2,599 research outputs found
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 -reduced Sobolev spaces in
We prove sharp embedding inequalities for certain reduced Sobolev spaces that
arise naturally in the context of Dirichlet problems with 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 .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
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