= Blow-up compactness proof of an interior Schauder estimate
A compactness proof of the <interior Schauder estimate> can magnify a pair of points where the <Hölder seminorm> of the <Hessian matrix> is nearly attained. If the proposed estimate fails, normalize that seminorm to one while lower norms and forcing tend to zero. The separation $r_j$ of the chosen pair tends to zero. Subtract the quadratic <Taylor polynomial> and divide by $r_j^{2+\alpha}$ after rescaling space by $r_j$.
The resulting functions have uniformly bounded Hessian <Hölder seminorms> on expanding domains and vanishing Taylor data at zero. <Compact embedding of Hölder spaces> yields an entire <harmonic function> whose second derivatives have finite global $\alpha$-Hölder seminorms, with $0<\alpha<1$. The <polynomial-growth Liouville theorem for harmonic functions> makes those derivatives constant, contradicting the normalized nonzero oscillation. The <Simon absorption lemma> removes the residual small multiple of the larger-ball seminorm.
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