Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R^n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.Comment: To appear in Advances in Mathematics. 54 Pages, minor changes and corrections in v2 (page 1, proof of Corollary 13, some typos). In v3 the more relevant changes/corrections were made on pages 9, 10, 27, 32, 34, 36, 40, 41, 47. Minor corrections in v

Optimal limiting embeddings for Δ\Delta-reduced Sobolev spaces in L1L^1

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with $L^1$ data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces $W_0^{k,n/k}$.Comment: In v3 the proof of the sharpness of (25) and (29) was corrected, and minor other changes and corrections were added. To appear in Ann. Inst. H. Poincar\'e Anal. Non Lin\'eair