Revista Matemática Complutense

, Volume 29, Issue 2, pp 271–294 | Cite as

New results concerning Moser-type inequalities in Lorentz-Sobolev spaces

  • Robert ČernýEmail author


Let \(n\in \mathbb {N}\), \(n\ge 2\), \(q\in (1,\infty ]\) and let \(\Omega \subset \mathbb {R}^n\) be an open bounded set. We obtain sharp constants concerning the Moser-type inequalities corresponding to the Lorentz-Sobolev space \(W_0^1L^{n,q}(\Omega )\) equipped with the norm
$$\begin{aligned} \Vert \nabla u\Vert _{(n,q)}:= {\left\{ \begin{array}{ll}\Vert t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{**}(t)\Vert _{L^q((0,\infty ))}&{}\text {for }q\in (1,\infty )\\ \sup _{t\in (0,\infty )}t^{\frac{1}{n}}|\nabla u|^{**}(t)&{}\text {for }q=\infty . \end{array}\right. } \end{aligned}$$
We also derive the key estimate for the Concentration-Compactness Principle in the case \(q\in (1,\infty )\) with respect to the above norm.


Sobolev spaces Lorentz-Sobolev spaces Moser-Trudinger inequality Concentration-Compactness Principle Sharp constants 

Mathematics Subject Classification

46E35 46E30 26D10 



The author was supported by the ERC CZ Grant LL1203 of the Czech Ministry of Education. The author would like to thank Luboš Pick for fruitful discussions.


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Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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