$L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

TL;DR

This paper develops $L^p$ estimates and Gaussian bounds for the square roots and associated functions of elliptic operators with BMO anti-symmetric parts, advancing the understanding of their analytical properties.

Contribution

The authors establish new $L^p$ bounds and Gaussian estimates for elliptic operators with BMO anti-symmetric coefficients, extending classical theory to less regular coefficient matrices.

Findings

Proved Gaussian estimates for kernels of $e^{-tL}$ and its derivatives.

Established $L^p$ bounds for the square root of $L$ and its relation to gradients.

Derived $L^p$ estimates for square functions associated with $e^{-tL}$.

Abstract

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space $BMO(\mathbb{R}^n)$, $n\ge2$. We prove the Gaussian estimates for the kernel of $e^{-tL}$, as well as that of $\partial_t^le^{-tL}$, for any $l\in\mathbb{N}$. We show that the square root of $L$ satisfies the $L^p$ estimates $\left\Vert{L^{1/2}f}\right\Vert_{L^p}\lesssim\left\Vert{\nabla f}\right\Vert_{L^p}$ for $1<p<\infty$, and $\left\Vert{\nabla f}\right\Vert_{L^p}\lesssim\left\Vert{L^{1/2}f}\right\Vert_{L^p}$ for $1<p<2+\epsilon$ for some $\epsilon>0$ depending on the ellipticity constant and the BMO semi-norm of the coefficients. Finally, we prove the $L^p$ estimates for square functions associated to $e^{-tL}$. In another article of the authors, these results are used to establish the solvability of the Dirichlet problem for elliptic equation ${\rm div}(A(x)\nabla u)=0$ in the upper half-space $(x,t)\in\mathbb{R}_+^{n+1}$ with the boundary data in $L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$.