The H\"ormander Multiplier Theorem III: The complete bilinear case via interpolation

TL;DR

This paper establishes an optimal bilinear H"ormander multiplier theorem using a novel multilinear interpolation approach, identifying the broadest range of Lebesgue space exponents for boundedness of bilinear operators with Sobolev symbols.

Contribution

It introduces a specialized multilinear complex interpolation theorem to prove the most comprehensive bilinear H"ormander multiplier result to date.

Findings

Identifies the largest open set of exponents for bounded bilinear multipliers.

Proves boundedness of bilinear operators with symbols in Sobolev spaces.

Provides a unified framework for bilinear multiplier theorems.

Abstract

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear H\"ormander multiplier theorem concerning symbols that lie in the Sobolev space $L^r_s(\mathbb R^{2n})$, $2\le r<\infty$, $rs>2n$, uniformly over all annuli. More precisely, given a smoothness index $s$, we find the largest open set of indices $(1/p_1,1/p_2 )$ for which we have boundedness for the associated bilinear multiplier operator from $L^{p_1}(\mathbb R^{ n})\times L^{p_2} (\mathbb R^{ n})$ to $ L^p(\mathbb R^{ n})$ when $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.