Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

Abstract

AbstractWe give an extension of Rubio de Francia’s extrapolation theorem for functions taking values in $${{\,\mathrm{UMD}\,}}$$ UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an m-(sub)linear operator $$\begin{aligned} T:L^{p_1}(w_1^{p_1})\times \cdots \times L^{p_m}(w_m^{p_m})\rightarrow L^p(w^p) \end{aligned}$$ T : L p 1 ( w 1 p 1 ) × ⋯ × L p m ( w m p m ) → L p ( w p ) for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces $$L^{p}(w^p;X)$$ 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.