Duality of weighted anisotropic Besov and Triebel-Lizorkin spaces

Li, Baode; Bownik, Marcin; Yang, Dachun; Yuan, Wen
June 2012
Positivity;Jun2012, Vol. 16 Issue 2, p213
Academic Journal
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 $${{\mathbb R}^n}$$.


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