Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Abstract

Let A be an expansive dilation on \({{\mathbb R}^n}\) and w a Muckenhoupt \({\mathcal A_\infty(A)}\) weight. In this paper, for all parameters \({\alpha\in{\mathbb R} }\) and \({p,q\in(0,\infty)}\), the authors identify the dual spaces of weighted anisotropic Besov spaces \({\dot B^\alpha_{p,q}(A;w)}\) and Triebel–Lizorkin spaces \({\dot F^\alpha_{p,q}(A;w)}\) with some new weighted Besov-type and Triebel–Lizorkin-type spaces. The corresponding results on anisotropic Besov spaces \({\dot B^\alpha_{p,q}(A; \mu)}\) and Triebel–Lizorkin spaces \({\dot F^\alpha_{p,q}(A; \mu)}\) associated with \({\rho_A}\) -doubling measure μ are also established. All results are new even for the classical weighted Besov and Triebel–Lizorkin spaces in the isotropic setting. In particular, the authors also obtain the \({\varphi}\) -transform characterization of the dual spaces of the classical weighted Hardy spaces on \({{\mathbb R}^n}\).