Vector-valued local approximation spaces

TL;DR

This paper establishes the equivalence between vector-valued Besov spaces and local approximation spaces in Banach spaces, and characterizes the embedding of Sobolev spaces into Besov spaces via martingale cotype properties.

Contribution

It extends previous results by characterizing vector-valued Besov and Sobolev spaces through local approximation and martingale cotype conditions.

Findings

Besov spaces coincide with local approximation spaces for functions into any Banach space.

Sobolev spaces embed into Besov spaces if and only if the Banach space has martingale cotype q.

Banach space geometry determines approximation properties of vector-valued function spaces.

Abstract

We prove that for every Banach space $Y$, the Besov spaces of functions from the $n$-dimensional Euclidean space to $Y$ agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type $q$ are continuously embedded in the Besov spaces of the same type 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). These two results combined give the 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.