Function spaces and their role in nonlinear partial equations 1000-1M25PF

The aim of this lecture is to discuss various types of function spaces that appear in partial equations and functional analysis. We will be particularly interested in absolutely continuous functions, Orlicz spaces, Sobolev spaces, and their various generalizations, including Sobolev spaces defined on metric spaces, such as fractals or topological groups, and Sobolev spaces with weights. Of particular interest here are the various types of inequalities that hold for these types of spaces: Young's, Hardie's, Sobolev's, and Gagliardo-Nirenberg's inequalities. The motivation for studying such spaces is partial equations. Therefore, we will emphasize the applications of these spaces to equations, discussing typical problems where such a space or inequality is needed. I invite all enthusiasts of functional analysis, partial differential equations, mathematical physics, and analysis in general. This theory holds many open questions.