Lemma 14.19.16.9C1A40570E78 In the setting of lemma 14.19.14, suppose that $A,\alpha >0$ and
\[ \omega _ u^ n(E)\leq A\operatorname {Cap}_\omega (E)^{1+\alpha } \qquad (E\subset X\text{ Borel}). \]
For every $0<D<\delta _\varepsilon $,
3CFE977F80E0
14.19.17
\begin{equation} D\leq C_\alpha (C_ HA)^{1/n} \operatorname {Cap}_\omega (U_ D)^{\alpha /n}, \qquad C_\alpha =\frac{2}{1-2^{-\alpha }}. \end{equation}
In particular, when $\alpha =1$,
623E09972ED8
14.19.18
\begin{equation} \operatorname {Cap}(U_ D)\geq \frac{D^ n}{4^ nC_ HA}. \end{equation}
Proof.
Put $a(s)=\operatorname {Cap}_\omega (U_ s)^{1/n}>0$ and $K=(C_ HA)^{1/n}$. The thin-sublevel estimate gives
\[ t\, a(s)\leq K a(s+t)^{1+\alpha } \qquad (0<s,t,\ s+t<\delta _\varepsilon ). \]
Suppose $C_\alpha K a(D)^\alpha <D$. Starting with $s_0=D$, set
\[ t_ j=2K a(s_ j)^\alpha ,\qquad s_{j+1}=s_ j-t_ j. \]
If $a(s_ i)\leq 2^{-i}a(D)$ for $0\leq i\leq j$, then
\[ \sum _{i=0}^ jt_ i \leq 2K a(D)^\alpha \sum _{i=0}^ j2^{-i\alpha } <C_\alpha K a(D)^\alpha <D. \]
Thus $s_{j+1}>0$, and the sublevel estimate, applied with $s=s_{j+1}$ and $t=t_ j$, gives $a(s_{j+1})\leq a(s_ j)/2$. Induction justifies every step. The decreasing sequence $s_ j$ has a limit
\[ s_\infty \geq D-C_\alpha K a(D)^\alpha >0. \]
Monotonicity now gives $0<a(s_\infty )\leq 2^{-j}a(D)$ for every $j$, a contradiction. This proves (14.19.17), including the case in which $U_ D=X$.
$\square$
Pome
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