Extremals for a Moser-jodeit Exponential Inequality
نویسندگان
چکیده
Trudinger and Moser, interested in certain nonlinear problems in differential geometry, showed that if |∇u|q is integrable on a bounded domain in R with q ≥ n ≥ 2, then u is exponentially integrable there. Symmetrization reduces the problem to a one-dimensional inequality, which Jodeit extended to q > 1. Carleson and Chang proved that this inequality has extremals when q ≥ 2 is an integer. Hence, so does the Moser-Trudinger inequality (with q = n). This paper extends the result of Carleson and Chang to all real numbers q > 1. An application and some related results involving noninteger q are also discussed.
منابع مشابه
Decomposition and Moser’s Lemma
Using the idea of the optimal decomposition developed in recent papers [EK2] by the same authors and in [CUK] we study the boundedness of the operator Tg(x) = ∫ 1 x g(u) du/u, x ∈ (0, 1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser’s lemma, O’Neil’s convolution inequality, and estimates for ...
متن کاملOn Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces
Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of co...
متن کاملSharp Singular Adams Inequalities in High Order Sobolev Spaces
In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in R. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend...
متن کاملA ug 2 00 9 Adams inequalities on measure spaces
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. 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 sati...
متن کامل2 9 Ju n 20 09 Adams inequalities on measure spaces
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. 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 sati...
متن کامل