Traces of some weighted function spaces and related non-standard real interpolation of Besov spaces

Abstract

We study traces of weighted Triebel-Lizorkin spaces $F_{p,q}^s({\mathbb{R}}^n,w)$ on hyperplanes ${\mathbb{R}}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight $w_{\alpha }(x)=|x_n{|}^{\alpha }$ when $|x_n|\le 1$, $x\in {\mathbb{R}}^n$, and $w_{\alpha }(x)=1$ otherwise, where $\alpha >-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space $(B_{p_1,p_1}^{s_1}({\mathbb{R}}^{n-k}),B_{p_2,p_2}^{s_2}({\mathbb{R}}^{n-k}){)}_{\theta ,r}$, $0<p_1<p_2<\infty$, $s_i=s-(\alpha +k)/p_i$, $i=1,2$, $s>0$ sufficiently large, $0<\theta <1$, $0<r\le \infty$. Apart from the case $1/r=(1-\theta )/p_1+\theta /p_2$ the question seems to be open for many years. Based on our first result we can now quickly solve this long-standing problem. Here we benefit from some very recent finding of Besoy, Cobos and Triebel.