Extrapolation for multilinear compact operators and applications

Cao, Mingming; Olivo, Andrea; Yabuta, Kôzô

doi:10.1090/tran/8645

Extrapolation for multilinear compact operators and applications
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by Mingming Cao, Andrea Olivo and Kôzô Yabuta PDF

Trans. Amer. Math. Soc. 375 (2022), 5011-5070 Request permission

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 Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to build a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear $\omega$-Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.

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