A Note on Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces
Martin Kell
In this note we show how to adjust some proofs of Koskela et. al 2003 and Jiang 2011 in order to prove that in certain spaces $(X,d,\mu)$, like $RCD(K,N)$-spaces, every Sobolev function with local $L^{p}$-Laplacian and $p>\dim\mu$ is locally Lipschitz continuous.
lip-13-07-17.tex
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\begin{document}

\title[A Note on Lipschitz Continuity of Solutions of Poisson Equations]{A Note on Lipschitz Continuity of Solutions of Poisson Equations
in Metric Measure Spaces}


\author{Martin Kell}


\email{mkell@mis.mpg.de}


\address{Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22,
D-04103 Leipzig, Germany}


\thanks{The author wants to thank is advisor Prof. J�rgen Jost and the MPI-MiS
for providing an inspiring research environment and the IMPRS ``Mathematics
in the Sciences'' for financial support.}


\keywords{$L^{p}$-Laplacian, Lipschitz regularity, $RCD(K,N)$-spaces}
\begin{abstract}
In this note we show how to adjust some proofs of Koskela et. al 2003
and Jiang 2011 in order to prove that in certain spaces $(X,d,\mu)$,
like $RCD(K,N)$-spaces, every Sobolev function with local $L^{p}$-Laplacian
and $p>\dim\mu$ is locally Lipschitz continuous.
\end{abstract}
\maketitle
In \cite{Koskela2003} and \cite{Jiang2011} Koskela et. al. and Jiang
showed that functions with local $L^{p}$-Laplacian, in particular
harmonic functions, are locally Lipschitz continuous if the space
is locally Ahlfors regular, a local uniform Poinca\'{e} inequality
holds and its heat flows admits a kind of Sobolev-Poincar\'{e} inequality.
In this note, we want to show that Ahlfors regularity can be replaced
by local uniform doubling and a weak upper bound on the volume growth.
In particular, every $RCD(K,N)$-space satisfies these conditions
(see \cite{Gigli2012} for nicely developed calculus and further references).
Since the proofs are almost the same as the ones in \cite{Jiang2011}
we will only show the necessary adjustments and refer to \cite{Koskela2003,Jiang2011}
for notation and precise statement.
\begin{rem*}
After this note was finished, we learnt that Jiang \cite{Jiang2013}
worked on an extension of \cite{Jiang2011}. Under similar assumptions,
but without the upper bound on the volume growth, he shows that functions
with local $L^{\infty}$-Laplacian are locally Lipschitz continuous
and additionally gives some gradient bounds for those functions. 
\end{rem*}
Throughout this note we assume that $(X,d,\mu)$ is a complete metric
measure space which is infinitesimal Hilbertian (see Gigli \cite{Gigli2012}).
For those spaces the Cheeger energy 
\[
\operatorname{Ch}(u)=\int|\nabla u|_{w}^{2}d\mu
\]
is a Dirichlet form, where $|\nabla u|_{w}$ is the minimal weak upper
gradient for $u$. We will drop the subscript $w$ and use the following
notation 
\[
\mathcal{E}(u,v)=\int\langle\nabla u,\nabla v\rangle d\mu
\]
such that 
\[
\mathcal{E}(u,u)=\operatorname{Ch}(u).
\]
Because $\operatorname{Ch}$ is a Dirichlet form the Sobolev space
\[
W^{1,2}(X)=\{u\in L^{2}(X)|\operatorname{Ch}(u)<\infty\}
\]
is a Hilbert space and for a dense subset of $W^{1,2}(X)$ we can
define the Laplacian $\Delta u\in L^{2}$ such that 
\[
\mathcal{E}(u,v)=-\int v\Delta ud\mu.
\]
Note that whereas $\langle\nabla u,\nabla v\rangle$ is a well-defined
object in $L^{1}$, $\nabla u$ is not defined. 
\begin{rem*}
Koskela et. al. \cite{Koskela2003} and Jiang \cite{Jiang2011} use
Cheeger derivatives, which are not necessary to develop the regularity
theory. Furthermore, it is not necessary to start with the Cheeger
energy. If we assume that a space is equipped with a Dirichlet form
$\mathcal{F}$ such that the induced metric $d_{\mathcal{F}}$ is
Lipschitz equivalent to the original metric, then the associated Laplacian
satisfies the Leibniz rule and a similar regularity theory could be
developed if doubling, Poincar\'{e} and Sobolev-Poincar\'{e} holds.
Furthermore, according to \cite{Koskela2012} if $\mathcal{F}$ satisfies
the Bakry-\'Emery condition then it is the Cheeger energy to $d_{\mathcal{F}}$
$ $ and the space $(X,d_{\mathcal{F}},\mu)$ is an $RCD(K,\infty)$-space.
However, if the Sobolev-Poincar\'{e} inequality holds uniformly independent
of $x$ then according to \cite{Bakry1997} the Bakry-\'Emery condition
holds.
\end{rem*}
We will assume the following doubling condition on the measure $\mu$:
for fixed $R>0$ and all $0<r<R$ there are constants $d_{R}>0$ and
$C_{R}>0$ such that 
\[
d_{R}r^{Q}\le\mu(B_{r}(x))
\]
 and 
\[
\mu(B_{2r}(x))\le C_{R}\mu(B_{r}(x)).
\]
Since we only want to show which adjustments are necessary, we will
further assume $Q\ge2$. This is no limitation on the spaces, since
every space satisfying the volume growth with $d_{R}$ and $Q$ will
satisfy it with $Q'\ge Q$ and some different $d_{R}^{'}$. We remark
that the case $Q\in(1,2)$ can be treated similar to \cite{Jiang2011},
but to keep this note short and simple we leave the details to the
reader. Furthermore, $d_{R}$ could depend on $x$ if it satisfies
certain growth condition (see below).

Every infinitesimal Hilbertian space $X$ admits a natural heat (semi)flow
$T_{t}:L^{2}\to L^{2}$ which is the gradient (semi)flow of its Cheeger
energy. Furthermore, this heat flow admits a heat kernel. In order
to get bounds on the heat kernel and H�lder regularity of functions
with $L^{p}$-Laplacian we also assume that $X$ satisfies a local
uniform Poincar\'{e} inequality. Under these assumptions Sturm \cite{Sturm1995}
showed that the following bounds hold:
\[
\frac{1}{C}\mathfrak{v}(t,x,y)e^{-\frac{d(x,y)^{2}}{C_{1}t}}\le p_{t}(x,y)\le C\mathfrak{v}(t,x,y)e^{-\frac{d(x,y)^{2}}{C_{2}t}}
\]
with 
\[
\mathfrak{v}(t,x,y):=\left(\mu(B_{\sqrt{t}}(x))\right)^{-\frac{1}{2}}\left(\mu(B_{\sqrt{t}}(y))\right)^{-\frac{1}{2}}.
\]


The following lemma will be useful to get estimates for $p_{t}(x,y)$
and $p_{lt}(x,y)$.
\begin{lem}
For any $l>0$ and any $(lt)^{2}<R$ the following is true
\[
\mathfrak{v}(t,x,y)\le\tilde{C}_{l,R}\mathfrak{v}(lt,x,y).
\]
\end{lem}
\begin{proof}
Note that by the doubling property we have 
\[
\mu(B_{\sqrt{lt}}(x))\le C_{l,R}\mu(B_{\sqrt{t}}(x))
\]
and thus 
\begin{eqnarray*}
\mathfrak{v}(t,x,y) & = & \left(\mu(B_{\sqrt{t}}(x))\mu(B_{\sqrt{t}}(y))\right)^{-\frac{1}{2}}\\
 & \le & \tilde{C}_{l,R}\left(\mu(B_{\sqrt{lt}}(x))\mu(B_{\sqrt{lt}}(y))\right)^{-\frac{1}{2}}=\tilde{C}_{l,R}\mathfrak{v}(lt,x,y).
\end{eqnarray*}
\end{proof}
\begin{cor}
\label{cor:heat bound}For any positive function $\varphi$ and $l=2C_{1}/C_{2}$
there is a constant $\hat{C}>0$ such that the following holds
\[
\int\varphi(y)p_{t}(x,y)d\mu(y)\le\hat{C}\int\varphi e^{-\frac{d(x,y)^{2}}{2C_{1}t}}p_{lt}(x,y)d\mu(y).
\]
\end{cor}
\begin{proof}
By the upper estimate for $p_{t}$ and the lower estimate for $p_{lt}$
we have 
\begin{eqnarray*}
\int\varphi(y)p_{t}(x,y)d\mu(y) & \le & C\int\varphi(y)\mathfrak{v}(t,x,y)e^{-\frac{d(x,y)^{2}}{2C_{1}t}}e^{-\frac{d(x,y)^{2}}{2C_{1}t}}d\mu(y)\\
 & \le & C\cdot\tilde{C}\int\varphi(y)e^{-\frac{d(x,y)^{2}}{2C_{1}t}}\mathfrak{v}(lt,x,y)e^{-\frac{d(x,y)^{2}}{C_{2}(lt)}}d\mu(y)\\
 & \le & C^{2}\cdot\tilde{C}\int\varphi(y)e^{-\frac{d(x,y)^{2}}{2C_{1}t}}p_{lt}(x,y)d\mu(y).
\end{eqnarray*}
\end{proof}
\begin{rem*}
Because of the term $e^{-\frac{d(x,y)^{2}}{2C_{1}t}}$, if $t$ is
chosen sufficiently small, we still get the same estimate with $\hat{C}$
depending also on $x$ if the doubling constants depend on the chosen
point but satisfy some growth condition, i.e. for each $x\in X$ there
is a $C_{R,x}$ such that 
\[
\mu(B_{2r}(x))<C_{R,x}\mu(B_{r}(x))
\]
and 
\[
C_{R,y}\le C_{R,x}e^{Dd(x,y)^{2}}.
\]

\end{rem*}
Note that in the following lemma, only H�lder continuity of functions
with $L^{p}$-Laplacian will be used further below. 
\begin{lem}
[{\cite[2.1,2.2]{Jiang2011}}]\label{lem:hoelder}Assume that $u\in W_{loc}^{1,2}(X)$
such that $\Delta u=f$ on $B_{2r}(x_{0})$ for some $f\in L_{loc}^{p}(X)$
for $p>Q\ge2$ such that $B_{2r}(x_{0})\subset\subset X$ then
\[
\sup_{B_{r}(x_{0})}|u|\le C\left\{ r^{-Q/2}\|u\|_{L^{2}(B_{2r}(x_{0}))}+r^{2-Q/p}\|f\|_{L^{p}(B_{2r}(x_{0})}\right\} .
\]
Furthermore, there is a $C'>0$ such that $u$ satisfy the above for
$B_{2R}(x_{0})$ and such that for all $0<r<R$ 
\[
\int_{B_{r}(x_{0})}|\nabla u|^{2}d\mu\le C'R^{2+Q(1-\frac{2}{p})}\|f\|_{L^{p}(B_{2R}(x_{0})}^{2}+\frac{C'}{(R-r)^{2}}\|u\|_{L^{2}(B_{2R}(x_{0}))}^{2}.
\]


In addition, $u$ is locally H�lder continuous with constants only
depending (locally) on $u$ and $f$ and the doubling constants.\end{lem}
\begin{proof}
The first two parts follow from \cite[Proposition 2.1, Proposition 2.2]{Jiang2011}
by noting that 
\[
\mu(B_{r}(x_{0}))^{-1/p}\le d_{R}^{-1/p}r^{-Q/p}.
\]
The last part follows from \cite[Theorem 5.13]{Biroli1995}.\end{proof}
\begin{rem*}
Note that by the same argument (i.e. ``lower Ahlfors regularity'')
Lemma 3.2 and 3.3 in \cite{Koskela2003} (resp. Lemma 2.4 and 2.5
in \cite{Jiang2011}) hold without changing the proofs. 
\end{rem*}
The proof of \cite[Proposition 3.4]{Koskela2003} (used also in \cite[Lemma 2.1]{Jiang2011})
only uses a very weak form of the ``upper Ahlfors regularity'' in
order to show that 
\[
r^{-1}(r\mu(B_{2r}(x)))^{\frac{1}{2}}=\left(\frac{\mu(B_{2r}(x))}{r^{\frac{1}{2}}}\right)^{\frac{1}{2}}
\]
is bounded for $r<1$ and converges to $0$. It is easy to see that
this holds if for some $\alpha>1$ and $R>0$ 
\[
\sup_{0<r<R}\frac{\mu(B_{r}(x))}{r^{\alpha}}<\infty,
\]
i.e. for some $D_{R}$ and all $0<r<R$ we have $\mu(B_{r}(x))<D_{R}r^{\alpha}$.
In the proposition below, we will show that an even weaker condition
is enough to prove the statement.

We say that $\phi:[0,T]\times X\to\mathbb{R}$ is a test function
if it is H�lder continuous, $\phi(t,\cdot)=\phi_{t}\in W^{1,2}(X),$
$(t,x)\mapsto|\nabla\phi(t,\cdot)|(x)\in L^{2}([0,T]\times X)$ and
$\phi(\cdot,x)$ is absolutely continuous on $[0,T]$ for $\mu$-almost
all $x\in X$.
\begin{prop}
Assume that for $\mu$-almost every $x$ and each $0<\alpha<1$ there
is a $C=C(x,\alpha)>0$ such that 
\[
\mu(B_{r}(x))\le C\cdot r^{\alpha}
\]
Then there exists a constant $K$ such that for every (H�lder continuous)
test function $\phi$ and almost every $x\in X$
\begin{eqnarray*}
\int_{0}^{T}\int\phi_{t}\Delta p_{t}(x,\cdot)d\mu dt & := & -\int_{0}^{T}\int\langle\nabla\phi_{t},\nabla p_{t}(x,\cdot)\rangle d\mu dt\\
 & = & \int_{0}^{T}\int\phi_{t}\partial_{t}p_{t}(x,\cdot)d\mu dt+K\phi_{0}(x).
\end{eqnarray*}
\end{prop}
\begin{rem*}
(1) A first version of this note contained a wrong proof. The usage
of the correct term and the correct adjustment is thanks to Jiang.
In \cite{Jiang2013} Jiang managed to avoid using \cite[Proposition 3.4]{Koskela2003}.

(2) The assumption holds, for example, if for $\mu$-almost every
$x$ 
\[
\dim\mu(x)=\limsup_{r\to0}\frac{\log\mu(B_{r}(x))}{\log r}\ge1,
\]
i.e. the Hausdorff dimension of $\mu$ is at least $1$. In particular,
it will hold for $RCD(K,N)$-spaces (see Lemma \ref{lem:rcd-dim}).\end{rem*}
\begin{proof}
Assume for some $0<\alpha<1$, to be chosen later on, we have 
\[
\mu(B_{r}(x))\le C\cdot r^{\alpha}
\]
for all $0<r<1$. Carefully checking the original proof, one notes
that it suffices to bound the following term for $0<r<1$ and show
that it converges to $0$ as $r$ does
\[
\begin{array}{l}
\frac{1}{r}\|\phi-\phi(0,x)\|_{L^{\infty}(B_{2r}(x)\times(0,2r))}\int_{0}^{2r}\int_{B_{2r}(x)}|\nabla p_{t}(x,\cdot)|d\mu dt\\
\le\frac{1}{r}\|\phi-\phi(0,x)\|_{L^{\infty}(B_{2r}(x)\times(0,2r))}(r\mu(B_{r}(x))^{\frac{1}{2}}\left(\int_{0}^{2r}\int_{B_{2r}(x)}|\nabla p_{t}(x,\cdot)|^{2}d\mu dt\right)^{\frac{1}{2}}\\
\le C^{\frac{1}{2}}\cdot C_{\phi}\cdot r^{-1+\beta+\nicefrac{1}{2}+\nicefrac{\alpha}{2}}\left(\int_{0}^{2r}\int_{B_{2r}(x)}|\nabla p_{t}(x,\cdot)|^{2}d\mu dt\right)^{\frac{1}{2}}
\end{array}
\]
where $\beta\in(0,1]$ is the H�lder exponent of $\phi$ and $C_{\phi}$
the H�lder constant. Choosing $\alpha\in(1-2\beta,1)$, the term is
dominated by $\tilde{C}r^{\gamma}$ for some $\gamma\in(0,1)$. In
particular, it is bounded and converges to $0$ as $r\to0$.

Furthermore, because the doubling and Poincar\'{e} constants only
depend on some $R$ we get by Sturm \cite[2.6]{Sturm1995} 
\[
|\partial_{t}p_{t}(x,y)|\le C\mathfrak{v}(t,x,y)\cdot t^{-1}e^{-\frac{d(x,y)^{2}}{C_{1}t}}.
\]


With these two facts the proof of \cite[3.4]{Koskela2003} can be
followed without any change.
\end{proof}
Finally we can state the main theorem. 
\begin{thm}
\label{thm:main}Assume the space satisfies the previous proposition
and that there is a constant $C>0$ and $T>0$ such that for every
$0<t<T$ and every $g\in W^{1,2}(X)$ the following Sobolev-Poincar\'{e}
inequality for $T_{t}$ holds 
\[
T_{t}(g^{2})(x)\le(2t+Ct^{2})T_{t}(|\nabla g|^{2})(x)+(T_{t}g(x))^{2}.
\]
Then any $u\in W_{loc}^{1,2}(X)$ with $\Delta u_{|\Omega}\in L_{loc}^{p}(\Omega)$
for some open $\Omega\subset X$ is locally Lipschitz in $\Omega$
if $p>Q\ge2$. \end{thm}
\begin{rem*}
1) Even though we made the assumption $Q\ge2$, similar to \cite{Jiang2011},
it is possible to show the same for $Q\in[1,2)$.

2) If the Sobolev-Poincar\'{e} inequality holds uniformly, i.e. $C$
does not depend on $x$ then $T_{t}$ satisfies the Bakry-\'Emery
condition and vice versa (see \cite{Bakry1997}). In the proof the
condition is only required to hold for $t$ sufficiently small and
for functions with support in a neighborhood of $x$, thus it might
be interpreted as a local curvature condition.\end{rem*}
\begin{proof}
Note that Ahlfors regularity is used in the proof of \cite[3.1, 3.2., 3.3]{Jiang2011}
only three times, namely Inequality $(3.4)$ on page 291, on the bottom
of page 294, and for Inequality $(3.12)$. We will only show how to
adjust these steps and leave out the details of the remaining parts.

Inequality $(3.4)$ on page $291$ of \cite{Jiang2011} can be proven
as follows:
\begin{eqnarray*}
|w(t,x)| & = & |u(x)\phi(x)-T_{t}(u\phi)(x_{0})|\\
 & = & |u(x)\phi(x)-u(x_{0})\phi(x_{0})+u(x_{0})\phi(x_{0})-T_{t}(u\phi)(x_{0})|\\
 & \le & C\cdot C(u,f)d(x,x_{0})^{\delta}+\int_{X}|u(x_{0})\phi(x_{0})-u(y)\phi(y)|p_{t}(x_{0},y)d\mu(y)\\
 & \le & C\cdot C(u,f)\big\{ d(x,x_{0})^{\delta}\\
 &  & +\int|u(x_{0})\phi(x_{0})-u(y)\phi(y)|e^{-\frac{d(x_{0},y)^{2}}{2C_{\text{1}}t}}p_{lt}(x_{0},y)d\mu(y)\big\}
\end{eqnarray*}
where we applied Corollary \ref{cor:heat bound}. Splitting the integral
as in \cite[p. 163]{Koskela2003} we get 
\begin{eqnarray*}
|w(t,x)| & \le & C\cdot C(u,f)\big\{ d(x,x_{0})^{\delta}\\
 &  & +\int_{B_{t^{\frac{1}{3}}}(x_{0})}d(x_{0},y)^{\delta}p_{lt}(x_{0},y)d\mu(y)\\
 &  & +2\|u\|_{\infty}e^{-\frac{t^{\frac{2}{3}}}{2C_{1}t}}\int_{X\backslash B_{t^{\frac{1}{3}}}(x_{0})}p_{lt}(x_{0},y)d\mu(y)\big\}\\
 & \le & C\cdot C(u,f)(d(x,x_{0})^{\delta}+t^{\delta/2}\}.
\end{eqnarray*}


Using the same argument we can derive the inequality on the bottom
of page 294 in \cite{Jiang2011}
\begin{eqnarray*}
\int_{X}|Dw(t,x)|^{2}p_{t}(x_{0},x)d\mu(x) & < & \frac{1}{2t}\int_{X}w^{2}(t,x)p_{t}(x_{0},x)d\mu(x)\\
 & \le & C\cdot C(u,f)^{2}\frac{1}{2t}\int_{X}(d(x,x_{0})^{\delta}+t^{\delta/2})^{2}p_{lt}(x_{0},x)d\mu(xy)\\
 & \le & C\cdot C(u,f)^{2}t^{\delta-1}.
\end{eqnarray*}


Finally, to get the lower bound on $\int_{0}^{T}\frac{d}{dt}J(t)dt$
we only have adjust inequality $(3.12)$ on page 295 of \cite{Jiang2011}.
The only term where Ahlfors regularity was used is the following 
\[
\int_{0}^{T}t^{-\epsilon}\left(\int p_{t}(x_{0},x)^{\frac{p}{p-2}}d\mu(x)\right)^{1-\frac{2}{p}}dt,
\]
where $\epsilon>0$ is chosen such that $\epsilon+\frac{Q}{p}<1$.

First note that $\frac{p}{p-2}=1+\frac{2}{p-2}$ and 
\[
p_{t}(x_{0,}x)^{\frac{2}{p-2}}\le(C\mu(B_{\sqrt{t}}(x_{0})^{-\frac{1}{2}}\mu(B_{\sqrt{t}}(x)^{-\frac{1}{2}})^{\frac{2}{p-2}}\le\tilde{C}t^{-\frac{Q}{p-2}}.
\]
Therefore,
\begin{eqnarray*}
\int_{0}^{T}t^{-\epsilon}\left(\int p_{t}(x_{0},x)^{\frac{p}{p-2}}d\mu(x)\right)^{1-\frac{2}{p}}dt & \le & \int_{0}^{T}t^{-\epsilon}\left(\tilde{C}t^{-\frac{Q}{p-2}}\int p_{t}(x_{0},x)d\mu(x)\right)^{\frac{p-2}{p}}dt\\
 & \le & \hat{C}\int_{0}^{T}t^{-\epsilon-\frac{Q}{p}}\left(\int p_{t}(x_{0},x)d\mu(x)\right)^{\frac{p-2}{p}}dt\\
 & \le & C'<\infty,
\end{eqnarray*}
where $C'$ depends only on $\hat{C}$, $T$ and $\epsilon+\frac{Q}{p}<1$,
which can be chosen uniformly in a neighborhood of $x_{0}$.

The remaining parts of the proof follow by directly copying Jiang's
proof of \cite[3.1,3.2, 3.3]{Jiang2011}.\end{proof}
\begin{cor}
Assume $X$ satisfies the $RCD(K,N)$ condition. Then any $u\in W_{loc}^{1,2}(X)$
with $\Delta u\in L_{loc}^{p}$ and $p>N$ is locally Lipschitz continuous.
\end{cor}
The corollary is a result of the following lemma which might be useful
in its own right. First let us state the Bishop-Gromov volume comparison
inequality: $X$ is said to satisfy $BG(K,N)$ if for each $0<r<R<\infty$
and $x\in X$ 
\[
\frac{\mu(B_{R}(x))}{\mu(B_{r}(x))}\le\frac{V_{K,N}(R)}{V_{K,N}(r)},
\]
where $V_{K,N}(r)$ is the volume of the ball of radius $r$ in the
$N$-dimensional model space $\mathbb{M}_{K,N}$ of constant curvature
$K$. Note that $V_{K,N}$ is non-decreasing and locally Lipschitz
in a neighborhood of every $r>0$. 
\begin{lem}
\label{lem:rcd-dim}Assume $X$ satisfies $BG(K,N)$. Then for every
$x\in X$ and $R_{0}>0$ there is a $C>0$ such that for every $0<r<R_{0}$
\[
\mu(B_{r}(x))\le C\cdot r.
\]
In particular, $C$ can be chosen uniformly in a neighborhood of $x$.\end{lem}
\begin{rem*}
This result is based on a proof of Lipschitz continuity of $x\mapsto\mu(B_{r}(x))$
by Ba\v{c}�k-Hua-Jost-Kell \cite{BHJK2013} based on Buckley's $\delta$-annular
decay property in \cite{Buckley1999}. Earlier, but independently,
Kitabeppu \cite[Lemma 3.1]{Kitabeppu2013} discovered this fact as
well.\end{rem*}
\begin{proof}
Take any $y\in X\backslash\{x\}$. Then for $0<\epsilon<r=d(x,y)$
\[
B_{\epsilon}(x)\subset B_{r+\epsilon}(y)\backslash B_{r-\epsilon}(y).
\]
Thus, by continuity of $\epsilon\mapsto\mu(B_{\epsilon}(x))$, it
suffices to show that 
\[
\frac{\mu(B_{r+\epsilon}(y)\backslash B_{r-\epsilon}(y))}{2\epsilon}
\]
is bounded for $0<\epsilon\ll1$. 

By the $BG(K,N)$ condition
\begin{eqnarray*}
\frac{\mu(B_{r+\epsilon}(y)\backslash B_{r-\epsilon}(y))}{\mu(B_{r+\epsilon}(y))} & = & \frac{\mu(B_{r+\epsilon}(y))-\mu(B_{r-\epsilon}(y))}{\mu(B_{r+\epsilon}(y))}=1-\frac{\mu(B_{r-\epsilon}(y))}{\mu(B_{r+\epsilon}(y))}\\
 & \le & 1-\frac{V_{K,N}(r-\epsilon)}{V_{K,N}(r+\epsilon)}=\frac{V_{K,N}(r+\epsilon)-V_{K,N}(r-\epsilon)}{V_{K,N}(r+\epsilon)}.
\end{eqnarray*}
Because $V_{K,N}$ is locally Lipschitz, i.e. $L=\operatorname{Lip}V_{K,N}|_{(r-\epsilon_{0},r+\epsilon_{0})}<\infty$
for some $\epsilon_{0}>0$, 
\[
\frac{\mu(B_{r+\epsilon}(y)\backslash B_{r-\epsilon}(y))}{2\epsilon}\le\frac{L\cdot\mu(B_{r+\epsilon_{0}}(x))}{V_{K,N}(r)}\le M<\infty
\]
for all $0<\epsilon<\epsilon_{0}$. Also, note that $M$ can be chosen
uniformly in a neighborhood of $x$.
\end{proof}
\bibliographystyle{amsalpha}
\bibliography{bib}

\end{document}

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