Lemma 5.10.569CE23EA1637

Proof. In this proof $C$ denotes constants that depend only on two cutoff functions $\chi $ and $\eta $ fixed below, and $C'$ constants that depend in addition on $p$. Since $v(0)>-\infty $, the function $v$ is not identically $-\infty $, and hence it is locally integrable by proposition 3.4.8. For $0<r<1$, the ball submean inequality (theorem 3.4.5) and $v\leq 0$ give

Let $\mu =\Delta v$ be the Riesz measure of $v$, and choose a smooth cutoff $\chi \geq 0$ supported in $D(0,19/20)$ and equal to one on $D(0,9/10)$. By the definition of the distributional Laplacian and (5.10.7),

Next choose a smooth cutoff $0\leq \eta \leq 1$ supported in $D(0,9/10)$ and equal to one on a neighborhood of $\overline{D(0,4/5)}$. The fundamental solution in the plane is $(2\pi )^{-1}\log |z|$, so theorem 3.4.14, applied with the measure $\nu =\eta \mu $, which is finite and supported in $\operatorname{Supp}\eta \subset D(0,9/10)$, gives

for every $z\in D(0,4/5)$, with $h$ harmonic on $D(0,4/5)$.

We estimate $P$. For $|\zeta |\leq 9/10$, the translated disc $D(0,4/5)-\zeta $ lies in $D(0,2)$, so

the integral $L_ p$ is finite because its part over $D$ equals $2\pi \int _0^1r|\log r|^ p\, \mathrm dr$, whose integrand is bounded on $(0,1]$, while $|\log |z||\leq \log 2$ on the rest of $D(0,2)$. Let $m=\nu (D)=\int \eta \, \mathrm d\mu $, so that $m\leq \mu (D(0,9/10))\leq C|v(0)|$. If $m=0$, then $P=0$. Otherwise, $|P(z)|\leq \frac m{2\pi }\int |\log |z-\zeta ||\, \mathrm d\nu (\zeta )/m$, and Hölder's inequality for the probability measure $\nu /m$ gives

and integrating over $z\in D(0,4/5)$ with Tonelli's theorem yields $\| P\| _{L^ p(D(0,4/5))}\leq L_ p^{1/p}m/(2\pi )\leq C'|v(0)|$.

We now estimate $h$. The case $p=1$ of the bound for $P$, in which $L_1$ is an absolute constant, and (5.10.7) give $\| h\| _{L^1(D(0,4/5))}\leq C|v(0)|$. For $|z|\leq 1/2$, the closed disc $\overline{D(z,1/4)}$ lies in $D(0,4/5)$, and the ball mean-value property of the harmonic function $h$ (proposition 3.1.5) gives $|h(z)|\leq (16/\pi )\| h\| _{L^1(D(0,4/5))}$. We have thus found a constant $C_0\geq 1$, depending only on the cutoff functions $\chi $ and $\eta $, such that $\nu (D)=m\leq C_0|v(0)|$ and $\sup _{\overline{D(0,1/2)}}|h|\leq C_0|v(0)|$; this proves (5.10.6). Adding the estimates for $P$ and $h$ on $D(0,1/2)$, we obtain

which proves the last assertion. $\square$