Recent zbMATH articles in MSC 30B50
https://www.zbmath.org/atom/cc/30B50
2022-05-16T20:40:13.078697Z
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Riesz projection and bounded mean oscillation for Dirichlet series
https://www.zbmath.org/1483.30009
2022-05-16T20:40:13.078697Z
"Konyagin, Sergei"
https://www.zbmath.org/authors/?q=ai:konyagin.sergey-v
"QueffĂ©lec, HervĂ©"
https://www.zbmath.org/authors/?q=ai:queffelec.herve
"Saksman, Eero"
https://www.zbmath.org/authors/?q=ai:saksman.eero
"Seip, Kristian"
https://www.zbmath.org/authors/?q=ai:seip.kristian
Summary: We prove that the norm of the Riesz projection from \(L^\infty (\mathbb{T}^n)\) to \(L^p(\mathbb{T}^n)\) is \(1\) for all \(n\ge 1\) only if \(p\le 2\), thus solving a problem posed by \textit{J. Marzo} and the fourth author [Bull. Sci. Math. 135, No. 3, 324--331 (2011; Zbl 1221.42013)]. This shows that \(H^p(\mathbb{T}^{\infty})\) does not contain the dual space of \(H^1(\mathbb{T}^{\infty})\) for any \(p > 2\). We then note that the dual of \(H^1(\mathbb{T}^{\infty})\) contains, via the Bohr lift, the space of Dirichlet series in BMOA of the right half-plane. We give several conditions showing how this BMOA space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on \(\mathbb{T}\), we compute its \(L^p\) norm when \(1 < p < \infty\), and we use this result to show that the \(L^\infty\) norm of the \(N\) th partial sum of a bounded Dirichlet series over \(d\)-smooth numbers is of order \(\log\log N\).
Riesz means in Hardy spaces on Dirichlet groups
https://www.zbmath.org/1483.43006
2022-05-16T20:40:13.078697Z
"Defant, Andreas"
https://www.zbmath.org/authors/?q=ai:defant.andreas
"Schoolmann, Ingo"
https://www.zbmath.org/authors/?q=ai:schoolmann.ingo
In 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)
Norms of composition operators on the \(H^2\) space of Dirichlet series
https://www.zbmath.org/1483.47045
2022-05-16T20:40:13.078697Z
"Brevig, Ole Fredrik"
https://www.zbmath.org/authors/?q=ai:brevig.ole-fredrik
"Perfekt, Karl-Mikael"
https://www.zbmath.org/authors/?q=ai:perfekt.karl-mikael
Summary: We consider composition operators \(\mathscr{C}_\varphi\) on the Hardy space of Dirichlet series \(\mathscr{H}^2\), generated by Dirichlet series symbols \(\varphi \). We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of \(\varphi \). The other concerns general symbols, and is based on a geometrical condition on the boundary values of \(\varphi \). Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of \textit{J. H. Shapiro} [Monatsh. Math. 130, No. 1, 57--70 (2000; Zbl 0951.47026)] on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that \(\varphi(s) = c + r 2^{- s} \). A~number of other examples are given.