On the interpolation of the spaces $W^{l,1}({\mathbb{R}}^d)$ and $W^{r,\infty }({\mathbb{R}}^d)$

Abstract

We study some properties of spaces obtained by interpolation of the Sobolev spaces $W^{k,1}({\mathbb{R}}^d)$ and $W^{l,\infty }({\mathbb{R}}^d)$, where $l$ and $r$ are nonnegative integers, and $d\ge 2$. We are concerned with the standard real and complex methods of interpolation. In the case of the real method, an old result of De Vore and Scherer (1979) gives that

where $\theta \in (0,1)$ and $1/p_{\theta }=1-\theta$. We complement this result by considering the case $l\ne r$. We prove that, when $l\ne r$,

where $\sigma :=(1-\theta )l+\theta r$ and $1/q=1-\theta$, if and only if $l-r\in \mathbb{R}\setminus [1,d]$. Also, we prove a similar fact when $W^{l,1}$ is replaced in $(\star )$ by a space $W^{s,p}$ where $s\ne r$ is a real number and $p\in (1,\infty )$. Several other problems like the boundedness of the Riesz transforms on interpolation spaces are also considered. In the case of the complex method, it was proved by M. Milman (1983) that, for any $1<p<\infty$,

where $1/p_{\theta }=(1-\theta )+\theta /p$. We show by simple arguments that $(\star \star )$ fails when $p=\infty$ and $l\ge 1$, answering a question of P. W. Jones (1984). As an immediate consequence of these arguments, we show that certain closed subspaces of $(C({\mathbb{T}}^d){)}^N$ (with $N\in {\mathbb{N}}^{\ast }$) that are described by Fourier multipliers are not complemented in $(C({\mathbb{T}}^d){)}^N$.