ScalingStacks

Proof. We induct on dimension. For $n=1$, the unit ball is already enclosed by an annulus, so $\sup_{B(1)}\Phi$ is bounded above, and the mean value property applied to all balls $B(p,2)$ with $1<|p|\leq 2$ gives a lower bound on $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}\Phi$. Thus the $L^{1}$-bound in $B(1)$ is clear.

For general $n$, notice by induction we can bound for each $i\leq n$,

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{\{1<|z_{i}|<4,|z_{j}|<4,\forall j\neq i\}}|\Phi|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{\{1<|z_{j}|<4,\forall j\}}|\Phi|,$$

so $\Phi$ is controlled in $L^{1}$ on an annulus enclosing $B(1)$, and we can bound $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}|\Phi|$ similar to the $n=1$ case. ∎