A Fourier multiplier theorem on anisotropic Hardy spaces associated with ball quasi-Banach function spaces

Abstract

Let A be a general expansive matrix. Let X be a ball quasi-Banach function space on \(\mathbb {R}^n\), which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first establish the boundedness of convolutional anisotropic Calderón–Zygmund operators on the Hardy space \(H_X^A(\mathbb {R}^n)\). As an application, the authors also obtain the boundedness of Fourier multipliers satisfying anisotropic Mihlin conditions on \(H_X^A(\mathbb {R}^n)\). All these results have a wide range of applications; in particular, when they are applied to Lebesgue spaces, all these results reduce back to the known best results and, even when they are applied to Lorentz spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces, the obtained results are also new.