Some multiplier theorems for anisotropic hardy spaces

Abstract

Let A be a symmetric expansive matrix and Hp (ℝ n) be the anisotropic Hardy space associated with A. For a function m in L∞ (ℝ n), an appropriately chosen function η in C ∞c (ℝ n) and j ∈ ℤ define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and \(\hat m_j \) belongs to the anisotropic non-homogeneous Herz space K 1/p−1,p 1 (ℝ n), then m is a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n). For p = 1, a similar result is obtained if the space K 0,1 1 (ℝ n) is replaced by a slightly smaller space K(ω). Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)∨} belongs to certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n).