Interior derivative estimates for harmonic functions

Work under [F8] and suppose first that $u$ is real-valued. Put $\rho :=r\mathrm{/}4>0$, so that $\overline{B_{2\rho }(x)}\subset B_r(x)\subset \mathrm{\Omega }$ and $u$ is harmonic, hence $C^2$, on a neighbourhood of $\overline{B_{2\rho }(x)}$.

Mean-value bound on the inner sphere. For $y\in \mathrm{\partial }{B}_{\rho }(x)$ and $w\in B_{\rho }(y)$ we have $\mathrm{\mid }w-x\mathrm{\mid }\le \mathrm{\mid }w-y\mathrm{\mid }+\mathrm{\mid }y-x\mathrm{\mid }<2\rho =r\mathrm{/}2<r$, so $B_{\rho }(y)\subset B_r(x)$; by [F1] and [F7], $\mathrm{\mid }u(y)\mathrm{\mid }=\mid \frac{1}{\mathrm{\mid }{B}_{\rho }(y)\mathrm{\mid }}{\int }_{B_{\rho }(y)}u\mid \le \frac{1}{\mathrm{\mid }{B}_{\rho }\mathrm{\mid }}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }=\frac{n}{{\omega }_{n-1}{\rho }^n}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }$.

Representation on the inner ball. For $n\ge 3$ put $g:=u{\mathrm{\mid }}_{\mathrm{\partial }{B}_{\rho }(x)}\in C(\mathrm{\partial }{B}_{\rho }(x))$ and note that $u\in C^2(B_{\rho }(x))\cap C(\overline{B_{\rho }(x)})$ is harmonic with trace $g$; the uniqueness clause of [F3] gives $u(z)=U_g(z)={\int }_{\mathrm{\partial }{B}_{\rho }(x)}{P}_{\rho ,x}(z,y)g(y)d{S}_y$ for $z\in B_{\rho }(x)$. For $n=2$: $u$ is continuous on $\mathrm{\Omega }$ and has the ball mean value property by [F1], so [F2] makes it $C^{\mathrm{\infty }}$; its restriction $g$ to the sphere is then real and $C^{\mathrm{\infty }}$, and $u$ is a $C^{\mathrm{\infty }}\cap C(\overline{B_{\rho }(x)})$ harmonic function with trace $g$, so uniqueness in [F4] gives $u(z)={\int }_{\mathrm{\partial }{B}_{\rho }(x)}\frac{{\rho }^2-\mathrm{\mid }z-x{\mathrm{\mid }}^2}{\rho {\omega }_{n-1}\mathrm{\mid }z-y{\mathrm{\mid }}^n}g(y)d{S}_y$ for $z\in B_{\rho }(x)$. Thus in both dimensions $u$ on $B_{\rho }(x)$ is the Poisson integral of $g$ with the same kernel.

Kernel derivative bound. Write $z=x+\rho \zeta$ and $y=x+\rho \eta$ with $\mathrm{\mid }\eta \mathrm{\mid }=1$; the kernel is $P_{\rho ,x}(z,y)={\rho }^{1-n}(1-\mathrm{\mid }\zeta {\mathrm{\mid }}^2)\mathrm{/}({\omega }_{n-1}\mathrm{\mid }\zeta -\eta {\mathrm{\mid }}^n)$, whose denominator is bounded below on the compact set $\{\mathrm{\mid }\zeta \mathrm{\mid }\le 1\mathrm{/}2\}\times \{\mathrm{\mid }\eta \mathrm{\mid }=1\}$ by $2^{-n}$. For every multi-index $\alpha$, the partial derivatives $D_{\zeta }^{\alpha }[(1-\mathrm{\mid }\zeta {\mathrm{\mid }}^2)\mathrm{\mid }\zeta -\eta {\mathrm{\mid }}^{-n}]$ are continuous by [F6] on that compact set and hence bounded in modulus by a constant $c_{n,\alpha }$ by [F7]; rescaling gives, for $\mathrm{\mid }\zeta \mathrm{\mid }\le 1\mathrm{/}2$, $\mid D_z^{\alpha }{P}_{\rho ,x}(z,y)\mid ={\rho }^{1-n-\mathrm{\mid }\alpha \mathrm{\mid }}\mid D_{\zeta }^{\alpha }[(1-\mathrm{\mid }\zeta {\mathrm{\mid }}^2)\mathrm{\mid }\zeta -\eta {\mathrm{\mid }}^{-n}]\mid \mathrm{/}{\omega }_{n-1}\le C_{n,\alpha }{\rho }^{1-n-\mathrm{\mid }\alpha \mathrm{\mid }}$.

Derivatives of $u$. By step 2.1, $u(z)={\int }_{\mathrm{\partial }{B}_{\rho }(x)}{P}_{\rho ,x}(z,y)g(y)d{S}_y$. On $\overline{B_{\rho \mathrm{/}2}(x)}\times \mathrm{\partial }{B}_{\rho }(x)$ every ordered $z$-derivative of the smooth kernel is continuous and bounded, by the compactness argument of step 2.2. Multiplying by the bounded continuous $g$ gives Borel integrable slices; the next coordinate derivative has an integrable constant majorant on the finite sphere. Thus [F5] applies on each sufficiently small open coordinate interval, with no exceptional points. Induction over ordered coordinate derivatives, with dominated convergence for their continuity, gives $D^{\alpha }u(z)={\int }_{\mathrm{\partial }{B}_{\rho }(x)}{D}_z^{\alpha }{P}_{\rho ,x}(z,y)g(y)d{S}_y$ for $z\in B_{\rho \mathrm{/}2}(x)$, using the canonical order for $D^{\alpha }$. At $z=x$ step 2.2 then yields $\mathrm{\mid }{D}^{\alpha }u(x)\mathrm{\mid }\le C_{n,\alpha }{\rho }^{1-n-\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\partial }{B}_{\rho }(x)}\mathrm{\mid }u(y)\mathrm{\mid }d{S}_y$.

Bounding the boundary integral by the sphere area, step 3.1 and [F7] give $\mathrm{\mid }{D}^{\alpha }u(x)\mathrm{\mid }\le C_{n,\alpha }{\rho }^{1-n-\mathrm{\mid }\alpha \mathrm{\mid }}\cdot {\omega }_{n-1}{\rho }^{n-1}\cdot {\sup }_{y\in \mathrm{\partial }{B}_{\rho }(x)}\mathrm{\mid }u(y)\mathrm{\mid }=C_{n,\alpha }{\omega }_{n-1}{\rho }^{-\mathrm{\mid }\alpha \mathrm{\mid }}{\sup }_{\mathrm{\partial }{B}_{\rho }(x)}\mathrm{\mid }u\mathrm{\mid }$.

Substituting the mean-value bound of step 1.2 into step 4.1 yields $\mathrm{\mid }{D}^{\alpha }u(x)\mathrm{\mid }\le C_{n,\alpha }{\omega }_{n-1}{\rho }^{-\mathrm{\mid }\alpha \mathrm{\mid }}\cdot \frac{n}{{\omega }_{n-1}{\rho }^n}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }=C_{n,\alpha }n{\rho }^{-n-\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }=C_{n,\alpha }n{4}^{n+\mathrm{\mid }\alpha \mathrm{\mid }}{r}^{-n-\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }$, and absorbing $n{4}^{n+\mathrm{\mid }\alpha \mathrm{\mid }}$ into the constant gives the displayed estimate with a constant depending only on $n$ and $\alpha$.

For complex $u$, apply steps 1.1–5.1 to $\mathrm{R}\mathrm{e}u$ and to $\mathrm{I}\mathrm{m}u$, which are real harmonic functions on $\mathrm{\Omega }$ with $\overline{B_r(x)}\subset \mathrm{\Omega }$: $\mathrm{\mid }{D}^{\alpha }u(x)\mathrm{\mid }\le \mathrm{\mid }{D}^{\alpha }\mathrm{R}\mathrm{e}u(x)\mathrm{\mid }+\mathrm{\mid }{D}^{\alpha }\mathrm{I}\mathrm{m}u(x)\mathrm{\mid }\le C_{n,\alpha }{r}^{-n-\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{B_r(x)}(\mathrm{\mid }\mathrm{R}\mathrm{e}u\mathrm{\mid }+\mathrm{\mid }\mathrm{I}\mathrm{m}u\mathrm{\mid })\le 2C_{n,\alpha }{r}^{-n-\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{B_r(x)}\mathrm{\mid }u\mathrm{\mid }$, and $2C_{n,\alpha }$ again depends only on $n$ and $\alpha$.

Steps 5.1 and 6.1 give the estimate for real and complex $u$ with a constant independent of $u,x,r,\mathrm{\Omega }$; the value $r>0$ is unavoidable because the estimate divides by $r$, and the hypothesis $\overline{B_r(x)}\subset \mathrm{\Omega }$ was used only to place $B_{2\rho }(x)$ and the mean-value balls inside $\mathrm{\Omega }$.