Pome

Corollary 15.4.3E429FB866D08

Corollary 15.4.3.E429FB866D08 Fix $C_0>0$. Let $u_ j,u\in \mathrm{PSH}(X,\theta )$ satisfy $-C_0\leq u_ j,u\leq 0$ and

\[ \mathrm{MA}_\theta (u_ j)=f_ j\, \mathrm{d}V,\qquad \mathrm{MA}_\theta (u)=f\, \mathrm{d}V, \qquad \sup _ j\| f_ j\| _ p+\| f\| _ p<+\infty . \]

If $u_ j\to u$ in $L^1$, then $u_ j\to u$ uniformly. More precisely, for every exponent $\eta $ as in the theorem,

\[ \| u_ j-u\| _\infty \leq C\| u_ j-u\| _{L^1(\mathrm{d}V)}^\eta . \]

Proof. Apply the theorem first with solution $u$ and comparison potential $u_ j$, and then with solution $u_ j$ and comparison potential $u$. The constants are uniform because of the common range and the common density bound. The larger of the two suprema is the uniform norm, and each integral of a positive part is at most the full $L^1$ difference. $\square$

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