On the sharpness of embeddings of H\"older spaces into Gaussian Besov spaces

TL;DR

This paper investigates the sharpness of embeddings of H"older spaces into Gaussian Besov spaces by constructing extremal subspaces using Rademacher randomization, confirming the optimality of these embeddings.

Contribution

It introduces a novel construction of extremal subspaces via Rademacher variables to demonstrate the sharpness of embeddings between H"older and Gaussian Besov spaces.

Findings

Constructed extremal subspaces isometric to ll_q^{( heta)}.

Verified extremal property for rescaled functions with periodicity.

Confirmed sharpness of natural embeddings from Hlder to Gaussian Besov spaces.

Abstract

For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1 \hookrightarrow E_0$ we use vectors $b_1,b_2,\ldots \in E_1$ that satisfy an extremal property with respect to the $J$- and $K$-functional to construct sub-spaces that are isometric to $\ell_q^{(\theta)}$. The construction is based on a randomisation using independent Rademacher variables. We verify that systems obtained by re-scaling a function with a certain periodicity property share this extreme property. This implies the sharpness of natural embeddings of H\"older spaces obtained by the real interpolation into the corresponding Gaussian Besov spaces.