Embeddings, Hardy operators and nonlinear problems
The author presents a survey on some recent developments involving embeddings between function spaces. He focuses first on variations of Sobolev inequalities analysing best constants and extremal functions using the technique of mass transportation which originated in the work of Brenier. The Sobolev inequality is also studied in the context of Banach function spaces covering Orlicz spaces and Lorentz spaces. Also formulations using Triebel-Lizorkin and Besov spaces and Lebesgue type with variable exponent are pointed out. A basic tool consists in reducing several embeddings to questions concerning the Hardy operator which is well understood in the setting of Banach function spaces. Also, considerations about entropy numbers and approximation numbers of compact operators are considered in the paper in terms of quantifying the compactness. These notions are used when analysing a nonlinear eigenvalue problem which leads to a series representation of compact operators acting not necessarily between Hilbert spaces but between Banach spaces with certain geometric properties. The Dirichlet problem for the \(p\)-Laplacian is also studied using some embedding maps.
scientific article; zbMATH DE number 5795229
scientific article