Vector-valued extensions of operators through multilinear limited range extrapolation

TL;DR

This paper extends Rubio de Francia's extrapolation theorem to vector-valued functions in UMD Banach spaces within a multilinear limited range setting, enabling broader boundedness results for various operators.

Contribution

It introduces a multilinear limited range extrapolation theorem for vector-valued functions in UMD Banach spaces, expanding the scope of boundedness results for multilinear operators.

Findings

Extended boundedness of multilinear operators to vector-valued functions in Banach spaces.

Derived new vector-valued bounds for the bilinear Hilbert transform and Fourier multipliers.

Applied the extrapolation to operators with sparse domination, broadening their applicability.

Abstract

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an $m$-(sub)linear operator \[T:L^{p_1}(w_1^{p_1})\times\cdots\times L^{p_m}(w_m^{p_m})\to L^p(w^p) \] for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces $L^{p}(w^p;X)$ for a wide class of Banach function spaces $X$, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.