Morrey inequality and Hölder regularity

Let $n<p<\infty$ and let $\Omega$ be a bounded Lipschitz domain. Every class in $W^{1,p}(\Omega)$ has a representative extending Hölder continuously to the closure with exponent $\alpha=1-n/p$. Its C⁰,ᵅ norm is bounded by a constant times the full W¹,ᵖ norm. Here the Hölder seminorm is the supremum of $|u(x)-u(y)|/|x-y|^\alpha$ over distinct points. The order-zero term controls additive constants, while the seminorm captures variation. A local version holds on interior balls without a boundary extension assumption.

Two continuous representatives agreeing almost everywhere on an open domain agree everywhere inside: if their difference were nonzero at a point, continuity would make it nonzero on a neighborhood of positive volume. Thus the continuous representative is canonical on the domain, and its extension to the closure determines boundary values. A different representative obtained by changing values at a dense null set need not be continuous, but it represents exactly the same Sobolev element. Embedding statements should therefore say that a continuous representative exists, rather than claiming every written pointwise function becomes continuous.

On an interval, the absolutely continuous representative satisfies $u(y)-u(x)=\int_x^yu'(t)\,dt$. Hölder's inequality bounds this by $|y-x|^{1-1/p}\|u'\|_{L^p(x,y)}$, and hence by the global derivative norm times the same power. This proves the one-dimensional Morrey exponent for $p>1$ directly. At $p=1$, absolute continuity still gives continuity but this argument provides no positive uniform power exponent from the L¹ norm alone. The strict inequality p greater than dimension is what produces a positive Hölder exponent in the general theorem.

In several dimensions, integrating along one chosen segment is not enough for an arbitrary Sobolev function, whose behavior on a fixed line may be unspecified. Standard proofs average line or ball estimates, use integral kernels, and exploit the integrability condition p greater than n. That distinction matters when translating a one-dimensional intuition into a higher-dimensional proof. Extension to the whole space then handles pairs whose connecting segment leaves a nonconvex domain, with constants depending on the geometry. An unqualified straight-line argument inside an arbitrary domain would overlook this issue.

A particular function may be smoother than the theorem guarantees. For $u(x)=x^\beta$ on $(0,1)$ with $0<\beta<1$, its derivative lies in Lᵖ exactly when $p(\beta-1)>-1$, or $\beta>1-1/p$. At equality the derivative integral diverges logarithmically. When membership holds, Morrey guarantees exponent $1-1/p$, whereas this power function actually has the larger exponent beta at zero. One example therefore cannot establish that the theorem's exponent is attained by every member, nor can mere Hölder continuity prove Sobolev membership.

A bounded family in W¹,ᵖ with p greater than dimension is uniformly bounded and equicontinuous through the common estimate. On a bounded Lipschitz domain, Arzelà–Ascoli then gives subsequential uniform convergence. Compactness into a smaller Hölder exponent is also available, but compactness at the exact endpoint exponent need not follow. Changing the exponent changes the topology: small uniform errors can still oscillate sharply at tiny scales. This separation between continuity and compactness anticipates the next lesson and is especially important when passing from energy approximations to nonlinear pointwise expressions.