Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

David Cruz-Uribe; José María Martell

doi:10.1007/s00208-018-1640-9

paper

Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

arXiv:1704.06833 · doi:10.1007/s00208-018-1640-9

Abstract

We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights $w_{1}, w_{2}$ for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form \[ BH : L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2}) \longrightarrow L^p(w^p), \] where $w = w_{1} w_{2}$ and $\frac{1}{p} = \frac{1}{p _{1}} + \frac{1}{p _{2}} < \frac{3}{2}$ . This improves the recent results of Culiuc et al. by increasing the families of weights for which this inequality holds and by pushing the lower bound on $p$ from $1$ down to $\frac{2}{3}$ , the critical index from the unweighted theory of the bilinear Hilbert transform. Second, as an easy consequence of our method we obtain that the bilinear Hilbert transform satisfies some vector-valued inequalities with Muckenhoupt weights. This reproves and generalizes some of the vector-valued estimates obtained by Benea and Muscalu in the unweighted case. We also generalize recent results of Carando, et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators.

In this version we correct and expand some of our results about vector-valued inequalities for the bilinear Hilbert transform

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Multiple Vector Valued Inequalities via the Helicoidal Method

Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

Abstract

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