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.
Highlights
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
They defined the uniform approximation by affine property of Lip(X, Y ), the space of Lipschitz functions f : X → Y, and showed that it is equivalent to the statement that (Y, · Y ) has an equivalent uniformly convex norm
One of our main results, Theorem 4 below, is of similar flavour; it states that Y admits an equivalent uniformly convex norm with modulus of convexity of power type q if and only if all weakly differentiable functions f : Rn → Y have good local approximations with polynomials, or more precisely, the Lq-type Sobolev spaces are continuously embedded in suitable local approximation spaces
Summary
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. We define the local approximation space by AspNq (Rn; Y ). The classical result states that the two spaces are equal with equivalent norms.
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