Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness

2018 ◽  
Vol 34 (4) ◽  
pp. 1541-1561 ◽  
Author(s):  
Jiao Chen ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Type Theorem ◽  
Fourier Multipliers

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.

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].

scholarly journals Universality on higher order Hardy spaces

2005 ◽  
Vol 71 (1) ◽  
pp. 17-28
Author(s):  
L. Bernal-González ◽  
A. Bonilla ◽  
M. C. Calderón-Moreno
Keyword(s):  
Unit Disk ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Type Theorem ◽  
Higher Order ◽  
Interpolating Sequences ◽  
Bounded Functions

We prove a Seidel-Walsh-type theorem about the universality of a sequence of derivation-composition operators generated by automorphisms of the unit disk in the setting of the higher order Hardy spaces. Moreover, some related positive or negative assertions involving interpolating sequences and sequences between two tangent circles are established for the class of bounded functions in the unit disk. Our statements improve earlier ones due to Herzog and to the first and third authors.

A T(b) type theorem on the weighted Herz-type Hardy spaces and applications

2015 ◽  
Vol 289 (8-9) ◽  
pp. 1112-1130
Author(s):  
Jianmiao Ruan ◽  
Lijing Sun ◽  
Huoxiong Wu ◽  
Xiao Yu
Keyword(s):  
Hardy Spaces ◽  
Type Theorem

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

On Multilinear Fourier Multipliers of Limited Smoothness

2013 ◽  
Vol 65 (2) ◽  
pp. 299-330 ◽  
Author(s):  
Loukas Grafakos ◽  
Akihiko Miyachi ◽  
Naohito Tomita
Keyword(s):  
Linear Theory ◽  
Hardy Spaces ◽  
Fourier Multipliers

AbstractIn this paper we prove a certain L2-estimate formultilinear Fouriermultiplier operators with multipliers of limited smoothness. As a consequence, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

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