H?rmander 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(R~n; A) associated with expensive dilation A:■Our main Theorem is the following: Assume that m(ξ) is a function on R~n satisfying ■with s > ζ(1/p-1/2). Then Tis bounded from H~p(R~n; A) to H~p(R~n; A) for all 0 < p ≤ 1 and ■where A~* denotes the transpose of A. Here we have used the notations m(ξ) = m(Aξ)φ(ξ) and φ(ξ) is a suitable cut-off function on R~n, and W~s(A~*) is an anisotropic Sobolev space associated with expansive dilation A~* on R~n.