论文标题
$ l^p $ -to- $ l^q $ comptators $ p> q $的紧凑型
The $L^p$-to-$L^q$ Compactness of Commutators with $p>q$
论文作者
论文摘要
令$ 1 <q <q <p <\ infty $，$ \ frac1r：= \ frac1q- \ frac1p $，而$ t $是一个非脱位calderón-zygmund操作员。我们表明，换向器$ [b，t] $从$ l^p（{\ mathbb r}^n）$到$ l^q（{\ mathbb r}^n）$ if，仅当符号$ b = a+c $带有$ b = with $ a a \ in l^r^r（{\ mathbb r}^n）$ c $和任何常数。由于相应的硬木 - 最大运算符和相应的calderón-zygmund最大运算符并未从$ l^p（{\ mathbb r}^n）$到$ l^q（{\ mathbb r}^n）$限制，因此我们对$ c._ compact的全部优势c}^\ infty（{\ Mathbb r}^n）$，对于许多相应的估计，当$ p \ leq q $中，但在$ p> q $时至关重要。我们还将结果扩展到多线性情况。
Let $1<q<p<\infty$, $\frac1r:=\frac1q-\frac1p$, and $T$ be a non-degenerate Calderón--Zygmund operator. We show that the commutator $[b,T]$ is compact from $L^p({\mathbb R}^n)$ to $L^q({\mathbb R}^n)$ if and only if the symbol $b=a+c$ with $a\in L^r({\mathbb R}^n)$ and $c$ being any constant. Since both the corresponding Hardy--Littlewood maximal operator and the corresponding Calderón--Zygmund maximal operator are not bounded from $L^p({\mathbb R}^n)$ to $L^q({\mathbb R}^n)$, we take the full advantage of the compact support of the approximation element in $C_{\rm c}^\infty({\mathbb R}^n)$, which seems to be redundant for many corresponding estimates when $p\leq q$ but to be crucial when $p>q$. We also extend the results to the multilinear case.
