论文标题
besov-sobolev空间和最佳嵌入二元组的框架表征
Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts
论文作者
论文摘要
我们在$ \ mathbb {r} $上研究了BESOV和Triebel-Lizorkin空间中HAAR系数的行为，对于HAAR系统不是无条件的参数范围。首先，我们获得一系列参数，扩展到平滑度$ s <1 $，其中$ f^s_ {p，q} $和$ b^s_ {p，q} $的特征是以双重超采样的HAAR系数（HAAR框架）为特征。其次，如果$ 1/p <s <1 $和b^s_ {p，q} $中的$ f \ $ f \，我们实际上证明了通常的HAAR系数规范，$ \ | \ | \ {2^j \ langle f，h_ {仍然等同于$ \ | f \ | _ {b^s_ {p，q}} $，即，经典的besov空间是其二元对应物的封闭子集。在端点情况下，$ s = 1 $和$ q = \ infty $，我们表明，这种表达式给出了Sobolev Space $ W^{1} _p（\ Mathbb {r}）$，$ 1 <p <\ infty $的等效范围，该$与Bočkarev相关的经典结果。最后，在几种端点情况下，我们阐明了二元与标准BESOV与Triebel-Lizorkin空间之间的关系。
We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $1/p<s<1$ and $f\in B^s_{p,q}$, we actually prove that the usual Haar coefficient norm, $\|\{2^j\langle f, h_{j,μ}\rangle\}_{j,μ}\|_{b^s_{p,q}}$ remains equivalent to $\|f\|_{B^s_{p,q}}$, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show that such an expression gives an equivalent norm for the Sobolev space $W^{1}_p(\mathbb{R})$, $1<p<\infty$, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.
