Elliptic measures for Dahlberg–Kenig–Pipher operators: asymptotically optimal estimates

Abstract

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal $A_\infty$ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the $A_\infty$ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems.

Appendix: An extension of Theorem 1.2

For a more general domain $\Omega$ (even if not graphical), we may still define an elliptic matrix in $\Omega$ satisfying the weak DKP condition (resp. with vanishing trace). In fact, let $A(\cdot )$ be an elliptic matrix in $\Omega$. We define the oscillation of $A(\cdot )$ similar as in Definition 2.5: For any $X\in \Omega$, let

(A.12)

where $\delta (X): = {\text {dist}}(X, \partial \Omega )$. We say $A(\cdot )$ satisfies the weak DKP condition (resp. with vanishing trace), if the measure

$$\begin{aligned} d\mu (X) = \frac{\alpha _2(X)^2}{\delta (X)} dX \end{aligned}$$

is a Carleson measure (resp. with vanishing trace).

Even though Theorem 1.2 is proven for the upper half space, it is not hard to show that the analogue also holds for more general classes of domains.

Theorem 6.13

Let $\Omega$ be a $C^1$-square Dini domain in ${{\mathbb {R}}}^{n+1}$. Let $A(\cdot )$ be an elliptic matrix in $\Omega$ which satisfies the weak DKP condition with vanishing trace. Then for any $X_0 \in \Omega$, the elliptic measure $\omega _\Omega ^{X_0}$ is absolute continuous with respect to the boundary surface measure $\sigma := {\mathcal {H}}^n|_{\partial \Omega }$, and moreover, the Poisson kernel $k_\Omega := \frac{d\omega _\Omega ^{X_0}}{d\sigma }$ satisfies $\log k_\Omega \in VMO_{loc}(\partial \Omega )$.

When we say $\Omega$ is a $C^1$-square Dini domain, it means there exist $R>0$, finitely many boundary points $x_i \in \partial \Omega$ and cylindrical regions ${\mathcal {C}}(x_i, R)$¹⁰ centered at $x_i$ such that $\Omega \subset \cup _i {\mathcal {C}}(x_i, R/2)$ and for each i, $\Omega \cap {\mathcal {C}}(x_i, R)$ is the region above the graph of a $C^1$-square Dini function $\varphi _i$. Assume without loss of generality that $X_0 \in {\mathcal {C}}(x_i, R) \setminus {\mathcal {C}}(x_i, R/2)$ for each i. It is not hard to see that (with the help of a cut-off function) we can extend $\varphi _i$ to a globally-defined function $\varphi : {{\mathbb {R}}}^n \rightarrow {{\mathbb {R}}}$, such that the modulus of continuity of $\nabla \varphi$ is also bounded above by $\theta$. Moreover, applying [3, Lemma 5.1] (in particular, see [3, Remark 5.11]) to $C^1$ domains, we can show that if two elliptic operators $L_1$ and $L_2$ agree in $\Omega \cap {\mathcal {C}}(x_i, R)$, then the ratio of their elliptic measures $d\omega _{L_2}^{X_0}/d\omega _{L_1}^{X_0}$ has small oscillation in a surface ball, whose radius is much smaller compared to R. In particular, it implies by a similar proof as that of Lemma 5.13 that

$$\begin{aligned} \log k_{L_1} \in VMO(\partial \Omega \cap {\mathcal {C}}(x_i, R/2)) \iff \log k_{L_2} \in VMO(\partial \Omega \cap {\mathcal {C}}(x_i, R/2)).\nonumber \\ \end{aligned}$$

(6.14)

Therefore the proof of Theorem 6.13 is reduced to the setting where $\Omega$ is the region above the graph of a single function $\varphi : {{\mathbb {R}}}^n \rightarrow {{\mathbb {R}}}$.

More precisely, let $\Omega \subset {{\mathbb {R}}}^{n+1}$ be the domain above the graph of $\varphi : {{\mathbb {R}}}^n \rightarrow {{\mathbb {R}}}$, where the modulus of continuity for $\nabla \varphi$ satisfies the square Dini condition (see (1.5)). Assume without loss of generality that $\varphi (0) = 0$ and $\nabla \varphi (0) = 0$. Let A(x, t) be a uniformly elliptic coefficient matrix in the graphical domain $\Omega$, which satisfies the weak DKP condition with vanishing trace, which means the following¹¹ For any $x_0 \in {{\mathbb {R}}}^n$ fixed, we define the Whitney region

$$\begin{aligned} W_\Omega (x_0, r) := \left\{ (x,t) \in {{\mathbb {R}}}^n \times {{\mathbb {R}}}: x\in \Delta _r(x_0), ~\varphi (x)+\frac{r}{2} < t \le \varphi (x) + r \right\} , \end{aligned}$$

and denote the $L^2$-oscillation of the matrix A as

$$\begin{aligned} \alpha _A(x_0,r) := \inf _{A_0 \in {\mathfrak {A}}(\Lambda )} \left( \frac{1}{|W_\Omega (x_0,r)|} \iint _{W_\Omega (x_0,r)} |A(x,t) - A_0|^2 dxdt \right) ^{1/2}, \end{aligned}$$

(6.15)

where, as in Definition 2.5 before, the infimum ranges over all constant coefficient matrices. We say A satisfies the weak DKP condition with vanishing trace, if

$$\begin{aligned} d\mu _A(x, r) = \alpha _A(x,r)^2 \frac{ dx dr}{r} \end{aligned}$$

(6.16)

is a Carleson measure in ${{\mathbb {R}}}^{n+1}_+$ with vanishing trace (see Definition 2.6). Let u be a solution to the elliptic equation

$$\begin{aligned} -{{\,\mathrm{div}\,}}(A(x,t) \nabla u) = 0 \text { in } \Omega . \end{aligned}$$

We consider the flattening map

$$\begin{aligned} \Phi : (y,s) \in {{\mathbb {R}}}^{n+1}_+ \mapsto (y,s+\varphi (y)) =: (x,t) \in \Omega , \end{aligned}$$

and a function ${\tilde{u}}: {{\mathbb {R}}}^{n+1}_+ \rightarrow {{\mathbb {R}}}$ defined by ${\tilde{u}}(y,s) := u \circ \Phi (y,s)$. A simple computation shows that ${\tilde{u}}$ is the solution to the elliptic operator $-{{\,\mathrm{div}\,}}(B(y,s) \nabla {\tilde{u}}) = 0$ in ${{\mathbb {R}}}^{n+1}_+$, where the coefficient matrix B(y, s) is given by

$$\begin{aligned} B(y,s)&= \det D\Phi \cdot \left( D\Phi (y,s) \right) ^{-1} A(\Phi (y,s)) \left( D\Phi ^T(y,s) \right) ^{-1} \nonumber \\&= \begin{pmatrix} {{\,\mathrm{Id}\,}}_{n} &{}\quad 0 \\ \left( -\nabla \varphi (y) \right) ^T &{}\quad 1 \end{pmatrix} A(\Phi (y,s)) \begin{pmatrix} {{\,\mathrm{Id}\,}}_{n} &{}\quad -\nabla \varphi (y) \\ 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

(6.17)

We may define the $L^2$-oscillation of the matrix B as in (6.15), except to replace the integration region by the corresponding Whitney region in ${{\mathbb {R}}}^{n+1}_+$

$$\begin{aligned} W(x_0, r) := \Delta _r(x_0) \times \left( \frac{r}{2}, r \right] . \end{aligned}$$

Let $A_0$ be a constant coefficient matrix which achieve the infimum for $\alpha _A(x_0, r)$. In particular, $A_0$ has the same constants of ellipticity as $A(\cdot )$. We define

$$\begin{aligned} B_0 := \begin{pmatrix} {{\,\mathrm{Id}\,}}_{d-1} &{}\quad 0 \\ \left( -\nabla \varphi (x_0) \right) ^T &{}\quad 1 \end{pmatrix} A_0 \begin{pmatrix} {{\,\mathrm{Id}\,}}_{d-1} &{}\quad -\nabla \varphi (x_0) \\ 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

For any $(y,s) \in W(x_0,r)$, by the formula (6.17) as well as (1.4) we get

$$\begin{aligned} \left| B(y,s) - B_0 \right|\lesssim & {} |A(\Phi (y,s)) - A_0| + |\nabla \varphi (y) - \nabla \varphi (x_0) ||A_0|\\\lesssim & {} |A(\Phi (y,s)) - A_0| + \theta (r), \end{aligned}$$

where the constant depends on $\Vert A(x,t)\Vert _\infty$. Therefore

$$\begin{aligned} \left| \alpha _B(x_0,r) \right| ^2&\le \frac{1}{|W(x_0,r)|} \iint _{W(x_0,r)} |B(y,s) - B_0|^2 dy ds \nonumber \\&\lesssim \frac{1}{|W(x_0,r)|} \iint _{W(x_0,r)} |A(\Phi (y,s)) - A_0|^2 dy ds + \theta (r)^2 \nonumber \\&\lesssim \frac{1}{|W_\Omega (x_0,r)|} \iint _{W_\Omega (x_0, r)} |A(x,t) - A_0|^2 dx dt + \theta (r)^2 \nonumber \\&= \left| \alpha _A(x_0, r) \right| ^2 + \theta (r)^2. \end{aligned}$$

(6.18)

In the penultimate inequality of (6.18), we use the fact that $\Phi (W(x_0,r)) = W_\Omega (x_0, r)$.

Similarly to (6.16), we define

$$\begin{aligned} d\mu _B(x,r) := \alpha _B(x,r)^2 \frac{dx dr}{r}. \end{aligned}$$

Then we may compute its Carleson norm on each surface ball $\Delta \subset \partial {{\mathbb {R}}}^{n+1}_+$ and

$$\begin{aligned} \Vert \mu _B\Vert _{{\mathcal {C}}(\Delta )} \lesssim \Vert \mu _A\Vert _{{\mathcal {C}}(\Delta )} + \int _0^{r_\Delta } \theta (r)^2 \frac{dr}{r}. \end{aligned}$$

In particular, if $\theta$ satisfies the square Dini condition, then

$$\begin{aligned} \Vert \mu _B\Vert _{{\mathcal {C}}(\Delta )} \rightarrow 0\quad \text { as } r_\Delta \rightarrow 0. \end{aligned}$$

However $\mu _B$ may not be a Carleson measure at large scales because of the extra $\theta (r)^2$ term. To remedy that, let $R>0$ be fixed and we use a similar construction as in Lemma 6.2 to define a new coefficient matrix ${\widetilde{B}}(\cdot )$, so that ${\widetilde{B}} \equiv B$ in $\Gamma (0,R/2)$, and ${\widetilde{B}}(\cdot )$ is a constant coefficient matrix in ${{\mathbb {R}}}^{n+1}_+ \setminus \Gamma (0,R)$. To be more precise, let ${\widetilde{B}}_0$ be a constant matrix which achieves the minimum of $\gamma (0, 100cR)$ for the matrix $B(\cdot )$. We define

$$\begin{aligned} {\widetilde{B}}(y,s) = \mathbb {1}_{\Gamma (0,R)}(y,s) \left[ (1- f(|y|))B(y,s) + f(|y|){\widetilde{B}}_0 \right] +\mathbb {1}_{\left( \Gamma (0,R)\right) ^c}(y,s) {\widetilde{B}}_0, \end{aligned}$$

where f is a piece-wise linear function defined as in (6.6). Then Lemma 6.2 implies ${\widetilde{B}}(\cdot )$ is indeed a Carleson measure with vanishing trace in ${{\mathbb {R}}}^{n+1}_+$.

Let $\omega _\Omega ^{X_0}$ denote the elliptic measures corresponding to the matrix A in $\Omega$. Let $\omega ^{Y_0}$ and ${\widetilde{\omega }}^{Y_0}$ denote the elliptic measure corresponding to the matrix B and ${\widetilde{B}}$, respectively, in ${{\mathbb {R}}}^{n+1}_+$. Theorem 1.2 gives that ${\widetilde{\omega }}^{Y_0} \ll {\mathcal {L}}^n = dx$ and the Poisson kernel ${\widetilde{k}}(x) := \frac{d{\widetilde{\omega }}^{Y_0}}{dx}(x)$ satisfies $\log {\widetilde{k}} \in VMO_{loc}({{\mathbb {R}}}^n)$. Similar to the discussions before (6.14), this implies that $\omega ^{Y_0} \ll {\mathcal {L}}^n$ in B(0, R/2), and moreover the Poisson kernel $k(x) = \frac{d\omega ^{Y_0}}{dx}(x)$ satisfies $\log k \in VMO({{\mathbb {R}}}^n \cap B(0, R/2))$.

A simple change of variable shows that

$$\begin{aligned} \omega _\Omega ^{X_0} (B_r(x,\varphi (x))) = \omega ^{\Phi ^{-1}(X_0)}(\Phi ^{-1}(B_r(x,\varphi (x)))). \end{aligned}$$

(6.19)

Besides, for each $x \in \overline{B(0, R/2)}$ there exists a constant $M>1$ which only depend on $\Vert \nabla \varphi \Vert _{L^\infty (\overline{B(0,R)})}$ such that

$$\begin{aligned} B_{r/M} (x, 0) \subset \Phi ^{-1}(B_r(x,\varphi (x))) \subset B_{Mr}(x,0). \end{aligned}$$

Let $\Pi _n: {{\mathbb {R}}}^{n+1} \rightarrow \partial {{\mathbb {R}}}^{n+1}_+ \approx {{\mathbb {R}}}^n$ denote the projection onto ${{\mathbb {R}}}^n$. Using (6.19), the fact that $\partial \Omega$ is a graph, and the Lebesgue differentiation theorem, we have

Therefore $\omega _\Omega ^{X_0} \ll {\mathcal {H}}^n_{\partial \Omega }$ and the corresponding Poisson kernel in $\Omega$

$$\begin{aligned} k_\Omega (x,\varphi (x)) := \frac{ d\omega _\Omega ^{X_0}}{ d{\mathcal {H}}^n_{\partial \Omega }}(x,\varphi (x)) \end{aligned}$$

satisfies

$$\begin{aligned} k_\Omega (x,\varphi (x)) = \frac{k(x)}{ \sqrt{1+|\nabla \varphi (x)|^2}}. \end{aligned}$$

Since $\sqrt{1+|\nabla \varphi (x)|^2}$ is continuous and (locally) bounded above and below, it follows that $\log k_\Omega \in VMO_{loc}(\partial \Omega \cap B(0, R/3))$. Therefore we have proven Theorem 6.13.