Ball-mean oscillation bound by the Riesz potential of the gradient

Spherical oscillation bound for a smooth function. Let $v\in C^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$, $B:=B(x,r)$, and fix $z\in B$ and $\rho >0$. For $y\in B$ the chain rule and the fundamental theorem [F3] applied to $t\mapsto v(ty+(1-t)z)$ give $v(y)-v(z)={\int }_0^{1}Dv(ty+(1-t)z)\cdot (y-z)dt$, hence $\mathrm{\mid }v(y)-v(z)\mathrm{\mid }\le \mathrm{\mid }y-z\mathrm{\mid }{\int }_0^1\mathrm{\mid }Dv(ty+(1-t)z)\mathrm{\mid }dt$. Writing $S_{\rho }:=\mathrm{\partial }B(z,\rho )$ for the sphere equipped with its surface measure, and using that the homothety $y\mapsto ty+(1-t)z$ maps $S_{\rho }$ onto $S_{t\rho }$ with surface element scaled by $t^{n-1}$; its image of $B\cap S_{\rho }$ is contained in $B\cap S_{t\rho }$ by convexity, so enlargement gives the inequality below (the surface measures are the polar measures of cones, so this is the linear change-of-variables property of [F1] and [F4]), ${\int }_{B\cap S_{\rho }}\mathrm{\mid }v(y)-v(z)\mathrm{\mid }dS(y)\le \rho {\int }_0^1{\int }_{B\cap S_{\rho }}\mathrm{\mid }Dv(ty+(1-t)z)\mathrm{\mid }dS(y)dt\le \rho {\int }_0^1{t}^{1-n}{\int }_{B\cap S_{t\rho }}\mathrm{\mid }Dv(w)\mathrm{\mid }dS(w)dt$. Since $\mathrm{\mid }w-z\mathrm{\mid }=t\rho$ on $S_{t\rho }$, the inner integral equals ${\rho }^{n-1}{t}^{n-1}{\int }_{B\cap S_{t\rho }}\mathrm{\mid }Dv(w)\mathrm{\mid }\mathrm{\mid }z-w{\mathrm{\mid }}^{1-n}dS(w)$; substituting $s=t\rho$, so that $dt=ds\mathrm{/}\rho$, the last display becomes ${\rho }^{n-1}{\int }_0^{\rho }{\int }_{B\cap S_s}\mathrm{\mid }Dv(w)\mathrm{\mid }\mathrm{\mid }z-w{\mathrm{\mid }}^{1-n}dS(w)ds={\rho }^{n-1}{\int }_{B\cap B(z,\rho )}\mathrm{\mid }Dv(w)\mathrm{\mid }\mathrm{\mid }z-w{\mathrm{\mid }}^{1-n}dw$, the final equality being the polar-coordinate formula [F1] for the nonnegative function $w\mapsto \mathrm{\mid }Dv(w)\mathrm{\mid }\mathrm{\mid }z-w{\mathrm{\mid }}^{1-n}$ on $B\cap B(z,\rho )$ (whose singularity at $w=z$ is integrable; a single point is null).

The oscillation bound for a smooth function. Let $v\in C^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ and keep $B=B(x,r)$. Since $u_B$ is the normalized integral, $\mathrm{\mid }v(z)-v_B\mathrm{\mid }=\mathrm{\mid }B{\mathrm{\mid }}^{-1}\mathrm{\mid }{\int }_B(v(z)-v(y))dy\mathrm{\mid }$; writing the integral over $B$ in polar coordinates around $z$ and using $B\subseteq B(z,2r)$, step 1.1 gives $\mathrm{\mid }v(z)-v_B\mathrm{\mid }\le \mathrm{\mid }B{\mathrm{\mid }}^{-1}{\int }_0^{2r}{\int }_{B\cap S_{\rho }}\mathrm{\mid }v(y)-v(z)\mathrm{\mid }dS(y)d\rho \le \mathrm{\mid }B{\mathrm{\mid }}^{-1}{\int }_0^{2r}{\rho }^{n-1}d\rho {\int }_B\mathrm{\mid }Dv(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy$ for every $z\in B$. By [F2], $\mathrm{\mid }B\mathrm{\mid }=\sigma (S^{n-1})r^n\mathrm{/}n$ and ${\int }_0^{2r}{\rho }^{n-1}d\rho =(2r{)}^n\mathrm{/}n$, so $\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\int }_0^{2r}{\rho }^{n-1}d\rho =2^n\mathrm{/}\sigma (S^{n-1})=:c(n)$; hence $\mathrm{\mid }v(z)-v_B\mathrm{\mid }\le c(n){\int }_B\mathrm{\mid }Dv(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy$ for every $z\in B$.

First pass to the Sobolev class on an inner ball $B_j:=B(x,r(1-1\mathrm{/}j))$, $j\ge 2$. Its closure lies in $B$, so [F6] supplies smooth mollifications $u_i\to u$ in $W^{1,p}(B_j)$. By [F8] their $B_j$ means converge. Apply step 2.1 on $B_j$. The potential operator on $B_j$ satisfies [F5], and $\mid {\int }_{B_j}\mathrm{\mid }D{u}_i(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy-{\int }_{B_j}\mathrm{\mid }Du(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy\mid \le {\int }_{B_j}\mathrm{\mid }D{u}_i-Du\mathrm{\mid }(y)\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy$ tends to zero in $L^p(B_j)$ by [F5]. Successive almost-everywhere subsequences from [F6] for $u_i$ and these potentials therefore give $\mathrm{\mid }u(z)-u_{B_j}\mathrm{\mid }\le c(n){\int }_{B_j}\mathrm{\mid }Du(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy$ for almost every $z\in B_j$.

Take the union of the countably many exceptional null sets from step 3.1. For $z$ outside this union, $z\in B_j$ for every sufficiently large $j$, and each right side is at most $c(n){\int }_B\mathrm{\mid }Du(y)\mathrm{\mid }\mathrm{\mid }z-y{\mathrm{\mid }}^{1-n}dy$. Since $u\in L^1(B)$, dominated convergence [F6] gives $u_{B_j}\to u_B$. Letting $j\to \mathrm{\infty }$ proves the asserted inequality on $B$, without any approximation claim at its boundary.