Trilinear embedding for divergence-form operators with complex coefficients

TL;DR

This paper establishes a novel dimension-free trilinear embedding for elliptic divergence-form operators with complex coefficients, enabling new inequalities and applications in analysis on arbitrary open sets.

Contribution

It introduces the first dimension-free trilinear embedding for such operators, utilizing Bellman functions and heat flows, with broad applications to inequalities and operator theory.

Findings

Proves a dimension-free $L^p$ embedding for triples of elliptic operators.

Derives new inequalities of Kato--Ponce type for complex coefficient operators.

Provides applications to paraproducts and square functions in this setting.

Abstract

We prove a dimension-free $L^p(\Omega)\times L^q(\Omega)\times L^r(\Omega)\rightarrow L^1(\Omega\times (0,\infty))$ embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on $\Omega$, and for triples of exponents $p,q,r\in(1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here $\Omega$ is allowed to be an arbitrary open subset of $\mathbb{R}^d$. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as $p$-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato--Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.