Hörmander Type Multipliers on Anisotropic Hardy Spaces.

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) = ∫ ℝ n m (ξ) f ^ (ξ) e 2 π i x ⋅ ξ d ξ . Our main Theorem is the following: Assume that m(ξ) is a function on ℝⁿ satisfying sup j ∈ ℤ ‖ m j ‖ W s (A ∗) < ∞ with s > ζ − − 1 (1 p − 1 2) Then T_m is bounded from H^p(ℝⁿ; A) to H^p(ℝⁿ; A) for all 0 < p ≤ 1 and ‖ T m ‖ H A p → H A p ≲ sup j ∈ ℤ ‖ m j ‖ 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. [ABSTRACT FROM AUTHOR]