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Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces

Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces AbstractThe main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} (0<p≤1{0<p\leq 1}):Tm⁢f⁢(x,y)=∫ℝn1×ℝn2m⁢(ξ,η)⁢f^⁢(ξ,η)⁢e2⁢π⁢i⁢(x⋅ξ+y⋅η)⁢d⁢ξ⁢d⁢η.T_{m}f(x,y)=\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}}m(\xi,\eta)\hat{f%}(\xi,\eta)e^{2\pi i(x\cdot\xi+y\cdot\eta)}\mathop{}\!d\xi\mathop{}\!d\eta.Our main theorem is the following: Assume that m⁢(ξ,η){m(\xi,\eta)} is a function on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} satisfyingsupj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→)<∞\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})}<\inftywith s1>ζ1,--1⁢(1p-12){s_{1}>\zeta_{1,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, s2>ζ2,--1⁢(1p-12){s_{2}>\zeta_{2,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, where ζ1,-{\zeta_{1,-}} and ζ2,-{\zeta_{2,-}} depend only on the eigenvalues and are defined in the first section.Then Tm{T_{m}} is bounded from Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} to Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} for all 0<p≤1{0<p\leq 1} and∥Tm∥Hp⁢(A→)→Hp⁢(A→)≤CA→,s1,s2,p⁢supj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→),\lVert T_{m}\rVert_{H^{p}(\vec{A})\to H^{p}(\vec{A})}\leq C_{{\vec{A},s_{1},s_%{2},p}}\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})},where W(s1,s2)⁢(A→){W^{(s_{1},s_{2})}(\vec{A})} is a bi-parameter anisotropic Sobolev space on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} with CA→,s1,s2,p{C_{{\vec{A},s_{1},s_{2},p}}} is a positive constant that depends on A→,s1,s2,p{\vec{A},s_{1},s_{2},p}.Here we use the notations mj,k⁢(ξ,η)=m⁢(A1∗j⁢ξ,A2∗k⁢η)⁢φ(1)⁢(ξ)⁢φ(2)⁢(η){m_{j,k}(\xi,\eta)=m(A_{1}^{\ast j}\xi,A_{2}^{\ast k}\eta)\varphi^{(1)}(\xi)%\varphi^{(2)}(\eta)}, where φ(1)⁢(ξ){\varphi^{(1)}(\xi)} is a suitable cut-off function on ℝn1{\mathbb{R}^{n_{1}}} and φ(2)⁢(η){\varphi^{(2)}(\eta)} is a suitable cut-off function on ℝn2{\mathbb{R}^{n_{2}}}, respectively. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces

Forum Mathematicum , Volume 32 (3): 18 – May 1, 2020

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References (38)

Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0268
Publisher site
See Article on Publisher Site

Abstract

AbstractThe main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} (0<p≤1{0<p\leq 1}):Tm⁢f⁢(x,y)=∫ℝn1×ℝn2m⁢(ξ,η)⁢f^⁢(ξ,η)⁢e2⁢π⁢i⁢(x⋅ξ+y⋅η)⁢d⁢ξ⁢d⁢η.T_{m}f(x,y)=\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}}m(\xi,\eta)\hat{f%}(\xi,\eta)e^{2\pi i(x\cdot\xi+y\cdot\eta)}\mathop{}\!d\xi\mathop{}\!d\eta.Our main theorem is the following: Assume that m⁢(ξ,η){m(\xi,\eta)} is a function on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} satisfyingsupj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→)<∞\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})}<\inftywith s1>ζ1,--1⁢(1p-12){s_{1}>\zeta_{1,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, s2>ζ2,--1⁢(1p-12){s_{2}>\zeta_{2,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, where ζ1,-{\zeta_{1,-}} and ζ2,-{\zeta_{2,-}} depend only on the eigenvalues and are defined in the first section.Then Tm{T_{m}} is bounded from Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} to Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} for all 0<p≤1{0<p\leq 1} and∥Tm∥Hp⁢(A→)→Hp⁢(A→)≤CA→,s1,s2,p⁢supj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→),\lVert T_{m}\rVert_{H^{p}(\vec{A})\to H^{p}(\vec{A})}\leq C_{{\vec{A},s_{1},s_%{2},p}}\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})},where W(s1,s2)⁢(A→){W^{(s_{1},s_{2})}(\vec{A})} is a bi-parameter anisotropic Sobolev space on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} with CA→,s1,s2,p{C_{{\vec{A},s_{1},s_{2},p}}} is a positive constant that depends on A→,s1,s2,p{\vec{A},s_{1},s_{2},p}.Here we use the notations mj,k⁢(ξ,η)=m⁢(A1∗j⁢ξ,A2∗k⁢η)⁢φ(1)⁢(ξ)⁢φ(2)⁢(η){m_{j,k}(\xi,\eta)=m(A_{1}^{\ast j}\xi,A_{2}^{\ast k}\eta)\varphi^{(1)}(\xi)%\varphi^{(2)}(\eta)}, where φ(1)⁢(ξ){\varphi^{(1)}(\xi)} is a suitable cut-off function on ℝn1{\mathbb{R}^{n_{1}}} and φ(2)⁢(η){\varphi^{(2)}(\eta)} is a suitable cut-off function on ℝn2{\mathbb{R}^{n_{2}}}, respectively.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2020

Keywords: Hörmander multiplier; Littlewood–Paley–Stein theory; bi-parameter anisotropic Hardy spaces; bi-parameter anisotropic Sobolev spaces; 42B15; 42B25

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