Abstract
We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the {{,mathrm{UMD},}} condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a {{,mathrm{UMD},}} condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.
Highlights
Vector-valued extensions of operators prevalent in the theory of harmonic analysis have been actively studied in the past decades
We point out that even in the linear case m = 1 the result of obtaining vector-valued extensions of operators in UMD Banach function spaces from sparse domination without appealing to a Rubio de Francia type extrapolation theorem is new
We introduce a rescaled multilinear analogue of the Hardy-Littlewood property and prove sparse domination of the multisublinear Hardy-Littlewood maximal operator using this condition
Summary
Vector-valued extensions of operators prevalent in the theory of harmonic analysis have been actively studied in the past decades. Combined this enabled Rubio de Francia to show a very general extension principle in [45], yielding vector-valued extensions for operators on L p(Rd ) satisfying bounds with respect to these Muckenhoupt weights to any UMD Banach function space This result was subsequently extended by Amenta, Veraar and the first author in [1] to a rescaled setting and by both authors in [36] to a multilinear setting. We point out that even in the linear case m = 1 the result of obtaining vector-valued extensions of operators in UMD Banach function spaces from sparse domination without appealing to a Rubio de Francia type extrapolation theorem is new. We say X is r -convex for r ∈ (0, ∞)m if X j is r j -convex for 1 ≤ j ≤ m
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