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arXiv · 2011.13191

Extrapolation for multilinear compact operators and applications

Mingming Cao·Andrea Olivo·Kôzô Yabuta
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Abstract

This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of $T$ from just one space to the full range of weighted spaces, whenever an $m$-linear operator $T$ is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights $A_{\vec{p}, \vec{r}}$, and the limited range of the $L^p$ scale. To show extrapolation theorems above, by means of a new weighted Fr\'{e}chet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to bulid a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear $\omega$-Calder\'{o}n-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calder\'{o}n commutators, and commutators of Riesz transforms related to Schr\"{o}dinger operators.

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Mingming Cao, Andrea Olivo, Kôzô Yabuta. 2020-11-26. Extrapolation for multilinear compact operators and applications. https://arxiv.org/abs/2011.13191

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