Pome89771FD1FD15Lemma 8.5.1.89771FD1FD15 Suppose $u_ j$ is smooth,
Then $u\in W^{2,p}(X)$ for every $1<p<\infty $, its distributional complex Hessian has bounded coefficients, and $u_ j\to u$ in $C^{1,\alpha }(X)$ for every $0<\alpha <1$. The same hypotheses give uniform $W^{2,p}$ bounds for the sequence.
Proof. On a coordinate ball the Euclidean Laplacian is $\Delta u_ j=4\sum _ i (u_ j)_{i\bar i}$, so it is uniformly bounded. The interior Laplace estimate on two nested balls gives
A finite cover gives the global bound. Weak $W^{2,p}$ limits coincide with the uniform limit. The bounded complex Hessians have weak-star limits in $L^\infty $; testing against compactly supported smooth functions identifies them with $H(u)$. Choose $p$ with $1-2n/p>\alpha $. The compact Sobolev embedding into $C^{1,\alpha }$ gives convergence of subsequences. Every such limit is $u$, hence the whole sequence converges. This estimates the full real Hessian in every finite $L^ p$ space, without asserting that it is bounded. $\square$
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