By Farit Avkhadiev, Ari Laptev (auth.), Ari Laptev (eds.)

International Mathematical sequence quantity 11

Around the study of Vladimir Ma'z'ya I

Function Spaces

Edited via Ari Laptev

Professor Maz'ya is among the most suitable professionals in numerous fields of useful research and partial differential equations. specifically, Maz'ya is a proiminent determine within the improvement of the speculation of Sobolev areas. he's the writer of the well known monograph Sobolev areas (Springer, 1985).

Professor Maz'ya is without doubt one of the most well known gurus in numerous fields of sensible research and partial differential equations. specifically, Maz'ya is a proiminent determine within the improvement of the idea of Sobolev areas. he's the writer of the well known monograph Sobolev areas (Springer, 1985). the next themes are mentioned during this quantity: Orlicz-Sobolev areas, weighted Sobolev areas, Besov areas with unfavourable exponents, Dirichlet areas and similar variational capacities, classical inequalities, together with Hardy inequalities (multidimensional types, the case of fractional Sobolev areas etc.), Hardy-Maz'ya-Sobolev inequalities, analogs of Maz'ya's isocapacitary inequalities in a measure-metric area environment, Hardy style, Sobolev, Poincare, and pseudo-Poincare inequalities in several contexts together with Riemannian manifolds, measure-metric areas, fractal domain names etc., Mazya's capacitary analogue of the coarea inequality in metric likelihood areas, sharp constants, extension operators, geometry of hypersurfaces in Carnot teams, Sobolev homeomorphisms, a speak to the Maz'ya inequality for capacities and functions of Maz'ya's skill method.

Contributors comprise: Farit Avkhadiev (Russia) and Ari Laptev (UK—Sweden); Sergey Bobkov (USA) and Boguslaw Zegarlinski (UK); Andrea Cianchi (Italy); Martin Costabel (France), Monique Dauge (France), and Serge Nicaise (France); Stathis Filippas (Greece), Achilles Tertikas (Greece), and Jesper Tidblom (Austria); Rupert L. Frank (USA) and Robert Seiringer (USA); Nicola Garofalo (USA-Italy) and Christina Selby (USA); Vladimir Gol'dshtein (Israel) and Aleksandr Ukhlov (Israel); Niels Jacob (UK) and Rene L. Schilling (Germany); Juha Kinnunen (Finland) and Riikka Korte (Finland); Pekka Koskela (Finland), Michele Miranda Jr. (Italy), and Nageswari Shanmugalingam (USA); Moshe Marcus (Israel) and Laurent Veron (France); Joaquim Martin (Spain) and Mario Milman (USA); Eric Mbakop (USA) and Umberto Mosco (USA ); Emanuel Milman (USA); Laurent Saloff-Coste (USA); Jie Xiao (USA)

Ari Laptev -Imperial collage London (UK) and Royal Institute of expertise (Sweden). Ari Laptev is a world-recognized expert in Spectral concept of Differential Operators. he's the President of the eu Mathematical Society for the interval 2007- 2010.

Tamara Rozhkovskaya - Sobolev Institute of arithmetic SB RAS (Russia) and an self reliant writer. Editors and Authors are completely invited to give a contribution to volumes highlighting fresh advances in numerous fields of arithmetic via the sequence Editor and a founding father of the IMS Tamara Rozhkovskaya.

Cover snapshot: Vladimir Maz'ya

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**Sample text**

It is clear that property 1) is fulfilled. If α 1, V is convex and the measure µ is log-concave. So, assume that 0 < α < 1, in which case V is not convex. It is easy to verify, the inequality of property 2’) holds for all t > 0 if and only it holds for t = 0, and then it reads as (α − 1) − ακn aα 0. Hence an optimal choice is κn = − 1−α αaα or, equivalently, κ=− 1−α αaα − n(1 − α) provided that αaα − n(1 − α) > 0. 2) Conclusion 1. 2). In other words, µ is convex only if the parameter a is sufficiently large.

It is easy to verify, the inequality of property 2’) holds for all t > 0 if and only it holds for t = 0, and then it reads as (α − 1) − ακn aα 0. Hence an optimal choice is κn = − 1−α αaα or, equivalently, κ=− 1−α αaα − n(1 − α) provided that αaα − n(1 − α) > 0. 2) Conclusion 1. 2). In other words, µ is convex only if the parameter a is sufficiently large. 3) 1/α , where C depends on the parameters a, b, α and the dimension n. 4) 38 S. Bobkov and B. 3) to hold with some rate function. 1. 3). 5) c1 w(x) c2 , x ∈ M, for some c1 , c2 > 0.

110, 55–67 (1999) 7. : Sharp Hardy–Sobolev inequalities. C. , Acad. Sci. Paris 339, no. 7, 483–486 (2004) 8. : Sharp two-sided heat kernel estimates for critical Schr¨ odinger operators on bounded domains. Commun. Math. Phys. 273, 237–281 (2007) 9. : Non-linear ground state representations and sharp Hardy inequalities. AP] 4 Mar. 2008 12 F. Avkhadiev and A. Laptev 10. : Vorlesungen u ¨ber Inhalt, Oberfl¨ asche und Isoperimetrie. Springer (1957) 11. : A geometrical version of Hardy’s inequalities.