Quantitative estimates and extrapolation for multilinear weight classes

Abstract

In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including ${\ell }^{\infty }$ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.

44 pages. Minor improvements

References in corpus (2)

• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• The Sparse T1 Theorem

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• Extrapolation of compactness on weighted spaces: Bilinear operators
• Sparse domination implies vector-valued sparse domination
• Bloom weighted bounds for sparse forms associated to commutators
• Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces
• Weighted estimates for Bilinear Bochner-Riesz means at the critical index
• Modern singular integral theory with mild kernel regularity
• Some new weighted estimates on product spaces
• Genuinely multilinear weighted estimates for singular integrals in product spaces
• Bilinear spherical maximal functions of product type
• Mixed-norm estimates via the helicoidal method
• Upper endpoint estimates and extrapolation for commutators
• Bilinear Wavelet Representation of Calderón-Zygmund Forms