Continuity properties of multi-parameter pseudodifferential operators

Abstract It is well known that an pseudodifferential operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> {T_{\sigma}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>S</m:mi> <m:mrow> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\sigma\in S^{m}_{\rho,\delta}(\mathbb{R}^{n})} is bounded on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{2}(\mathbb{R}^{n})} if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>≤</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>δ</m:mi> <m:mo>-</m:mo> <m:mi>ρ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>&lt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> </m:math> {m\leq 0,m+\frac{n}{2}(\delta-\rho)&lt;0} which was established by Hörmander. In this paper, we generalize this result to multi-parameter pseudodifferential operators. Using the atomic decomposition theory, we obtain the boundedness from <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 stretchy="false">(</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 stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to <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 stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mrow> <m:msub> <m:mi>n</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{p}(\mathbb{R}^{n_{1}+n_{2}})} of bi-parameter pseudodifferential operators 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>