The Schauder and $W^{2,p}$ scales are different, not interchangeable

On a bounded domain, $C^{2,\alpha }(\bar{\mathrm{\Omega }})$ embeds strictly into $W^{2,p}(\mathrm{\Omega })$ for every finite $p$, and there is no reverse inclusion. Thus the spaces are not equivalent at top order. The estimate theorems on this page also use different forcing-data hypotheses, as item (iii) records.

• (i) Inclusion $C^{2,\alpha }\subseteq W^{2,p}$ for all finite $p$. If $\mathrm{\Omega }$ is bounded and $u\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$, then each derivative $D^{\beta }u$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 2$ is continuous on the compact set $\bar{\mathrm{\Omega }}$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,p}(\mathrm{\Omega })}\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}p}{\sum }_{\mathrm{\mid }\beta \mathrm{\mid }\le 2}{\sup }_{\mathrm{\Omega }}\mathrm{\mid }{D}^{\beta }u\mathrm{\mid }\le C(\mathrm{\Omega })\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})},$ so $C^{2,\alpha }(\bar{\mathrm{\Omega }})\subseteq W^{2,p}(\mathrm{\Omega })$ with a norm bound depending on the volume and on $p$ through $\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}p}$; the inclusion is strict, and the Schauder scale is the stronger hypothesis at the top order.
• (ii) No reverse inclusion for any finite $p$. For every $1<p<\mathrm{\infty }$ the space $W^{2,p}(\mathrm{\Omega })$ is not contained in $C^{2,\alpha }(\bar{\mathrm{\Omega }})$: on the unit ball the function $u(x)=(x_1{)}_{+}^2$ belongs to $W^{2,p}$ for every finite $p$ while $D_{11}u={\mathbf{21}}_{\{x_1>0\}}$ is discontinuous, as recorded with proof in $W^{2,p}$ regularity implies classical or H"older regularity when $p$ is large. More generally, under the Axiom of Choice, for $n\ge 2$, and on a bounded $W^{2,p}$-extension domain, the Sobolev embedding gives at most one H"older derivative, with exponent strictly below $1-n\mathrm{/}p$ when $p>n$; finite $p$ never gives the two-derivative H"older estimate.
• (iii) The data classes differ in the same direction. The $W^{2,p}$ estimate of Interior $W^{2,p}$ estimate for uniformly elliptic equations with continuous coefficients accepts forcing $Lu\in L^p$ and concludes an $L^p$ bound for $D^{2}u$, while the Schauder estimate of Interior Schauder estimate for uniformly elliptic equations requires $Lu\in C^{0,\alpha }$ and concludes a H"older bound; since $C^{0,\alpha }(\bar{\mathrm{\Omega }})\subseteq L^p(\mathrm{\Omega })$ on a bounded domain with equality false, the Schauder theorem assumes strictly more on the data and concludes strictly more on the solution.
• (iv) The endpoints are genuine restrictions of the two theories. The strict range $0<\alpha <1$ in the Schauder scale is not a technicality: at the endpoint $\alpha =1$ the interior estimate fails, and the sharp modulus of $D^{2}Nf$ for a Lipschitz source is $\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }\log \mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }$ rather than $\mathrm{\mid }x\mathrm{\mid }$, with the explicit witness recorded on the companion page (The Schauder estimate fails at the H"older endpoint $\alpha =1$ ↗). The range $1<p<\mathrm{\infty }$ is the range of the Riesz-multiplier and singular-integral arguments used on this page for the Sobolev estimates; no endpoint $p=1$ or $p=\mathrm{\infty }$ version is asserted here.
• (v) Neither scale is a boundary regularity theorem by itself. The $W^{2,p}$ estimate is a priori and assumes $u\in W^{2,p}$; a weak solution on a merely Lipschitz domain can fail to reach $W^{2,p}$ altogether at a reentrant corner (Boundary $W^{2,p}$ regularity needs more than Lipschitz boundary ↗), and the radius bookkeeping of both estimates is exercised by The Schauder estimate on a quadratic Poisson solution: radius powers balance ↗.

Thus the two scales should be used according to the data: rough $L^p$ forcing is treated by the Sobolev scale at the price of losing H"older regularity at the top order, while $C^{0,\alpha }$ forcing with controlled coefficients is treated by the Schauder scale, which gives two H"older derivatives but no improvement at the endpoint $\alpha =1$ and no statement for rough coefficients.

• The comparison is local in nature: on an infinite-volume domain, boundedness of $u$ and its derivatives does not imply $L^p$ integrability: $u\equiv 1$ on ${\mathbb{R}}^n$ is a counterexample. An inclusion there requires additional integrability, and on domains with corners both scales require corresponding boundary hypotheses.
• The statement of this remark carries no proof obligation of its own: each itemized claim is proved or witnessed in the cited item, and the companion examples page holds the endpoint counterexamples for both scales.