On the sharpness of embeddings of Hölder spaces into Gaussian Besov spaces

Abstract

For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1\hookrightarrow E_0$ we use vectors $b_1,b_2,\dots \in E_1$ 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 Hölder spaces obtained by the real interpolation into the corresponding Gaussian Besov spaces.

References in corpus (1)

• On fractional smoothness and $L_p$-approximation on the Gaussian space