Vector-valued local approximation spaces

Tuomas Hytönen, Jori Merikoski

Published 2016-12-30Version 1

We prove that for every Banach space $Y,$ the Besov space $B^{s}_{pq}( \mathbb{R}^n; Y)$ of functions $f: \mathbb{R}^n \to Y$ agrees with a suitable local approximation space $A^{s}_{pq}( \mathbb{R}^n; Y)$ with equivalent norms. In adddition, we prove that the Sobolev space $W^{k,q}( \mathbb{R}^n; Y)$ is continuously embedded in the Besov space $B^k_{qq}( \mathbb{R}^n; Y)$ if and only if $Y$ has martingale cotype $q.$ We interpret this as an extension of earlier results of Xu (1998), and Mart\'inez, Torrea and Xu (2006), which correspond to the case $k=0.$ These two results combined give the satisfactory characterization that $Y$ admits an equivalent norm with modulus of convexity of power type q if and only if weakly differentiable functions have good local approximations with polynomials.