Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces RN, hyperbolic spaces and other settings, and such sharp inequalities of Lions type. On the other hand, we present and prove some new results on sharp singular Moser-Trudinger and Adams type inequalities with exact growth condition and their affine analogues of such inequalities (Theorems 1.1, 1.2 and 1.3). We also establish a sharpened version of the classical Moser-Trudinger inequality on finite balls (Theorem 1.4).