What is sobolev embedding?

Sobolev embedding is a theorem that relates the regularity of functions to their smoothness. It provides a way to estimate norm of a function in terms of its derivatives. More specifically, the theorem states that if a function u belongs to the Sobolev space W^{k,p}(Ω), where Ω is a bounded domain in R^n, and p > n/k, then u is continuous on the closure of Ω and there exists a constant C such that ||u||{L^q(Ω)} ≤ C||u||{W^{k,p}(Ω)}, where q is the critical exponent given by q = np/(n-kp).

The Sobolev embedding theorem has important applications in various fields such as elliptic partial differential equations, harmonic analysis, and geometric analysis. It is used to prove existence and uniqueness of solutions to certain differential equations, as well as to derive estimates and regularity properties of solutions.

The theorem has several variations depending on the type of Sobolev space and the domain considered. For instance, there are special embeddings for functions defined on unbounded domains or manifolds with boundary. Moreover, the theorem can be generalized to include fractional spaces and Besov spaces, which are function spaces that are intermediate between Sobolev spaces and classical function spaces.