论文标题
在加权空间上的紧凑性外推
Extrapolation of compactness on weighted spaces
论文作者
论文摘要
令$ t $是一个线性操作员，对于某些$ p_1 \（1，\ infty）$，在$ l^{p_1}（\ tilde w）$上限制了所有$ \ tilde w \ in a_ {p_1}中的$ \ tilde w \ in a_ {p_1}（\ nathb r^d）$，以及$ l^d_1 $ l l^_1}（for $ l l^^w iN）（ A_ {P_1}（\ Mathbb r^d）$。然后，$ t $在（1，\ infty）$中的所有$ p \ in $ l^p（w）$上紧凑并紧凑，而a_p（\ mathbb r^d）$中的所有$ w \ in All $ W \。卢比奥·德·弗兰西亚（Rubio de Francia）著名的加权推断定理的“紧凑版”源于一方面的加权空间的插值和外推理论的结合，另一方面，紧凑的操作员在抽象的空间上的紧凑型操作员。此外，对于从一个空间到另一个空间（“偏外估计”）或仅在有限的$ l^p $ scale范围内，可以获得这种紧凑性外推的概括。作为应用程序，我们很容易地恢复了有关奇异积分运算符，分数积分和伪差异操作员的加权紧凑型的最新结果，并获得了有关Bochner-Riesz乘数的加权紧凑性的新结果。
Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}(\mathbb R^d)$. Then $T$ is bounded and compact on $L^p(w)$ for all $p\in(1,\infty)$ and all $w\in A_p(\mathbb R^d)$. This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the $L^p$ scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers.
