scholarly journals Fourier Multipliers for Hardy Spaces of Dirichlet Series

2013 ◽  
Vol 2014 (16) ◽  
pp. 4368-4378 ◽  
Author(s):  
Alexandru Aleman ◽  
Jan-Fredrik Olsen ◽  
Eero Saksman
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Fourier Multipliers

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Hp-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

2007 ◽  
Vol 59 (6) ◽  
pp. 1207-1222 ◽  
Author(s):  
Shangquan Bu ◽  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Hardy Spaces ◽  
Maximal Regularity ◽  
Type Theorem ◽  
Closed Operator ◽  
Fourier Multipliers ◽  
The Cauchy Problem

AbstractWe consider maximal regularity in the Hp sense for the Cauchy problem u′(t) + Au(t) = f(t) (t ∈ ℝ), where A is a closed operator on a Banach space X and f is an X-valued function defined on ℝ. We prove that if X is an AUMD Banach space, then A satisfies Hp-maximal regularity if and only if A is Rademacher sectorial of type < . Moreover we find an operator A with Hp-maximal regularity that does not have the classical Lp-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces Hp(ℝ X), in the case when X is an AUMD Banach space.

Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence

2019 ◽  
Author(s):  
Maxime Bailleul ◽  
Pascal Lefèvre ◽  
Luis Rodríguez-Piazza
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Upper Bound ◽  
Interpolation Problem ◽  
Orlicz Spaces ◽  
Elementary Method ◽  
Abscissa Of Convergence ◽  
Upper Bound Estimate

Abstract The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p\geq 1$) was initiated in [7] when $p=2$ and $p=\infty $, and in [2] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^\psi $. We focus on the case $\psi =\psi _q(t)=\exp (t^q)-1,$ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\,,1/2\}$ filling the gap between the case $\mathscr{H}^\infty $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$. The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6].

Fourier Multipliers in Hardy Spaces in Tubes over Open Cones

2014 ◽  
Vol 14 (4) ◽  
pp. 681-719 ◽  
Author(s):  
Alexander V. Tovstolis
Keyword(s):  
Hardy Spaces ◽  
Fourier Multipliers

scholarly journals Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

2016 ◽  
Vol 119 (2) ◽  
pp. 237
Author(s):  
Ole Fredrik Brevig
Keyword(s):  
Hilbert Space ◽  
Real Number ◽  
Dirichlet Series ◽  
Hilbert Spaces ◽  
Hardy Spaces ◽  
Weighted Spaces ◽  
Number Of Divisors ◽  
Zeros Of Functions

For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_\alpha$ to the case $\alpha>0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p <2$.

Interpolation of Hardy spaces of Dirichlet series

2019 ◽  
Vol 277 (3) ◽  
pp. 786-805
Author(s):  
Frédéric Bayart ◽  
Mieczysław Mastyło
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces
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