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

International Mathematical sequence quantity 11

Around the learn of Vladimir Ma'z'ya I

Function Spaces

Edited by way of Ari Laptev

Professor Maz'ya is among the ultimate gurus in a number of fields of practical 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).

Professor Maz'ya is among the premiere experts in a number of fields of practical 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). the subsequent subject matters are mentioned during this quantity: Orlicz-Sobolev areas, weighted Sobolev areas, Besov areas with unfavorable 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 surroundings, Hardy sort, 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 communicate to the Maz'ya inequality for capacities and functions of Maz'ya's ability method.

Contributors contain: 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 university London (UK) and Royal Institute of expertise (Sweden). Ari Laptev is a world-recognized expert in Spectral thought of Differential Operators. he's the President of the ecu Mathematical Society for the interval 2007- 2010.

Tamara Rozhkovskaya - Sobolev Institute of arithmetic SB RAS (Russia) and an self sustaining writer. Editors and Authors are solely invited to give a contribution to volumes highlighting contemporary advances in a number of fields of arithmetic through the sequence Editor and a founding father of the IMS Tamara Rozhkovskaya.

Cover photo: Vladimir Maz'ya

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**Example text**

Then, using the H¨older inequality with exponents α, β > 1 such that α1 + β1 = 1, we get Ef p = E (f p/r )r A(r)r A(r)r p r r p r p r r/2 p E f 2( r −1) |∇f |2 E f 2α ( r −1) r/2α E |∇f |2β r/2β . 7) 1 = 1r − p1 . , 2α Since q > 2, we have r < p, so α > 0. Moreover, α > 1 ⇔ 1r < 12 + p1 which 1 = 1q , so that β = 2q > 1. 7) turns into is fulfilled. Also, put 2β Ef p p r A(r)r r (E f p )r/2α (E |∇f |q )r/2β , which is equivalent to f p A(p, q) ∇f q. 8) In the general case, we split f = f + − f − with f + = max{f, 0} and f = max{−f, 0}.

S a2 b2/α 2 α log 2/α 2 αe e−2/α = 1 s 2/α 1 log2/α 1 s . e−2/α , then 1 αe = a2 b2/α ϕ(ε0 ) 2 2/α log2/α 1 s e1/e a2 (be)2/α log2/α 1 . s Note that, since s < 1/4, the requirement s e−2/α is automatically fulfilled, as long as α 1/ log 2. 7) is thus proved with constants β0 = e1/e and β1 = e. Now, let α < 1/ log 2 and s e−2/α . Then ϕ is increasing and is maximized on [0,1] at ε = 1, which gives β(s) a2 b2/α 1s . So, we need the bound 1 s A log2/α 1 s s 1/4. Since the function t log 1t is decreasing in in the interval e−2/α t 1/e, the optimal value of A is attained at s = 1/4, so A = 4/ log2/α 4.

Avkhadiev was supported by the Russian Foundation for Basic Research (grant 08-01-00381) and by the G¨oran Gustafssons Stiftelse in Sweden. References 1. : Hardy type inequalities in higher dimensions with explicit estimate of constants. Lobachevskii J. Math. 21, 3–31 (2006) 2. : Hardy-type inequalities on planar and spatial open sets. Proc. Steklov Inst. Math. 255, no. 1, 2–12 (2006) 3. : Unified Poincar´e and Hardy inequalities with sharp constants for convex domains. Z. Angew. Math. Mech. 87, 632–642 (2007) 4.