Recent zbMATH articles in MSC 30H10https://www.zbmath.org/atom/cc/30H102022-05-16T20:40:13.078697ZWerkzeugPaatero's \(V(k)\) space and a claim by Pinchukhttps://www.zbmath.org/1483.300342022-05-16T20:40:13.078697Z"Andreev, Valentin V."https://www.zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://www.zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://www.zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we obtain a factorization theorem for the functions in Paatero's \(V(k)\) space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in \(V(k)\) and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials.Composition of analytic paraproductshttps://www.zbmath.org/1483.300962022-05-16T20:40:13.078697Z"Aleman, Alexandru"https://www.zbmath.org/authors/?q=ai:aleman.alexandru"Cascante, Carme"https://www.zbmath.org/authors/?q=ai:cascante.carme"Fàbrega, Joan"https://www.zbmath.org/authors/?q=ai:fabrega.joan"Pascuas, Daniel"https://www.zbmath.org/authors/?q=ai:pascuas.daniel"Peláez, José Ángel"https://www.zbmath.org/authors/?q=ai:pelaez.jose-angelSummary: For a fixed analytic function \(g\) on the unit disc \(\mathbb{D}\), we consider the analytic paraproducts induced by \(g\), which are defined by \(T_g f(z)=\int_0^z f(\zeta)g^\prime(\zeta)d\zeta\), \(S_g f(z)=\int_0^z f^\prime(\zeta) g(\zeta)d\zeta\), and \(M_g f(z)=f(z) g(z)\). The boundedness of these operators on various spaces of analytic functions on \(\mathbb{D}\) is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example \(T_g^2\), \(T_gS_g\), \(M_g T_g\), etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol \(g\). In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol \(g\) than the case of a single paraproduct.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)The factorisation property of \(\ell^\infty(X_k)\)https://www.zbmath.org/1483.460142022-05-16T20:40:13.078697Z"Lechner, Richard"https://www.zbmath.org/authors/?q=ai:lechner.richard"Motakis, Pavlos"https://www.zbmath.org/authors/?q=ai:motakis.pavlos"Müller, Paul F. X."https://www.zbmath.org/authors/?q=ai:muller.paul-f-x"Schlumprecht, Thomas"https://www.zbmath.org/authors/?q=ai:schlumprecht.thomasSummary: In this paper we consider the following problem: let \(X_k\) be a Banach space with a normalised basis \((e_{(k, j)})_j\), whose biorthogonals are denoted by \((e_{(k,j)}^*)_j\), for \(k\in\mathbb{N}\), let \(Z=\ell^\infty(X_k:k\in\mathbb{N})\) be their \(\ell^\infty\)-sum, and let \(T:Z\to Z\) be a bounded linear operator with a large diagonal, i.e.,
\[
\inf\limits_{k,j}\left|e^*_{(k,j)}(T(e_{(k,j)})\right|>0.
\]
Under which condition does the identity on \(Z\) factor through \(T\)? The purpose of this paper is to formulate general conditions for which the answer is positive.RKH spaces of Brownian type defined by Cesàro-Hardy operatorshttps://www.zbmath.org/1483.460232022-05-16T20:40:13.078697Z"Galé, José E."https://www.zbmath.org/authors/?q=ai:gale.jose-e"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-j"Sánchez-Lajusticia, Luis"https://www.zbmath.org/authors/?q=ai:sanchez-lajusticia.luisSummary: We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.\(C\)-selfadjointness of the product of a composition operator and a maximal differentiation operatorhttps://www.zbmath.org/1483.470632022-05-16T20:40:13.078697Z"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-haji"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Hai, Pham Viet"https://www.zbmath.org/authors/?q=ai:pham-viet-hai.Summary: Let \(\varphi\) be an automorphism of \(\mathbb{D}\). In this paper, we consider the operator \(C_\varphi D_{\psi_0,\psi_1}\) on the Hardy space \(H^2\) which is the product of composition and the maximal differential operator. We characterize these operators which are \(C\)-selfadjoint with respect to some conjugations \(C\). Moreover, we find all Hermitian operators \(C_\varphi D_{\psi_0,\psi_1}\), when \(\varphi\) is a rotation.