Interior gradient bound for Poisson solutions

Work under [F9], let $x_0\in B_{r\mathrm{/}2}(a)$ be arbitrary, and put $\rho :=r\mathrm{/}4$, so that $\overline{B_{\rho }(x_0)}\subset B_r(a)$; write $M_u:=\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)}$ and $M_f:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)}$. Both are finite: $M_u<\mathrm{\infty }$ is given, and the finite Hölder seminorm bounds $\mathrm{\mid }f(x)\mathrm{\mid }\le \mathrm{\mid }f(a)\mathrm{\mid }+[f{]}_{0,\alpha ;B_r(a)}(2r{)}^{\alpha }$ for every $x\in B_r(a)$.

Kernel integrals. By [F1] and [F2], $\mathrm{\mid }D\mathrm{\Phi }(z)\mathrm{\mid }\le C_n\mathrm{\mid }z{\mathrm{\mid }}^{1-n}$ and ${\int }_{B_s}\mathrm{\mid }D\mathrm{\Phi }(z)\mathrm{\mid }dz\le C_{n}s$ for every $s>0$. For the fixed scale $\rho$ of step 1.1, a change of variables $z=\rho \zeta$ in the power-kernel case $n\ge 3$, and the identity $\mathrm{\Phi }(\rho \zeta )-\mathrm{\Phi }(\rho )=-(2\pi {)}^{-1}\log \mathrm{\mid }\zeta \mathrm{\mid }$ when $n=2$, give ${\int }_{B_{3\rho \mathrm{/}2}}\mathrm{\mid }\mathrm{\Phi }(z)-\mathrm{\Phi }(\rho )\mathrm{\mid }dz\le C_n{\rho }^2\mathrm{.}$ The scaled integral is finite in every dimension by polar coordinates; constants depend only on $n$.

Cutoff and normalized potential at $x_0$. By [F6] fix a smooth cutoff $\eta$ with $\eta =1$ on ${\bar{B}}_{3\rho \mathrm{/}4}(x_0)$ and $\mathrm{supp}\eta \subseteq B_{\rho }(x_0)$, and put $G:=\eta f$ on $B_{\rho }(x_0)$, extended by $0$ to ${\mathbb{R}}^n$. Then $G$ is continuous, compactly supported and has finite $\alpha$-Hölder seminorm. Define $w(x):={\int }_{{\mathbb{R}}^n}(\mathrm{\Phi }(x-y)-\mathrm{\Phi }(\rho ))G(y)dy=NG(x)-\mathrm{\Phi }(\rho ){\int }_{{\mathbb{R}}^n}G(y)dy\mathrm{.}$ By [F3] and [F4], $w\in C^2({\mathbb{R}}^n)$ and $-\mathrm{\Delta }w=G$ pointwise; the subtracted term is constant in $x$.

The remainder $h:=u-w$ is harmonic on $B_{3\rho \mathrm{/}4}(x_0)$: there $\eta =1$, so $G=f$ and $-\mathrm{\Delta }h=-\mathrm{\Delta }u+\mathrm{\Delta }w=f-G=0$ by the hypothesis and step 2.2.

Explicit form and bound for $Dw$. Fix $x\in B_{3\rho \mathrm{/}4}(x_0)$ and a coordinate $k$. For $y$ with $\mathrm{\mid }x-y\mathrm{\mid }>2\mathrm{\mid }t\mathrm{\mid }$, the real mean value theorem [F7] and $\mathrm{\mid }D\mathrm{\Phi }(z)\mathrm{\mid }\le C_n\mathrm{\mid }z{\mathrm{\mid }}^{1-n}$ give $\mid \frac{\mathrm{\Phi }(x+t{e}_k-y)-\mathrm{\Phi }(x-y)}{t}\mid \le C_n\mathrm{\mid }x-y{\mathrm{\mid }}^{1-n},$ since every point on the segment between $x-y$ and $x+t{e}_k-y$ has norm at least $\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}2$. The right side is integrable on the bounded support of $G$. On this far region the quotients converge pointwise for $y\mathrm{\ne }x$ to ${\mathrm{\partial }}_k\mathrm{\Phi }(x-y)$, so dominated convergence [F5], with the indicator of $\mathrm{\mid }x-y\mathrm{\mid }>2\mathrm{\mid }t\mathrm{\mid }$, gives convergence of the far-region integrals to $\int {\mathrm{\partial }}_k\mathrm{\Phi }(x-y)G(y)dy$. On the near region $\mathrm{\mid }x-y\mathrm{\mid }\le 2\mathrm{\mid }t\mathrm{\mid }$, the quotient integral is bounded by $\frac{M_f}{\mathrm{\mid }t\mathrm{\mid }}({\int }_{B_{2\mathrm{\mid }t\mathrm{\mid }}(t{e}_k)}\mathrm{\mid }\mathrm{\Phi }(z)\mathrm{\mid }dz+{\int }_{B_{2\mathrm{\mid }t\mathrm{\mid }}(0)}\mathrm{\mid }\mathrm{\Phi }(z)\mathrm{\mid }dz),$ which tends to zero: it is $O(\mathrm{\mid }t\mathrm{\mid })$ for $n\ge 3$ and $O(\mathrm{\mid }t\mathrm{\mid }(1+\mathrm{\mid }\log \mathrm{\mid }t\mathrm{\mid }\mathrm{\mid }))$ for $n=2$, by polar coordinates [F1, F2]. The integral of $\mathrm{\mid }D\mathrm{\Phi }(x-y)G(y)\mathrm{\mid }$ over that near region is $O(M_f\mathrm{\mid }t\mathrm{\mid })$ by step 2.1. Hence ${\mathrm{\partial }}_{k}w(x)=\int {\mathrm{\partial }}_k\mathrm{\Phi }(x-y)G(y)dy$. At $x=x_0$, this yields $\mathrm{\mid }Dw(x_0)\mathrm{\mid }\le M_f{\int }_{B_{\rho }(x_0)}\mathrm{\mid }D\mathrm{\Phi }(x_0-y)\mathrm{\mid }dy\le C_n{M}_f\rho$. For $x\in B_{\rho \mathrm{/}2}(x_0)$, the normalized kernel and the inclusion $B_{\rho }(x_0)\subset B_{3\rho \mathrm{/}2}(x)$ give $\mathrm{\mid }w(x)\mathrm{\mid }\le M_f{\int }_{B_{3\rho \mathrm{/}2}}\mathrm{\mid }\mathrm{\Phi }(z)-\mathrm{\Phi }(\rho )\mathrm{\mid }dz\le C_n{M}_f{\rho }^2$ by step 2.1.

Harmonic gradient bound. Since $h$ is harmonic on $B_{3\rho \mathrm{/}4}(x_0)\supseteq \overline{B_{\rho \mathrm{/}2}(x_0)}$, apply [F8] separately to each coordinate derivative ${\mathrm{\partial }}_{i}h(x_0)$, $0\le i<n$. The vector norm satisfies $\mathrm{\mid }Dh(x_0)\mathrm{\mid }\le \sqrt{n}{\max }_i\mathrm{\mid }{\mathrm{\partial }}_{i}h(x_0)\mathrm{\mid }$, so, absorbing $\sqrt{n}$ into $C_n^{\mathrm{\prime }}$, $\mathrm{\mid }Dh(x_0)\mathrm{\mid }\le C_n^{\mathrm{\prime }}(2\mathrm{/}\rho ){\sup }_{B_{\rho \mathrm{/}2}(x_0)}\mathrm{\mid }h\mathrm{\mid }\le C_n^{\mathrm{\prime }}(2\mathrm{/}\rho )(M_u+C_n{M}_f{\rho }^2)$ by steps 3.1 and 3.2.

Combining steps 3.2 and 4.1 at the point $x_0$, $\mathrm{\mid }Du(x_0)\mathrm{\mid }\le \mathrm{\mid }Dw(x_0)\mathrm{\mid }+\mathrm{\mid }Dh(x_0)\mathrm{\mid }\le C_n{M}_f\rho +C_n^{\mathrm{\prime }}(2\mathrm{/}\rho )(M_u+C_n{M}_f{\rho }^2)\le C_n^{\mathrm{\prime }\mathrm{\prime }}({\rho }^{-1}{M}_u+\rho M_f)=C_n^{\mathrm{\prime }\mathrm{\prime }}(4r^{-1}{M}_u+\frac{1}{4}r{M}_f)$, and absorbing the numerical factors into $C_n^{\mathrm{\prime }\mathrm{\prime }}$ gives $\mathrm{\mid }Du(x_0)\mathrm{\mid }\le C_n(r^{-1}{M}_u+r{M}_f)$.

Since $x_0\in B_{r\mathrm{/}2}(a)$ was arbitrary, taking the supremum over $x_0$ gives $\mathrm{\parallel }Du{\mathrm{\parallel }}_{\mathrm{\infty };B_{r\mathrm{/}2}(a)}\le C_n(r^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)}+r\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)})$, the displayed estimate; the constants encountered in steps 2.1, 3.2 and 4.1 depend only on $n$, and the Hölder seminorm of $f$ entered only through the qualitative $C^2$ clause of [F4] used to define $w$ and $h$, never through a quantitative bound. The argument covers complex-valued $u$ and $f$ by applying the real case to real and imaginary parts.