1 Sobolev spaces and embeddings

Sobolev spaces are spaces of functions that have certain derivatives in some L^p norm. For example, W^{1,p}(U) consists of functions u that belong to L^p(U) and have weak derivatives u_x that also belong to L^p(U), where u_x denotes the partial derivative of u with respect to x. The norm of u in W^{1,p}(U) is defined as ||u||_{W^{1,p}(U)} = ||u||_{L^p(U)} + ||u_x||_{L^p(U)}. Sobolev spaces are useful for studying partial differential equations, variational problems, and harmonic analysis. One of the key questions in Sobolev theory is how to embed Sobolev spaces into other function spaces, such as L^q spaces or Holder spaces. Embeddings are useful for estimating the regularity, continuity, or integrability of Sobolev functions.

2 Rellich-Kondrachov theorem

The Rellich-Kondrachov theorem is one of the most important embedding theorems in Sobolev theory. It states that if U is a bounded domain in R^n with Lipschitz boundary, and 1 \leq p < n, then the embedding of W^{1,p}(U) into L^q(U) is compact for any q < np / (n - p). This means that any bounded and weakly convergent sequence in W^{1,p}(U) has a strongly convergent subsequence in L^q(U). The theorem also holds for p = n, in which case q can be any number less than infinity. The proof of the theorem relies on several tools, such as the Poincare inequality, the Sobolev inequality, and the Rellich lemma. The theorem can be seen as a generalization of the Arzela-Ascoli theorem, which deals with compactness of sequences of continuous functions.

3 Higher order Sobolev spaces

The Rellich-Kondrachov theorem can be extended to higher order Sobolev spaces, such as W^{k,p}(U), where k is a positive integer. These are spaces of functions that have weak derivatives up to order k in L^p norm. The embedding of W^{k,p}(U) into L^q(U) is compact for any q < n p / (n - k p), provided that U is a bounded domain in R^n with C^k boundary. The proof follows the same strategy as the original theorem, but uses higher order versions of the Poincare and Sobolev inequalities. The compactness of these embeddings has applications to elliptic and parabolic equations, as well as to nonlinear analysis and calculus of variations.

4 Fractional Sobolev spaces

Another way to generalize the Rellich-Kondrachov theorem is to consider fractional Sobolev spaces, such as W^{s,p}(U), where s is a positive real number. These are spaces of functions that have fractional derivatives in L^p norm. Fractional derivatives are defined using Fourier transforms or integral operators, and measure the smoothness of functions at different scales. The embedding of W^{s,p}(U) into L^q(U) is compact for any q < n p / (n - s p), provided that U is a bounded domain in R^n with Lipschitz boundary. The proof uses a fractional version of the Rellich lemma, as well as interpolation and duality arguments. The compactness of these embeddings has applications to fractional differential equations, as well as to geometric and functional inequalities.

5 Different norms and spaces

The Rellich-Kondrachov theorem can also be adapted to different norms and spaces, such as weighted L^p spaces, Lorentz spaces, Orlicz spaces, or Besov spaces. These are spaces of functions that have different measures of integrability, smoothness, or decay. The embedding of W^{1,p}(U) into these spaces is compact under suitable conditions on the weights, the norms, or the domains. The proof usually involves some modifications of the Rellich lemma, as well as some comparison or approximation techniques. The compactness of these embeddings has implications for nonlinear analysis, harmonic analysis, and functional analysis.