Sobolev embedding and critical exponents

Let $n\ge2$ and $1\le p<n$, and define $p^*=np/(n-p)$. The whole-space Sobolev inequality bounds the Lᵖ-star norm of a compactly supported smooth function by its gradient Lᵖ norm; by density it applies to the appropriate W¹,ᵖ functions. On a bounded Lipschitz domain, a bounded extension gives the continuous embedding $W^{1,p}(\Omega)\hookrightarrow L^{p^*}(\Omega)$ with the full Sobolev norm on the right. If the function has zero trace, the gradient-only version is available through zero extension and Poincaré control.

On a bounded domain, finite measure also gives embeddings into smaller finite exponents. On the whole space, finite measure is unavailable, so a stronger Lᵖ-star bound alone does not automatically imply every smaller Lᑫ bound. Combining the original Lᵖ membership with the Sobolev gain yields intermediate exponents. The endpoint $p=1$ is included when $n\ge2$, but formulas at $p=n$ require a separate statement because the denominator in p-star vanishes. Treating infinity as a number obtained by substitution would conceal an actual change in the theorem.

Choose a nonzero smooth bump supported in a ball and consider $u_\lambda(x)=\lambda^{n/p-1}\phi(\lambda x)$. Change of variables shows its gradient Lᵖ norm is independent of $\lambda$, while its Lᵠ norm scales as $\lambda^{n/p-1-n/q}$. The exponent vanishes precisely at $q=p^*$. For larger q, this norm grows as the bump concentrates, even though the first-order Sobolev norm stays bounded on a fixed surrounding domain for large $\lambda$. Thus a proposed larger-exponent embedding with one uniform constant cannot hold.

At the critical exponent, the concentrated bumps keep a nonzero Lᵖ-star norm while their support shrinks. This does not contradict continuity of the embedding, which only bounds that norm. It does show why compactness at the critical target generally fails: the sequence can converge almost everywhere to zero without its target norm tending to zero. Below the critical exponent, the same scaling makes the target norm decrease. A scaling computation is a sharp diagnostic and a counterexample generator; it does not by itself prove that the surviving exponent actually satisfies the positive embedding theorem.

On a bounded Lipschitz domain in dimension $n\ge2$, $W^{1,n}$ embeds into every finite $L^q$ space, but generally not into L∞. The constants can depend on q and need not stay bounded as q grows. Near the origin, a suitably localized radial function like $\log\log(e/|x|)$ has a derivative of magnitude comparable to $1/(|x|\log(e/|x|))$. Its nth-power radial derivative integral converges when n is greater than one, while the function is unbounded. This demonstrates why the finite-q family cannot simply be passed to infinity with a uniform constant.

When $p>n$, Morrey's theorem instead supplies a Hölder-continuous representative with exponent $1-n/p$, under the corresponding domain hypotheses. This is a different form of regularity, not just a larger Lᑫ estimate. Dimension one has its own endpoint feature: W¹,¹ on a bounded interval has an absolutely continuous, hence bounded, representative. Always state dimension, derivative order, integrability, and boundary geometry together. A theorem copied from a whole-space homogeneous setting can otherwise lose either its order-zero term or a necessary domain condition when applied to a bounded-domain PDE.