论文标题
某些加权功能空间的痕迹和相关的非标准空间的实际插值
Traces of some weighted function spaces and related non-standard real interpolation of Besov spaces
论文作者
论文摘要
我们研究了加权triebel-lizorkin空间的痕迹$ f^s_ {p，q}（{\ mathbb r}^n，w）$上的超平面$ {\ mathbb r}^{n-k} $，其中的权重是muckenhoupt型。我们专注于示例重量$w_α（x）= | x_n |^α$当$ | x_n | \ leq 1 $，$ x \ in {\ mathbb r}^n $，$w_α（x）= 1 $，否则，$α> -1 $。在这里，我们使用一些精致的原子分解参数以及相应（未加权）BESOV空间中的适当小波表示。第二个主要结果是对实际插值空间的描述$（b^{s_1} _ {p_1，p_1}（{\ Mathbb r}^{n-k}），b^{s_2} _ {p_2，p_2，p_2，p_2}（ $ 0 <p_1 <p_2 <\ infty $，$ s_i = s-（α+k）/{p_i} $，$ i = 1,2 $，$ s> 0 $ $ a $足够大，$ 0 <θ<1 $，$ 0 <r \ leq \ leq \ infty $。除了情况外，$ 1/r =（1-θ）/{p_1}+θ/{p_2} $这个问题似乎已经开放了很多年。根据我们的第一个结果，我们现在可以快速解决这个长期存在的问题。在这里，我们从Besoy，Cobos和Triebel的一些最新发现中受益。
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 $w_α(x) = |x_n|^α$ when $|x_n|\leq 1$, $x\in{\mathbb R}^n$, and $w_α(x)=1$ otherwise, where $α>-1$. 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 $(B^{s_1}_{p_1,p_1}({\mathbb R}^{n-k}), B^{s_2}_{p_2,p_2}({\mathbb R}^{n-k}))_{θ,r}$, $0<p_1<p_2<\infty$, $s_i=s-(α+k)/{p_i}$, $i=1,2$, $s>0$ sufficiently large, $0<θ<1$, $0<r\leq\infty$. Apart from the case $1/r= (1-θ)/{p_1}+ θ/{p_2}$ 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.
