Potential space estimates in local Hardy spaces for Green potentials in convex domains

2004 ◽  
Vol 20 (4) ◽  
pp. 342-349
Author(s):  
Wang Henggeng ◽  
Jia Houyu
Keyword(s):  
Hardy Spaces ◽  
Potential Space ◽  
Convex Domains ◽  
Local Hardy Spaces ◽  
Green Potentials

scholarly journals The poisson problem on Lipschitz domains

2005 ◽  
Author(s):  
◽  
Svitlana Mayboroda
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Open Problem ◽  
Neumann Boundary ◽  
Lipschitz Domains ◽  
Convex Domains ◽  
Poisson Problem ◽  
Strengthened Form ◽  
Green Potentials ◽  
Well Posed

The aim of this work is to describe the sharp ranges of indices, for which the Poisson problem for Laplacian with Dirichlet or Neumann boundary conditions is well-posed on the scales of Besov and Triebel-Lizorkin spaces on arbitrary Lipschitz domains. The main theorems we prove extend the work of D. Jerison and C. Kenig [JFA, 95], whose methods and results are largely restricted to the case p_ 1, and answer the open problem #3.2.21 on p. 121 in C. Kenig's book in the most complete fashion. When specialized to Hardy spaces, our results provide a solution of a (strengthened form of a) conjecture made by D.-C. Chang, S.Krantz and E. Stein regarding the regularity of the Green potentials on Hardy spaces in Lipschitz domains. The corollaries of our main results include new proofs and various extensions of: Hardy space estimates for Green potentials in convex domains due to V. Adolfsson, B.Dahlberg, S. Fromm, D. Jerison, G.Verchota and T.Wolff and the Lp - Lq estimates for the gradients of Green potentials in Lipschitz domains, due to B. Dahlberg.

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 Weighted local Hardy spaces and their applications

2012 ◽  
Vol 56 (2) ◽  
pp. 453-495 ◽  
Author(s):  
Lin Tang
Keyword(s):  
Hardy Spaces ◽  
Local Hardy Spaces

scholarly journals Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates

2019 ◽  
Vol 30 (3) ◽  
pp. 3275-3330 ◽  
Author(s):  
Víctor Almeida ◽  
Jorge J. Betancor ◽  
Estefanía Dalmasso ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Hardy Spaces ◽  
Variable Exponents ◽  
Local Hardy Spaces ◽  
Adjoint Operators ◽  
Gaussian Estimates

Notes on Embedding Theorems for Local Hardy spaces

1994 ◽  
Vol 165 (1) ◽  
pp. 231-244
Author(s):  
Takahiro Mizuhara
Keyword(s):  
Hardy Spaces ◽  
Embedding Theorems ◽  
Local Hardy Spaces
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