Hörmander Type Multipliers on Anisotropic Hardy Spaces

Abstract

The main purpose of this paper is to establish, using the Littlewood-Paley-Stein theory (in particular, the Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces H^p (ℝⁿ; A) associated with expensive dilation A:

$${T_m}f(x) = \int_{{\mathbb{R}^n}} {m(\xi)\hat f(\xi){{\rm{e}}^{{\rm{2}}\pi {\rm{i}}x \cdot \xi}}d\xi}.$$

Our main Theorem is the following: Assume that m(ξ) is a function on ℝⁿ satisfying

$$\mathop {\sup}\limits_{j \in \mathbb{Z}} {\left\| {{m_j}} \right\|_{{W^s}({A^ *})}} < \infty $$

with $s > \zeta _ - ^{- 1}\left({{1 \over p} - {1 \over 2}} \right)$ Then T_m is bounded from H^p(ℝⁿ; A) to H^p(ℝⁿ; A) for all 0 < p ≤ 1 and

$${\left\| {{T_m}} \right\|_{H_A^p \to H_A^p}} \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} \mathop {\sup}\limits_{j \in \mathbb{Z}} {\left\| {{m_j}} \right\|_{{W^s}({A^ *})}},$$

where A* denotes the transpose of A. Here we haveusedthe notations m_j (ξ)= m(A*^jξ)φ(ξ)and is a suitable cut-off function on ℝⁿ, and W^s(A*) is an anisotropic Sobolev space associated with expansive dilation A* on ℝⁿ.