In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.

Let W^k,p(Rⁿ) denote the Sobolev space consisting of all real-valued functions on Rⁿ whose weak derivatives up to order k are functions in L^p. Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ are two real numbers such that

(given ${\displaystyle n}$, ${\displaystyle p}$, ${\displaystyle k}$ and ${\displaystyle \ell }$ this is satisfied for some ${\displaystyle q\in [1,\infty )}$ provided ${\displaystyle (k-\ell )p<n}$), then

and the embedding is continuous: for every ${\displaystyle f\in W^{k,p}(\mathbf {R} ^{n})}$, one has ${\displaystyle f\in W^{l,q}(\mathbf {R} ^{n})}$, and

In the special case of k = 1 and ℓ = 0, Sobolev embedding gives

where p^∗ is the Sobolev conjugate of p, given by

and for every ${\displaystyle f\in W^{1,p}(\mathbf {R} ^{n})}$, one has ${\displaystyle f\in L^{p^{*}}(\mathbf {R} ^{n})}$ and

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality. The result should be interpreted as saying that if a function ${\displaystyle f}$ in ${\displaystyle L^{p}(\mathbf {R} ^{n})}$ has one derivative in ${\displaystyle L^{p}}$, then ${\displaystyle f}$ itself has improved local behavior, meaning that it belongs to the space ${\displaystyle L^{p^{*}}}$ where ${\displaystyle p^{*}>p}$. (Note that ${\displaystyle 1/p^{*}<1/p}$, so that ${\displaystyle p^{*}>p}$.) Thus, any local singularities in ${\displaystyle f}$ must be more mild than for a typical function in ${\displaystyle L^{p}}$.