# Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

## TL;DR

This paper develops a new multilinear extrapolation theorem and applies it to improve weighted norm inequalities for the bilinear Hilbert transform, extending the range of weights and exponents for which these inequalities hold.

## Contribution

It introduces a limited range multilinear extrapolation theorem and applies it to enhance weighted inequalities for the bilinear Hilbert transform, including vector-valued and Marcinkiewicz-Zygmund estimates.

## Key findings

- Extended the range of weights for bilinear Hilbert transform inequalities.
- Lowered the exponent bound from 1 to 2/3 for weighted inequalities.
- Generalized vector-valued and Marcinkiewicz-Zygmund estimates.

## 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 $w=w_1w_2$ 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 Calder\'on-Zygmund operators.