$L^p$ 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 A is an $n\times n$ matrix of real coefficients and satisfies the ellipticity condition, with $n\ge 2$. We assume that the coefficients of the symmetric part of A are in $L^\infty ({\mathbb {R}}^n)$, and those of the anti-symmetric part of A only belong to the space $BMO({\mathbb {R}}^n)$. We create a complete narrative of the $L^p$ theory for the square root of L and show that it satisfies the $L^p$ estimates $\left\| {\sqrt{L}f}\right\| _{L^p}\lesssim \left\| {\nabla f}\right\| _{L^p}$ for $1<p<\infty$, and $\left\| {\nabla f}\right\| _{L^p}\lesssim \left\| {\sqrt{L}f}\right\| _{L^p}$ for $1<p<2+\epsilon$ for some $\epsilon >0$ depending on the ellipticity constant and the BMO semi-norm of the coefficients. Moreover, we prove the $L^p$ estimates for some vertical 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 $p\in (1,\infty )$.

Appendix

We include some frequently used results in this appendix for reader’s convenience.

Lemma A.1

([7] Chapter V Proposition 1.1) Let Q be a cube in ${\mathbb {R}}^n$. Let $g\in L^q(Q)$, $q>1$, and $f\in L^s(Q)$, $s>q$, be two nonnegative functions. Suppose

$$\begin{aligned} \fint _{Q_R(x_0)}g^qdx\le b\left( \fint _{Q_{2R}(x_0)}gdx\right) ^q+\fint _{Q_{2R}(x_0)}f^qdx+\theta \fint _{Q_{2R}(x_0)}g^qdx \end{aligned}$$

for each $x_0\in Q$ and each $R<\min \left\{ \frac{1}{2}{{\,\mathrm{dist}\,}}(x_0,\partial Q),R_0\right\}$, where $R_0$, b, $\theta$ are constants with $b>1$, $R_0>0$, $0\le \theta <1$. Then $g\in L_{{{\,\mathrm{loc\ }\,}}}^p(Q)$ for $p\in [q,q+\epsilon )$ and

$$\begin{aligned} \left( \fint _{Q_R}g^pdx\right) ^{1/p}\le c\Bigg \{\left( \fint _{Q_{2R}}g^qdx\right) ^{1/q}+\left( \fint _{Q_{2R}}f^pdx\right) ^{1/p}\Bigg \} \end{aligned}$$

for $Q_{2R}\subset Q$, $R<R_0$, where c and $\epsilon$ are positive constants depending only on b, $\theta$, q, n (and s).

Lemma A.2

Suppose $u,v\in L^2\left( (0,T),W^{1,2}({\mathbb {R}}^n)\right)$ with $\partial _tu, \partial _tv \in L^2\left( (0,T),\widetilde{W}^{-1,2}({\mathbb {R}}^n)\right)$. Then

• (i) $u\in C\left( [0,T], L^2({\mathbb {R}}^n)\right)$;

• (ii) The mapping $t\mapsto \left\| u(\cdot ,t)\right\| _{L^2({\mathbb {R}}^n)}$ is absolutely continuous, with

  $$\begin{aligned} \frac{d}{dt}\left\| u(\cdot ,t)\right\| _{L^2({\mathbb {R}}^n)}^2=2\Re \langle \partial _tu(\cdot ,t),u(\cdot ,t)\rangle _{{\widetilde{W}}^{-1,2},W^{1,2}} \quad \text {for a.e. }t\in [0,T]. \end{aligned}$$

  As a consequence,

  $$\begin{aligned} \frac{d}{dt}\left( u(\cdot ,t),v(\cdot ,t)\right) _{L^2({\mathbb {R}}^n)}=\langle \partial _tu(\cdot ,t),v(\cdot ,t)\rangle _{{\widetilde{W}}^{-1,2},W^{1,2}}+\overline{\langle {\partial _tv(\cdot ,t),u(\cdot ,t)\rangle }}_{\widetilde{W}^{-1,2},W^{1,2}}\quad \text {a.e.}. \end{aligned}$$

For its proof see e.g. [6] Section 5.9.2 Theorem 3.

Lemma A.3

([15] Chapter V, Theorem 3.35) For any bounded linear operator B on $L^2$, if $BL=LB$ in D(L), then $L^{1/2}B=BL^{1/2}$ in $D(L^{1/2})$.