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

Abstract

We consider the operator $L=-\mathrm{div}(A\nabla )$, 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 $BMO({\mathbb{R}}^n)$, $n\ge 2$. We prove the Gaussian estimates for the kernel of $e^{-tL}$, as well as that of ${\partial }_t^l{e}^{-tL}$, for any $l\in \mathbb{N}$. We show that the square root of $L$ satisfies the $L^p$ estimates ${\Vert L^{1/2}f\Vert }_{L^p}\lesssim {\Vert \nabla f\Vert }_{L^p}$ for $1<p<\infty$, and ${\Vert \nabla f\Vert }_{L^p}\lesssim {\Vert L^{1/2}f\Vert }_{L^p}$ 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 $\mathrm{div}(A(x)\nabla u)=0$ 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 $p\in (1,\infty )$.

41 pages. To appear in Mathematische Zeitschrift

References in corpus (2)

• The Dirichlet problem for elliptic operators having a BMO anti-symmetric part
• Fundamental solutions for second order parabolic systems with drift terms

Cited by in corpus (1)

• The Dirichlet problem for elliptic operators having a BMO anti-symmetric part