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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}):Tmf(x,y)=∫ℝn1×ℝn2m(ξ,η)f^(ξ,η)e2πi(x⋅ξ+y⋅η)dξdη.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,psupj,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∗jξ,A2∗kη)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.
Forum Mathematicum – de 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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