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

TL;DR

This paper investigates the traces of weighted Triebel-Lizorkin spaces on hyperplanes, especially with Muckenhoupt weights, and characterizes certain real interpolation spaces, solving a long-standing open problem in the field.

Contribution

It provides a detailed analysis of weighted Triebel-Lizorkin space traces and describes the real interpolation spaces, addressing an open question for specific parameter ranges.

Findings

Derived atomic decomposition and wavelet representation for weighted spaces.

Characterized the trace spaces on hyperplanes with Muckenhoupt weights.

Solved a long-standing open problem on real interpolation of Besov spaces.

Abstract

We study traces of weighted Triebel-Lizorkin spaces $F^s_{p,q}({\mathbb R}^n,w)$ 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 $|x_n|\leq 1$, $x\in{\mathbb R}^n$, 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 $(B^{s_1}_{p_1,p_1}({\mathbb R}^{n-k}), B^{s_2}_{p_2,p_2}({\mathbb R}^{n-k}))_{\theta,r}$, $0<p_1<p_2<\infty$, $s_i=s-(\alpha+k)/{p_i}$, $i=1,2$, $s>0$ sufficiently large, $0<\theta<1$, $0<r\leq\infty$. 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.