Functional analysis Classical Banach spaces of sequences and functions Theorem

Sobolev embedding theorem

Statement

Let $n \ge 1$ and $1 \le p < \infty$ . Let $W^{1,p}(\R^n)$ be the Sobolev space of functions $u \in L^p(\R^n)$ whose first weak derivatives lie in $L^p(\R^n)$ .

If $p < n$ , let $p^* = np/(n - p)$ . There is a constant $C = C(n, p)$ such that $\norm{u}_{L^{p^*}} \le C \norm{\nabla u}_{L^p}$ for all $u \in W^{1,p}(\R^n)$ . So $W^{1,p}(\R^n)$ embeds continuously in the Lp space $L^{p^*}(\R^n)$ .
If $p > n$ , let $\gamma = 1 - n/p$ . Each $u \in W^{1,p}(\R^n)$ agrees almost everywhere with a bounded function that is Hölder continuous with exponent $\gamma$ . Its supremum norm and Hölder constant are at most $C(n, p) \norm{u}_{W^{1,p}}$ .

Remarks

Part 1 is the Gagliardo–Nirenberg–Sobolev inequality, and part 2 is Morrey's inequality. For $p < n$ , the space $W^{1,p}(\R^n)$ also embeds in $L^q(\R^n)$ for every $q$ between $p$ and $p^*$ , by interpolation.

The exponent $p^*$ is forced by scaling. For $u_\lambda(x) = u(\lambda x)$ , one has $\norm{u_\lambda}_{L^q} = \lambda^{-n/q} \norm{u}_{L^q}$ and $\norm{\nabla u_\lambda}_{L^p} = \lambda^{1 - n/p} \norm{\nabla u}_{L^p}$ . The inequality of part 1 can hold for all $\lambda > 0$ only if $-n/q = 1 - n/p$ , that is $q = p^*$ .

For $p = n \ge 2$ , functions in $W^{1,n}(\R^n)$ need not be bounded. On the ball $\abs{x} < 1/2$ in $\R^2$ , the function $\log \log (1/\abs{x})$ is in $W^{1,2}$ but unbounded near $0$ .

Going deeper

For a student

Sobolev embeddings are the main tool to get regularity of weak solutions of partial differential equations. One first finds a solution in a Sobolev space by energy methods, then uses embeddings to show that it is continuous or smooth. On bounded domains with regular boundary the same embeddings hold, and the Rellich–Kondrachov theorem makes them compact for $q < p^*$ .

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Tag	`GZTT6FT18G`
URI	`m2://GZTT6FT18G`
Kind	statement, theorem
Writing	W0 stub
MSC	46E35
Status	This copy of the site has no build data. Run `m2 status GZTT6FT18G`.
Uses	Lp space, Hölder continuity
Changed	2 November 2026