Some multiplier theorems for anisotropic hardy spaces

Abstract

Let A be a symmetric expansive matrix and H^p (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 m_j(ξ) = m(A^jξ)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 H^p (ℝ ⁿ) to L^p (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 {(m_j)^∨} belongs to certain mixednorm space, depending on p, then m is also a Fourier multiplier from H^p (ℝ ⁿ) to L^p (ℝ ⁿ).