Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform

Abstract

In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasi-Banach range. As a direct consequence, we obtain weighted vector-valued inequalities reproving some of the results by Benea and Muscalu. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators. Finally, our last application gives new weighted estimates for the commutators of multilinear Calderón-Zygmund operators and for the bilinear Hilbert transform with BMO functions using ideas from Bényi et al.

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Cited by in corpus (16)

• Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach
• Quantitative estimates and extrapolation for multilinear weight classes
• Extrapolation in general quasi-Banach function spaces
• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• Extrapolation of compactness on weighted spaces: Bilinear operators
• Sparse domination implies vector-valued sparse domination
• Vector-valued extensions of operators through multilinear limited range extrapolation
• Modern singular integral theory with mild kernel regularity
• Some new weighted estimates on product spaces
• Bilinear spherical maximal functions of product type
• Sparse domination and weighted estimates for rough bilinear singular integrals
• Genuinely multilinear weighted estimates for singular integrals in product spaces
• Weighted mixed weak-type inequalities for multilinear fractional operators
• Weak and strong type estimates for the multilinear Littlewood-Paley operators
• Bilinear Wavelet Representation of Calderón-Zygmund Forms
• Upper endpoint estimates and extrapolation for commutators