Embeddings, Hardy operators and nonlinear problems

We survey some of the recent developments involving embeddings between function spaces. Emphasis is placed on improvements of classical Sobolev inequalities, the reduction of embedding questions to problems involving Hardy operators, and quantitative estimates of compactness of embeddings that have applications to the spectral theory of operators. We also consider a nonlinear eigenvalue problem which leads to a series representation of compact linear operators acting between Banach spaces, under mild restrictions on the spaces, thus establishing a complete analogue of E. Schmidt's classical Hilbert space theorem for compact operators. Information about relevant embedding maps enables the Dirichlet problem for the p-Laplacian to be studied, and a brief discussion is given of the generalizations of the trigonometric functions that appear naturally in this connection.