have
\begin{multline*}
\lim_{n\rightarrow\infty} \frac1{\lvert F_n\rvert} \bigl\lvert\bigl\{m\in\mathbb N: \{m,m+1,\ldots,m+k-1\}\subset F_n\mbox{ and } \\
(x_m,x_{m+1},\ldots,x_{m+k-1})=w\bigr\}\bigr\rvert =2^{-k}.
\end{multline*}
Considering only Følner sequences does not pose a restriction, since there do not exist \((F_n)\)-normal sequences with respect to non-Følner sequences \((F_n)\).
As the next step, the authors consider countably infinite amenable cancellative semigroups \(G\); such groups admit (left) Følner sequences and are used in place of the index set \(\mathbb N\) of the sequence \(x\). More precisely, let \((F_n)\) be a Følner sequence in \(G\). Suppose that \(x=(x_g)_{g\in G}:G\rightarrow\{0,1\}\), that \(K\subset G\) is finite, and \(B\in\{0,1\}^K\). Set
\[
\mathsf N(B,x,F_n) =\bigl\lvert\bigl\{g\in G\cup\{e\}:(\text{ for all } h\in K)hg\in F_n\mbox{ and }x_{hg}=B(h)\bigr\}\bigr\rvert.
\]
A \((F_n)\)-normal sequence is defined by the property that for any nonempty finite \(K\subset G\) and every \(B\in\{0,1\}^K\),
\[
\lim_{n\rightarrow\infty}\frac 1{\lvert F_n\rvert} \mathsf N(B,x,F_n)=2^{-\lvert K\rvert}.
\]
Under the condition that \(\lvert F_n\rvert\) is strictly increasing (in particular), it is shown in Theorem~4.2 that \(\lambda\)-almost every \(x\in\{0,1\}^G\) is \((F_n)\)-normal, where \(\lambda\) is the uniform product measure on \(\{0,1\}^G\). In a topological sense however, the set of \((F_n)\)-normal sequences \(x\in\{0,1\}^G\) is small, that is, of first category (Proposition~4.7).
In Section~5, the authors generalize the Champernowne-construction of a normal number to this general setting.
In the remaining sections, special emphasis is given to the semigroups \((\mathbb N,+)\) and \((\mathbb N,\times)\). In Section~6, a more transparent construction of a Champernowne-like normal sequence is given for the case \((\mathbb N,\times)\). Section~7 is concerned with normal subsets of \(\mathbb N\), and combinatorial and Diophantine properties of such sets are proved. Finally, in Section~8, the existence of \((F_n)\)-normal Liouville numbers is established, and a set of such numbers is constructed.
Reviewer: Lukas Spiegelhofer (Wien)Sharp nonzero lower bounds for the Schur product theoremhttps://www.zbmath.org/1483.150102022-05-16T20:40:13.078697Z"Khare, Apoorva"https://www.zbmath.org/authors/?q=ai:khare.apoorvaTo sketch the background of this interesting paper, let us recall a few basic facts from matrix analysis and linear algebra. Fix \(m, n \in \mathbb{N}\). Let \(M_{m,n}(\mathbb{F}) \equiv \mathbb{F}^{m \times n}\) denote the set of all \(m \times n\) matrices with entries in \(\mathbb{F}\), where \(\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}\). A matrix \(A \in M_n(\mathbb{F}) \equiv M_{n,n}(\mathbb{F})\) is positive semidefinite (resp., positive definite) in \(\mathbb{M}_n(\mathbb{F})\) if and only if the following two conditions are satisfied:
\begin{itemize}
\item[(i)] \(A = A^\ast\) (i.e., \(A\) is Hermitian);
\item[(ii)] \(x^\ast A x \geq 0\) (respectively \(x^\ast A x > 0\)) for all \(x \in {\mathbb{F}^n}\setminus\{0\}\).
\end{itemize}
If \(A \in \mathbb{M}_n(\mathbb{R})\), then \(A^\ast = A^\top\). In the complex case, condition (i) is unnecessary. However, the inclusion of (i) guarantees that \(A \in M_n(\mathbb{R})\) is positive semidefinite in \(M_n(\mathbb{R})\) if and only if \(A \in M_n(\mathbb{R}) \subseteq M_n(\mathbb{C})\) is positive semidefinite in \(M_n(\mathbb{C})\). To recognise this fact, we just have to examine the example of the non-symmetric real \(2 \times 2\)-matrix \(\left(\begin{smallmatrix} 0 & 1\\ -1 & 0 \end{smallmatrix}\right)\).
Let \(\mathbb{P}_n(\mathbb{F}) \subseteq \mathbb{M}_n(\mathbb{F})\) denote the set of all positive semidefinite \(n \times n\) matrices with entries in \(\mathbb{F}\) (in the following simply abbreviated as PSD).
The considerably rich structure of the convex cone \(\mathbb{P}_n(\mathbb{F})\) is not only essential in linear algebra and matrix analysis itself. \(\mathbb{P}_n(\mathbb{F})\) also plays a key role in conic optimisation (in particular, semidefinite programming), quantum information theory, computational complexity and spectral graph theory.
If \(A, B \in \mathbb{P}_n(\mathbb{F})\), it is a well-known fact that in general the standard matrix product \(AB\) is not positive semidefinite. In fact, \(AB \in \mathbb{P}_n(\mathbb{F})\) if and only if \(AB\) is Hermitian, which is equivalent to \(AB = BA\), i.e., \(A\) and \(B\) commute (see [\textit{A. R. Meenakshi} and \textit{C. Rajian}, Linear Algebra Appl. 295, No. 1--3, 3--6 (1999; Zbl 0940.15022)]).
The situation changes completely when the Hadamard product of matrices is considered instead. If \(C = (c_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\) and \(D = (d_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\), the \textit{Hadamard product of \(C\) and \(D\)} is defined as
\[
C \circ D : = (c_{ij}\,d_{ij}) \quad ((i,j) \in [m] \times [n])\,.
\]
The Hadamard product is sometimes called the \textit{entrywise product} for obvious reasons, or the \textit{Schur product}, because of some early and basic results about the product obtained by Schur (see [\textit{R. A. Horn} and \textit{C. R. Johnson}, Topics in matrix analysis. Cambridge etc.: Cambridge University Press (1991; Zbl 0729.15001)]).
(NB: In my opinion, the symbolic notation \(\circ\) perhaps could lead to a minor ambiguity, since quite regularly, \(\circ\) denotes composition of mappings. In [\textit{V. Paulsen}, Completely bounded maps and operator algebras. Cambridge: Cambridge University Press (2002; Zbl 1029.47003)], the symbol \(\ast\) is used instead. Also \(\odot\) could be a useful substitution for \(\circ\).)
Like the usual matrix product, the distributive law also holds for the Hadamard product: \(A\circ(B + C) = A \circ B + A \circ C\). Unlike the usual matrix product, the Hadamard product is commutative: \(A \circ B = B \circ A\).
In the real subspace of Hermitian matrices, the most common order relation is the \textit{Loewner partial order}. It is induced by the cone \(\mathbb{P}_n(\mathbb{F})\). By definition, \(A \geq B\) if and only if both \(A, B\) are Hermitian and \(A-B \in \mathbb{P}_n(\mathbb{F})\). A seminal result by \textit{I. Schur} [J. Reine Angew. Math. 140, 1--28 (1911; JFM 42.0367.01)], nowadays known as ``Schur product theorem'', asserts that if \(A \geq 0\) and \(B \geq 0\), then also \(A \circ B \geq 0\) (see [Horn and Johnson, loc. cit.], Chapter 5.2). Moreover, if we implement the (unique) positive semidefinite root of a PSD matrix (\(A = A^{1/2}\,A^{1/2}\) for all \(A \in \mathbb{P}_n(\mathbb{F})\)) into the trace, we reobtain the well-known fact that \(\mathbb{P}_n(\mathbb{F})\) is a self-dual cone (for both fields), meaning that
\[
A \geq 0 \text{ if and only if } \langle A, B\rangle_F : = \operatorname{tr}(A B^\ast) = \operatorname{tr}(A B) \geq 0 \ \text{ for all } B \in \mathbb{P}_n(\mathbb{F}).
\]
Thereby,
\[
\mathbb{M}_{m,n}(\mathbb{F}) \times \mathbb{M}_{m,n}(\mathbb{F}) \ni (C, D) \mapsto \langle C, D\rangle_F : = \operatorname{tr}(C D^\ast) = \operatorname{tr}(D^\ast\,C)
\]
denotes the Frobenius inner product. Because of Hölder's inequality, the matrix \(C \in \mathbb{M}_{m,n}(\mathbb{F})\) can be identified as bounded linear operator from \(l_p^n\) to \(l_q^m\) (of finite rank) which satisfies \(\Vert A \Vert \leq \big(\sum_{i = 1}^m \sum_{j = 1}^n \vert a_{ij}\vert^q\big)^{1/q}\), for \(1 \leq p, q \leq \infty\) with \(\frac{1}{p} + \frac{1}{q} = 1\). Thus, if we view \(C\) as Hilbert-Schmidt operator from the Hilbert space \(l_2^n\) to the Hilbert space \(l_2^m\), the Frobenius inner product coincides with the Hilbert-Schmidt inner product of \(C \in \mathcal{S}_2(l_2^n, l_2^m)\).
The paper consists of three parts. In the first part, the main result (Theorem A) enhances a result of \textit{J. Vybíral}, developed in [Adv. Math. 368, Article ID 107140, 8 p. (2020; Zbl 1441.15024), Theorem 1]. The author improves Vybíral's \textit{positive} lower bound (with respect to the Loewner partial order, induced by PSD matrices). That lower bound appears in a stronger version of the Schur product theorem, also introduced by Vybíral. Moreover, Theorem A unveils that the emerging positive constant as part of the lower bound is the maximum possible one. It cannot be further increased.
Besides an application of the Cauchy-Schwarz inequality with respect to the Frobenius inner product, the trace equality (1.12) plays a significant role in the proof of Theorem A.
We would like to emphasize that (1.12) itself can be deduced from the following simple fact, which however reveals a further direct link between the Hadamard product and the standard matrix product:
\[
M \circ uv^\top = D_u\,M\,D_v \text{ for all } M \in \mathbb{M}_{m,n}(\mathbb{F}), (u,v) \in \mathbb{F}^m \times \mathbb{F}^n\,,
\]
where \(D_x\) denotes the \(p \times p\)-diagonal matrix, whose \((i,j)\)'th entry is given by \(\delta_{ij}\,x_j\), for any \(x = (x_1, \ldots, x_p)^\top \in \mathbb{F}^p\). Here \(\delta_{ij}\) is the Kronecker delta.
The first part concludes with a few (fairly technical) refinements of Theorem A including the determination of an upper bound and an in-depth discussion of special cases of Theorem A which were published in the past, primarily by Vybíral.
In the second part, the author develops a ``suitably modified'' version of Theorem A which produces a non-trivial positive lower bound in the case of Hilbert-Schmidt operators between arbitrary, not necessarily finite-dimensional Hilbert spaces (see Theorem 2.4).
In the third and last part a few applications of Theorem A, including the important entrywise calculus on classes of positive matrices are touched briefly. The latter also plays a crucial role regarding an approximation of the upper bound of the real and complex Grothendieck constant in the famous Grothendieck inequality; a fact, based on my own research activities, not covered here. Complex kernels with lower bounds are introduced, and related non-trivial PSD matrices are listed. Even indications for future research problems are sketched. In this respect, I would like to add a further research question: could Theorem A and its applications to the entrywise calculus on classes of positive matrices even become useful to improve the already existing \textit{lower} bounds of the complex and real Grothendieck constant?
Finally, as a minor and weak ``criticism'', let me point out that it would be very helpful for the reader to see explicitly on which field \(\mathbb{F} \in \{\mathbb{R},\mathbb{C}\}\) a statement about \(\mathbb{P}_n(\mathbb{F})\) is referred to, mainly to work out what results hold for both fields simultaneously. In my view, the primarily considered field is \(\mathbb{C}\).
Reviewer: Frank Oertel (London)Computability of Følner setshttps://www.zbmath.org/1483.200602022-05-16T20:40:13.078697Z"Cavaleri, Matteo"https://www.zbmath.org/authors/?q=ai:cavaleri.matteoThe \(L^2\)-torsion polytope of amenable groupshttps://www.zbmath.org/1483.200782022-05-16T20:40:13.078697Z"Funke, Florian"https://www.zbmath.org/authors/?q=ai:funke.florianSummary: We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the \(L^2\)-torsion polytope among \(G\)-CW-complexes for these groups. As another application we prove that the \(L^2\)-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.Tempered homogeneous spaces. IIIhttps://www.zbmath.org/1483.220092022-05-16T20:40:13.078697Z"Benoist, Yves"https://www.zbmath.org/authors/?q=ai:benoist.yves"Kobayashi, Toshiyuki"https://www.zbmath.org/authors/?q=ai:kobayashi.toshiyukiLet \(G\) be a real semisimple algebraic group. Let \(H\) be a real reductive algebraic subgroup of \(G\). The paper under review characterizes the pairs \((G, H)\) such that the natural unitary representation of \(G\) on the Hilbert space \(L^2(G/H)\) is tempered. Using the earlier temperedness criterion given in Theorem 4.1 of the authors in [J. Eur. Math. Soc. (JEMS) 17, No. 12, 3015--3036 (2015; Zbl 1332.22015)], the current paper gives a necessary condition as well as a sufficient condition for the representation \(L^2(G/H)\) to be tempered. See Theorem 1.1. Moreover, the paper also completely lists the pairs \((G, H)\) such that \(L^2(G/H)\) is \textit{not} tempered.
For Part IV, see \url{arxiv:2009.10391}
Reviewer: Chao-Ping Dong (Changsha)Negative definite functions on the infinite dimensional special linear grouphttps://www.zbmath.org/1483.220112022-05-16T20:40:13.078697Z"Rabaoui, Marouane"https://www.zbmath.org/authors/?q=ai:rabaoui.marouaneIn this paper the author investigate the connection between the boundedness of negative definite functions and Kazhdan's property (T) in the framework of Olshanski spherical pairs.
Let \(\mathbb{F}=\mathbb{R}, \mathbb{C}\) or \(\mathbb{H}\) be the quaternion field, and let \(SL_n(\mathbb{F})\) be the special linear group with \(n\geq 3\). Let \(K_n\) be a compact subgroup of \(SL_n(\mathbb{F})\) such that \((SL_n(\mathbb{F}), K_n)\) is a Gelfand pair. Consider the inductive limit as \(n \rightarrow \infty\), the Olshanski spherical pair \((SL_\infty(\mathbb{F}), K_\infty)\). The author proves that the group \(SL_\infty(\mathbb{F})\) has the Kazhdan property (T), from which he deduces that every continuous negative definite function on \(SL_\infty(\mathbb{F})\) is bounded.
An integral representation of \(K_\infty\)-bi-invariant continuous negative definite function on \(SL_\infty(\mathbb{F})\) is also given.
Reviewer: Abdelhamid Boussejra (Kénitra)Graphop mean-field limits for Kuramoto-type modelshttps://www.zbmath.org/1483.352702022-05-16T20:40:13.078697Z"Gkogkas, Marios Antonios"https://www.zbmath.org/authors/?q=ai:gkogkas.marios-antonios"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christianAverages and maximal averages over Product \(j\)-varieties in finite fieldshttps://www.zbmath.org/1483.420072022-05-16T20:40:13.078697Z"Koh, Doowon"https://www.zbmath.org/authors/?q=ai:koh.doowon"Lee, Sujin"https://www.zbmath.org/authors/?q=ai:lee.sujinSummary: We study both averaging and maximal averaging problems for Product \(j\)-varieties defined by \(\prod_j = \{ x\in \mathbb{F}_q^d : \prod_{k=1}^d x_k = j \}\) for \(j \in \mathbb{F}_q^{\ast}\), where \(\mathbb{F}_q^d\) denotes a \(d\)-dimensional vector space over the finite field \(\mathbb{F}_q\) with \(q\) elements. We prove the sharp \(L^p \to L^r\) boundedness of averaging operators associated to Product \(j\)-varieties. We also obtain the optimal \(L^p\) estimate for a maximal averaging operator related to a family of Product \(j\)-varieties \(\{\prod_j\}_{j \in \mathbb{F}_q^{\ast}}\).Boundedness of commutators of \(\theta\)-type Calderón-Zygmund operators on generalized weighted Morrey spaces over RD-spaceshttps://www.zbmath.org/1483.420092022-05-16T20:40:13.078697Z"Li, Qiumeng"https://www.zbmath.org/authors/?q=ai:li.qiumeng"Lin, Haibo"https://www.zbmath.org/authors/?q=ai:lin.haibo.1|lin.haibo"Wang, Xinyu"https://www.zbmath.org/authors/?q=ai:wang.xinyuIn this article the authors studied the boundedness of the commutators generated by the \(\theta\)-type Calderón-Zygmund operators and \(BMO\) functions on generalized weighted Morrey spaces over \(RD\)-spaces. By assuming slightly weaker conditions, the authors obtained the bounds for the above operators on the generalized weighted Morrey spaces \(\widetilde{\mathcal{M}}^{p,\psi}(w)\) and the generalized weighted Morrey spaces of \(L\ln L\) type \(\widetilde{\mathcal{M}}_{L\ln L}^{1,\psi}(w)\) over the \(RD\)-spaces.
Personally speaking, this paper is very interesting and very well written. This paper involves a large amount of definitions, notation and references, which increases its richness. Overall, this article is a nice piece of work.
Reviewer: Feng Liu (Qingdao)Frames associated with shift invariant spaces on positive half linehttps://www.zbmath.org/1483.420172022-05-16T20:40:13.078697Z"Ahmad, Owais"https://www.zbmath.org/authors/?q=ai:ahmad.owais"Ahmad, Mobin"https://www.zbmath.org/authors/?q=ai:ahmad.mobin"Ahmad, Neyaz"https://www.zbmath.org/authors/?q=ai:ahmad.neyazThe authors show some necessary and sufficient conditions under which shift-invariant systems become frames on positive half line. Applications to Gabor frames and wavelet frames on positive half line are also presented.
Reviewer: Paşc Găvruţă (Timişoara)Inversion problem in measure and Fourier-Stieltjes algebrashttps://www.zbmath.org/1483.430012022-05-16T20:40:13.078697Z"Ohrysko, Przemysław"https://www.zbmath.org/authors/?q=ai:ohrysko.przemyslaw"Wasilewski, Mateusz"https://www.zbmath.org/authors/?q=ai:wasilewski.mateuszThe following two important problems were investigated by N. Nikolski: Let \(G\) be a locally compact abelian group, and let \(\mu\) be a complex-valued Borel regular measure on \(G\) such that \(\|\mu\|\leq 1\) and \(\inf_{\gamma\in \widehat{G}} |\widehat{\mu}(\gamma)|=\delta >0\). What is the minimal value \(\delta_0>0\) such that for any \(\delta>\delta_0\) the measure \(\mu\) is automatically invertible? What can be said about the norm of the inverse?
In the article the authors completely solve the first problem by showing that \(\delta_0=\tfrac{1}{2}\). In some important partial cases the authors obtain answers also to the second problem, which are considerable improvements of the existent estimations. Also they consider analogous problems for Fourier-Stieltjes algebras.
Reviewer: Saak S. Gabriyelyan (Beer-Sheva)On norm almost periodic measureshttps://www.zbmath.org/1483.430022022-05-16T20:40:13.078697Z"Spindeler, Timo"https://www.zbmath.org/authors/?q=ai:spindeler.timo"Strungaru, Nicolae"https://www.zbmath.org/authors/?q=ai:strungaru.nicolaeSummary: In this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of \(\mu\) is equivalent to the equi-Bohr almost periodicity of \(\mu *g\) for all \(g\) in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon-Nikodym density.Amenability of semigroups and the Ore condition for semigroup ringshttps://www.zbmath.org/1483.430032022-05-16T20:40:13.078697Z"Guba, V. S."https://www.zbmath.org/authors/?q=ai:guba.victor-sSummary: It is known that if a cancellative monoid \(M\) is left amenable then the monoid ring \(K[M]\) satisfies the Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring.
\textit{J. Donnelly} [Semigroup Forum 81, No.~2, 389--392 (2010; Zbl 1201.20065)]
shows that a partial converse to this statement is true. Namely, if the monoid \(\mathbb{Z}^+[M]\) of all elements of \(\mathbb{Z}[M]\) with positive coefficients has nonzero common right multiples, then \(M\) is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of groups. If \(M\) is a free metabelian group, then \(M\) is amenable but the Ore condition fails for \(\mathbb{Z}^+[M]\). Besides, we study the case of the monoid \(M\) of positive elements of R.~Thompson's group~\(F\). The amenability problem for \(F\) is a famous open question. It is equivalent to left amenability of the monoid \(M\). We show that for this case the monoid \(\mathbb{Z}^+[M]\) does not satisfy the Ore condition. That is, even if \(F\) is amenable, this cannot be shown using the above sufficient condition.When is an invariant mean the limit of a Følner net?https://www.zbmath.org/1483.430042022-05-16T20:40:13.078697Z"Hopfensperger, John"https://www.zbmath.org/authors/?q=ai:hopfensperger.johnThe author deals with the topological invariant mean on \(G\), say \(TLIM(G)\), and the limit points of Følner-nets, say \(TLIM_{0}(G)\), where \(G\) is an amenable locally compact group. The author also discusses whether \(TLIM(G)=TLIM_{0}(G)\). Using these facts some previous results like [\textit{C. Chou}, Trans. Am. Math. Soc. 151, 443--456 (1970; Zbl 0202.14001)] and [\textit{N. Hindman} and \textit{D. Strauss}, Topology Appl. 156, No. 16, 2614--2628 (2009; Zbl 1189.22003)] are improved.
Reviewer: Amir Sahami (Tehran)A bijection of invariant means on an amenable group with those on a lattice subgrouphttps://www.zbmath.org/1483.430052022-05-16T20:40:13.078697Z"Hopfensperger, John"https://www.zbmath.org/authors/?q=ai:hopfensperger.johnLet \(G=\mathbb{R}^{d}\) and \(\Gamma=\mathbb{Z}^{d}\). Grosvenor showed that there exists a natural affine injection \(i\) from \(LIM(\Gamma)\) into \(TLIM(G)\) such that is also surjection, where \(LIM\) denotes for the set of left invariant means and \(TLIM\) denotes the set of all topologically left invariant means. In the present paper the author improves this result for amenable groups $G$ and shows that \(i\) is a surjection if and only if \(\frac{G}{\Gamma}\) is compact.
Reviewer: Amir Sahami (Tehran)Riesz means in Hardy spaces on Dirichlet groupshttps://www.zbmath.org/1483.430062022-05-16T20:40:13.078697Z"Defant, Andreas"https://www.zbmath.org/authors/?q=ai:defant.andreas"Schoolmann, Ingo"https://www.zbmath.org/authors/?q=ai:schoolmann.ingoIn a recent series of papers the authors have developed a theory of Hardy spaces of general Dirichlet series, closely connected with harmonic analysis on groups. Given a frequency \(\lambda = (\lambda_{n})_{n}\) (i.e., strictly increasing and unbounded), they introduced in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No. 6, 3220--3258 (2019; Zbl 1429.43004)] the notion of \(\lambda\)-Dirichlet group (which defines a family of characters \((h_{\lambda_{n}})_{n}\)). For such a group \(G\), they also defined the Hardy space \(H_{p}^{\lambda}(G)\) for \(1 \leq p \leq \infty\). Here they deal with the convergence of the Riesz means for functions in these spaces.
Given \(f \in H_{1}^{\lambda}(G)\), the first \((\lambda,k)\)-Riesz sum of length \(x >0\) is defined as
\[
R^{\lambda,k}_{x}(f) = \sum_{\lambda_{n}<x} \hat{f}(h_{\lambda_{n}}) \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} \,.
\]
The main result of the paper shows that, for every \(k>0\), the expression
\[
R^{\lambda,k}_{\max}(f) (\omega) = \sup_{x >0} \big\vert R^{\lambda,k}_{x}(f) (\omega) \big\vert \,,
\]
for \(f \in H_{1}^{\lambda}(G)\) and \(\omega \in G\), defines a bounded sublinear operator
\[
R^{\lambda,k}_{\max} : H_{1}^{\lambda}(G) \to L_{1,\infty}(G)
\]
and
\[
R^{\lambda,k}_{\max} : H_{p}^{\lambda}(G) \to L_{p}(G) \text{ for } 1 < p \leq \infty .
\]
As a consequence, \(R^{\lambda,k}_{x}(f)(\omega)\) converges (in \(x\)) to \(f(\omega)\) for almost every \(\omega\).
When horizontal translations are considered, the situation improves. It is shown that for \(u,k > 0\), there exists a constant \(C=C(u,k)\) so that for every frequency \(\lambda\), all \(1 \leq p \leq \infty\) and \(f \in H_{p}(G)^{\lambda}\) we have
\[
\bigg( \int_{G} \sup_{x >0} \Big\vert \sum_{\lambda_{n} < x} \hat{f} (h_{\lambda_{n}}) e^{-u\lambda_{n}} \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} (\omega) \Big\vert^{p} d \omega \bigg)^{1/p} \leq C \Vert f \Vert_{p} \,.
\]
Note that in this case the inequality holds even for \(p=1\), and that the constant does not depend on \(p\).
One of the main tools to prove the main result is a maximal Hardy-Littlewood operator, adapted to this setting. If \((G, \beta)\) is a Dirichlet group and \(f \in L_{1}(G)\), then for almost every \(\omega \in G\) the function defined by \(f_{\omega}(t) = f(\omega \beta(t))\) is locally integrable on \(\mathbb{R}\). It is proved that the adapted Hardy-Littlewood maximal operator, given by
\[
\overline{M}(f) (\omega) = \sup_{\genfrac{}{}{0pt}{2}{I \subset \mathbb{R}}{\text{interval}}} \frac{1}{\vert I \vert} \int_{I} \vert f_{\omega} (t) \vert dt
\]
defines a sublinear bounded operator \(\overline{M}: L_{1}(G) \to L_{1,\infty}(G)\) and \(\overline{M}: L_{p}(G) \to L_{p}(G)\) for \(1 < p \leq \infty\).
It is known that, for \(1 < p < \infty\) and any frequency \(\lambda\), the sequence \((h_{\lambda_{n}})\) is a Schauder basis of \(H_{p}^{\lambda}(G)\) and, therefore the Riesz means of any function \(f\) converge (in norm) to \(f\). Here it is proved that this is also the case for \(p=1\), that is
\[
\lim_{x \to \infty} \big\Vert R^{\lambda, k}_{x}(f) - f \Vert_{1} =0
\]
for every \(k>0\) and every \(f \in H_{1}^{\lambda}(G)\).
Applications of all these are given to general Dirichlet series and to almost periodic functions.
Reviewer: Pablo Sevilla Peris (Valencia)Lipschitz conditions in Damek-Ricci spaceshttps://www.zbmath.org/1483.430072022-05-16T20:40:13.078697Z"El Ouadih, Salah"https://www.zbmath.org/authors/?q=ai:el-ouadih.salah"Daher, Radouan"https://www.zbmath.org/authors/?q=ai:daher.radouanThe paper gives a complete characterization of the (Dini-) Lipschitz conditions for $L^2$ space on Damek-Ricci spaces by estimating the norm of the Fourier-Helgason transform. A necessary characterization of the (Dini-) Lipschitz conditions for the $L^p$ space when $1< p< 2$ is nicely presented. Furthermore, the norm of the Fourier-Helgason transform is also estimated for functions in the (Dini-) Lipschitz class. The results greatly improve the ones obtained by \textit{E. C. Titchmarsh} [Introduction to the theory of Fourier integrals. Oxford: Clarendon Press (1937; Zbl 0017.40404)].
Reviewer: Jinsong Wu (Harbin)Wavelet sets and scaling sets in local fieldshttps://www.zbmath.org/1483.430082022-05-16T20:40:13.078697Z"Behera, Biswaranjan"https://www.zbmath.org/authors/?q=ai:behera.biswaranjanThe author introduces the concepts of MSF (minimally supported frequency) multiwavelets, multiwavelet sets, generalized scaling sets, and scaling sets in a local field of positive characteristic and provides their characterizations. Some examples of unbounded wavelet sets are constructed. It is shown that the corresponding wavelets are not associated with multiresolution analyses. In particular, the author shows the existence of non-MRA wavelets on the Cantor dyadic group (for examples of MRA-wavelets on this group, see [\textit{Yu. A. Farkov} et al., Construction of wavelets through Walsh functions. Singapore: Springer (2019; Zbl 1418.42002)]).
Reviewer: Yuri A. Farkov (Moskva)Singular integrals on regular curves in the Heisenberg grouphttps://www.zbmath.org/1483.430092022-05-16T20:40:13.078697Z"Fässler, Katrin"https://www.zbmath.org/authors/?q=ai:fassler.katrin-s"Orponen, Tuomas"https://www.zbmath.org/authors/?q=ai:orponen.tuomasSummary: Let \(\mathbb{H}\) be the first Heisenberg group, and let \(k\in C^\infty(\mathbb{H}\smallsetminus \{0\})\) be a kernel which is either odd or horizontally odd, and satisfies
\[
|\nabla_{\mathbb{H}}^nk(p)|\leq C_n\|p\|^{-1-n},\quad p\in\mathbb{H} \smallsetminus \{0 \},\, n\geq 0.
\]
The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel \(k(p)=\nabla_{\mathbb{H}}\log \|p\|\). We prove that convolution with \(k\), as above, yields an \(L^2\)-bounded operator on regular curves in \(\mathbb{H}\). This extends a theorem of G. David to the Heisenberg group.
As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield \(L^2\) bounded operators on \textit{Lipschitz flags} in \(\mathbb{H}\). This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by \textit{V. Chousionis} and \textit{S. Li} [Anal. PDE 10, No. 6, 1407--1428 (2017; Zbl 1369.28004)].Computational bounds for doing harmonic analysis on permutation modules of finite groupshttps://www.zbmath.org/1483.430102022-05-16T20:40:13.078697Z"Hansen, Michael"https://www.zbmath.org/authors/?q=ai:hansen.michael-alan|hansen.michael-edberg|hansen.michael-reichhardt|hansen.michael-pilegaard"Koyama, Masanori"https://www.zbmath.org/authors/?q=ai:koyama.masanori"McDermott, Matthew B. A."https://www.zbmath.org/authors/?q=ai:mcdermott.matthew-b-a"Orrison, Michael E."https://www.zbmath.org/authors/?q=ai:orrison.michael-e"Wolff, Sarah"https://www.zbmath.org/authors/?q=ai:wolff.sarahSummary: We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.Almost contractive maps between \(C^{\ast}\)-algebras with applications to Fourier algebrashttps://www.zbmath.org/1483.471232022-05-16T20:40:13.078697Z"Ricard, É."https://www.zbmath.org/authors/?q=ai:ricard.eric"Roydor, J."https://www.zbmath.org/authors/?q=ai:roydor.jeanSummary: We prove that unital almost contractive maps between \(C^\ast\)-algebras enjoy approximately certain properties of unital positive maps (such as selfadjointness or the Kadison-Schwarz inequality). Our main application is a description of almost contractive homomorphisms between Fourier algebras and Fourier-Stieltjes algebras: they are actually contractive if their norm is smaller than 1.00018. For surjective isomorphisms of Fourier algebras, the bound 1.0005 is sufficient in order to obtain an isometry.