Abstract
In this paper, we review some of the developments on the Moser–Trudinger and Adams inequalities in both Euclidean space and on Riemannian manifolds. We will also describe some of the closely related problems.
Highlights
Sobolev inequalities which describe the embedding of Sobolev spaces into Lp spaces or Holder spaces play a major role in the analysis of partial differential equations.These inequalities are crucial in proving the existence or nonexistence of solutions, qualitative properties of solutions, etc
In many important problems coming from geometry, physics, engineering, etc. sharp forms of these inequalities, knowledge of the best constants, existence and classification of extremals, etc. are required and these problems are intimately connected with many other problems in subjects like geometry and harmonic analysis
The theorem is not true if we remove the connectedness. We proved this theorem as a consequence of the Moser–Trudinger inequality in conformal discs, so we postpone an outline of the proof to Sec. 5 after proving Theorem 5.8
Summary
Sobolev inequalities which describe the embedding of Sobolev spaces into Lp spaces or Holder spaces play a major role in the analysis of partial differential equations. Other questions addressed are improvements of these inequalities, existence of extremal functions, compactness properties of these embeddings, etc Many of these questions addressed are motivated by questions from nonlinear PDE and geometry, establishing sharp inequalities and related problems have become a subject of independent interest requiring many tools from subjects like harmonic analysis and geometry. We refer to [55] for a detailed survey on some of the developments on the Moser– Trudinger–Adams inequalities in the elliptic and subelliptic setting and related PDEs. We will discuss the initial result of Trudinger, Moser and Adams in little detail as these are some of the important points any student of the subject should necessarily know.
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