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

Abstract

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mo>∂</m:mo> <m:mo>¯</m:mo> </m:mover> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{p}({\mathbb{R}^{n}})} bounded for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {1&lt;p&lt;\infty} , but only bounded on local Hardy spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0&lt;p\leq 1} . Though much work has been done on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:msub> <m:mi>n</m:mi> <m:mn>1</m:mn> </m:msub> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:msub> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msub> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {1&lt;p&lt;\infty} and Hardy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:msub> <m:mi>n</m:mi> <m:mn>1</m:mn> </m:msub> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:msub> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msub> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow>

Topics & Concepts