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

Jiao Chen, Wei Ding and Guozhen Lu
From the journal Forum Mathematicum

Abstract

After 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 ¯ 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 Lp(n) bounded for 1<p<, but only bounded on local Hardy spaces hp(n) introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0<p1. Though much work has been done on the Lp(n1×n2) boundedness for 1<p< and Hardy Hp(n1×n2) boundedness for 0<p1 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<p1. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces hp(n1×n2) for 0<p1 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].


Communicated by Christopher D. Sogge


Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11801049

Award Identifier / Grant number: 11501308

Award Identifier / Grant number: 11771223

Funding source: Natural Science Foundation of Chongqing

Award Identifier / Grant number: cstc2019jcyj-msxmX0374

Award Identifier / Grant number: cstc2019jcyj-msxmX0295

Funding source: Simons Foundation

Award Identifier / Grant number: 519099

Funding statement: The first two authors were supported in part by grants from NNSF of China (Grant numbers 11801049, 11501308, 11771223) and the Natural Science Foundation of Chongqing (Grant numbers cstc2019jcyj-msxmX0374, cstc2019jcyj-msxmX0295). The third author was partly supported by the Simons foundation (Grant number 519099).

Acknowledgements

The authors would like to thank the referee for his/her very careful reading and helpful comments which have improved the exposition of the paper.

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Received: 2019-11-18
Revised: 2020-02-10
Published Online: 2020-04-15
Published in Print: 2020-07-01

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