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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.

Use the paper's classical norm convention: $|u{|}_{2;U}$ is the derivative supremum norm through order two, not an $L^2$ norm. Write $H(u;U)=[D^{2}u{]}_{\alpha ;U}$ and $Q(u,f)=|u{|}_{2;B_1}+|f{|}_{0,\alpha ;B_1}$, where the latter is the full Hölder norm of the forcing.

Fix $0<\delta <1$. If the asserted estimate failed, rescaling the functions by their global Hessian matrix Hölder seminorm would produce $u_j,f_j$ withIn particular $Q(u_j,f_j)\to 0$. Choose $x_j,y_j\in B_{1/2}$ for whichThe supremum norm of $D^2{u}_j$ tends to zero, so $r_j=|y_j-x_j|\to 0$. Let $P_j$ be the quadratic Taylor polynomial of $u_j$ at $x_j$, and defineThe domains contain $B_{1/(2r_j)}$, and $U_j,D{U}_j,D^2{U}_j$ vanish at zero. MoreoverThe right side is bounded by $[f_j{]}_{\alpha }|z{|}^{\alpha }$ and tends to zero locally uniformly. The normalized Hessian matrix bound controls $|D^2{U}_j(z)|\le |z{|}^{\alpha }$; integration along line segments then bounds $D{U}_j,U_j$ on every compact Euclidean ball. Arzelà-Ascoli theorem and a diagonal subsequence argument give convergence in $C^2$ on compact subsets to an entire harmonic function $U$, with $[D^{2}U{]}_{\alpha ;{\mathbb{R}}^n}\le 1$ and $D^{2}U(0)=0$.

After a further subsequence $z_j=(y_j-x_j)/r_j\to z_{\ast }$ with $|z_{\ast }|=1$. The normalization ensures $|D^{2}U(z_{\ast })|\ge \delta /2$. But each second derivative of $U$ is harmonic and has finite global Hölder seminorm with exponent below one. The supplied Liouville theorem makes each derivative constant; it is the globally Hölder case of the polynomial-growth Liouville theorem for harmonic functions. Its value at zero makes it zero, a contradiction. This blow-up compactness proof of an interior Schauder estimate establishes

To absorb the term on the larger Euclidean ball, rescale this estimate to Euclidean balls contained in $B_s$. For $1/2\le r<s<1$, pairs separated by less than $(s-r)/2$ are controlled in a Euclidean ball centred at their first point; farther pairs are controlled directly by $\Vert D^{2}u{\Vert }_{\infty }$. ThusThis is the covering step in the Simon absorption lemma. Take $r_j=1-2^{-j-1}$ and fix $\delta <2^{-2-\alpha }$. Iterating makes the remainder ${\delta }^{j}H(u;B_{r_j})$ tend to zero, since the global seminorm is finite; the error terms form a convergent geometric series. Consequently

For the requested failure of the weaker estimate, take $u_m=m^{-2}\sin (m{x}_1)$ and $f_m=-\sin (m{x}_1)$. The derivative supremum norm $|u_m{|}_2$ and $\Vert f_m{\Vert }_{\infty }$ remain bounded, but comparison at $0$ and $\pi e_1/(2m)$, both in $B_{1/2}$ for large $m$, givesThis proves the necessity of Hölder forcing for Schauder estimates. Every member of the sequence is smooth; failure comes from the increasing frequency, not from any lack of individual regularity.

Write $K(u,f)=\Vert u{\Vert }_{C^2(B_1)}+\Vert f{\Vert }_{C^{0,\alpha }(B_1)}$; the paper's $|u{|}_{2;B_1}$ denotes a $C^2$ norm, not an $L^2$ norm. We use a blow-up compactness proof of an interior Schauder estimate. If the estimate failed for fixed $\delta >0$, normalization would produceChoose $x_j,y_j\in B_{1/2}$ with $d_j=|y_j-x_j|$ and $|D^2{u}_j(y_j)-D^2{u}_j(x_j)|\ge \frac{1}{2}{d}_j^{\alpha }$. Since $\Vert D^2{u}_j{\Vert }_{\infty }\to 0$, necessarily $d_j\to 0$. Subtract the quadratic Taylor polynomial $P_j$ at $x_j$ and setThe domains expand to ${\mathbb{R}}^n$. At zero, $w_j,D{w}_j,D^2{w}_j$ vanish; the Hölder seminorm of $D^2{w}_j$ is at most ${\delta }^{-1}$. Integration along segments gives uniform $C^{2,\alpha }$ bounds on each fixed ball. Alsolocally uniformly, since $[f_j{]}_{\alpha }\to 0$. By compact embedding of Hölder spaces and a diagonal subsequence, $w_j$ converges locally in $C^{2,\beta }$ to an entire harmonic function $w$, with $[D^{2}w{]}_{\alpha }\le {\delta }^{-1}$ and $D^{2}w(0)=0$. The unit vectors $(y_j-x_j)/d_j$ have a convergent subsequence; its limit $e$ satisfies $|D^{2}w(e)|\ge 1/2$.

Each second derivative of $w$ is an entire harmonic function with finite global Hölder seminorm. The supplied polynomial-growth Liouville theorem for harmonic functions, at growth exponent $\alpha <1$, makes it constant, and its value at zero makes it zero. This contradicts the nonzero Hessian at $e$, proving the estimate with the small $\delta$ term.

To remove that term, rescale the estimate to all contained balls and apply the Simon absorption lemma. Explicitly, with $d(x)=1-|x|$ and $d_{xy}=\min (d(x),d(y))$, defineFor $|x-y|<d_{xy}/8$, apply the rescaled estimate on $B_{d(x)/2}(x)$; its larger-ball seminorm is controlled by $M$ because all points there have boundary distance at least $d(x)/2$. For more separated pairs use $2\Vert D^{2}u{\Vert }_{\infty }$. The result is $M\le C_0\delta M+C_{\delta }K(u,f)$. Choose $C_0\delta <1/2$ and use $d_{xy}\ge 1/2$ on $B_{1/2}$ to obtainThe forcing's Hölder seminorm cannot be discarded. For integers $k\to \infty$, takeTheir $C^2$ and forcing supremum norms stay bounded, but the $11$ entry of the Hessian matrix differs by two at $x_1=\pm \pi /(2k)$, giving $[D^2{u}_k{]}_{\alpha ;B_{1/2}}\ge 2(k/\pi {)}^{\alpha }\to \infty$.

One useful form of the Simon absorption lemma is the following. Let $S$ be nonnegative and subadditive on balls contained in $B_R$, and suppose that for every such $B_r$,where $A$ is also subadditive and $\epsilon$ is smaller than a constant depending only on $n$ and $\gamma$. ThenMore generally, an additional controlled remainder on the right remains with a dimensional constant. The proof covers $B_{R/2}$ by finitely many smaller balls, sums by subadditivity, and iterates so that the $S$ term is absorbed.

The interior Schauder estimate iswhere $C$ depends only on $n$, $\alpha$, the ellipticity constant, and the stated coefficient bounds.

It is enough first to prove the estimate for the Hölder seminorm of $D^{2}u$. The Interpolation inequality in Holder spaces controls $Du$ and $D^{2}u$ by a small multiple of $[D^{2}u{]}_{\alpha }$ plus $\Vert u{\Vert }_0$; the small term is later absorbed. Freezing $a^{ij}$ at the centre of each ball and moving coefficient differences and lower-order terms to the right reduces the local estimate toOnce this is known, the Simon absorption lemma gives the desired estimate on $B_{1/2}$.

For completeness, prove the frozen-coefficient estimate by contradiction. If it failed, choose solutions $u_k$, points $x_k$, and scales ${\rho }_k\downarrow 0$ at which the scale-invariant Hölder quotient is almost maximal. Let $q_k$ be the quadratic Taylor polynomial of $u_k$ at $x_k$ and defineThen $v_k(0)=D{v}_k(0)=D^2{v}_k(0)=0$, the Hessians have uniformly bounded local $C^{0,\alpha }$ seminorms, and the normalization makes their oscillation nonzero on a fixed ball. The localized equation iswhere the coefficient matrices converge locally uniformly to one constant positive-definite matrix and the given normalized error $g_k$ tends locally uniformly to zero.

The Arzelà-Ascoli theorem produces a locally $C^2$ convergent subsequence with limit $v$ satisfying a constant-coefficient elliptic equation on ${\mathbb{R}}^n$. A linear rotation and scaling turn it into a harmonic function. The normalization gives $D^{2}v(0)=0$ but a nonconstant Hessian, while maximality of the scaled quotient gives growth at most $C(1+|y{|}^{2+\alpha })$. Applying the Liouville theorem to derivatives shows that every second derivative is constant because $0<\alpha <1$. This contradicts the normalized Hessian oscillation. The frozen estimate follows, and interpolation plus absorption completes the proof.

The boundary Schauder estimate iswhere $C$ depends only on the dimension, $\alpha$, the ellipticity constants, and the coefficient Hölder norms.

Subtract a $C^{2,\alpha }$ extension of $\phi$ to reduce to a function $v$ with zero data on the flat boundary; this changes the forcing by a controlled $C^{0,\alpha }$ term. The key local estimate is that for every $\delta >0$,To prove it, argue by contradiction. A failing normalized sequence has points $x_k,y_k$ at which the second-derivative Hölder quotient stays nonzero. Set $r_k=|x_k-y_k|$, subtract the appropriate second-order Taylor polynomial, and rescale space by $r_k$ and the functions by $r_k^{2+\alpha }[D^2{v}_k{]}_{\alpha }$. Necessarily $r_k\to 0$.

If the rescaled distance to the flat boundary tends to infinity, the domains converge to all of ${\mathbb{R}}^n$; otherwise they converge to a half-space. Coefficient compactness freezes the principal matrix, while the normalized right-hand sides converge locally uniformly to zero. A linear change of variables turns every limiting equation into the Laplace equation. In the whole-space case the polynomial-growth Liouville theorem for harmonic functions makes the limit a polynomial of degree at most two. In the half-space case the zero boundary data permit odd reflection across the flat boundary, after which the same theorem applies. The subtracted Taylor normalization forces that polynomial to vanish to second order, contradicting the nonzero limiting Hölder oscillation.

Apply the local estimate on all interior balls and boundary half-balls. The interior and boundary forms of the Simon absorption lemma absorb the term multiplied by $\delta$, while the Hölder interpolation inequality absorbs the remaining lower $C^2$ norm into the top seminorm and the $C^0$ norm. Restoring $\phi$ yields the displayed estimate.

For $y$ in the closed upper half-space, write $B_r^{+}(y)=B_r(y)\cap \overline{{\mathbb{R}}_{+}^n}$. The boundary Simon absorption lemma has the following scaled form. Let $S$ be a nonnegative set functional that is monotone and subadditive on finite covers by such half-balls. Given $\lambda \ge 0$ and $0<\theta <1$, there is ${\delta }_0={\delta }_0(n,\lambda ,\theta )>0$ such that, if $0<\delta \le {\delta }_0$ and every admissible nested pair satisfiesthenThe same conclusion holds for half-balls centred on the flat boundary and truncated balls meeting it. Iteration over a finite covering absorbs the small first term into the left-hand side.