scholarly journals Local Hardy Spaces of Differential Forms on Riemannian Manifolds

2011 ◽  
Vol 23 (1) ◽  
pp. 106-169 ◽  
Author(s):  
Andrea Carbonaro ◽  
Alan McIntosh ◽  
Andrew J. Morris
Keyword(s):  
Riemannian Manifolds ◽  
Hardy Spaces ◽  
Differential Forms ◽  
Local Hardy Spaces

scholarly journals Hardy Spaces of Differential Forms on Riemannian Manifolds

2007 ◽  
Vol 18 (1) ◽  
pp. 192-248 ◽  
Author(s):  
Pascal Auscher ◽  
Alan McIntosh ◽  
Emmanuel Russ
Keyword(s):  
Riemannian Manifolds ◽  
Hardy Spaces ◽  
Differential Forms

scholarly journals Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds

2007 ◽  
Vol 344 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Pascal Auscher ◽  
Alan McIntosh ◽  
Emmanuel Russ
Keyword(s):  
Riemannian Manifolds ◽  
Hardy Spaces ◽  
Differential Forms ◽  
Riesz Transforms

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 Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite

2017 ◽  
Vol 9 (4) ◽  
pp. 1
Author(s):  
Lina Wu
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Differential Forms ◽  
Computational Method ◽  
Liouville Type ◽  
Balanced Growth ◽  
Curved Manifolds ◽  
Research Findings ◽  
Liouville Type Results ◽  
Flat Manifolds

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincaré-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincaré-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincaré-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of $q$-energy from finite to infinite. At this point, a technique of $p$-balanced growth has been introduced to overcome difficulties of broadening from finite $q$-energy in $L^q$ spaces to infinite $q$-energy in non-$L^q$ spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincaré-Sobolev Inequality approaching to infinite $q$-energy growth.

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