# Download e-book for kindle: Around the Research of Vladimir Maz'ya I: Function Spaces by Farit Avkhadiev, Ari Laptev (auth.), Ari Laptev (eds.)

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

ISBN-10: 1441913408

ISBN-13: 9781441913401

ISBN-10: 1441913416

ISBN-13: 9781441913418

ISBN-10: 5901873416

ISBN-13: 9785901873410

International Mathematical sequence quantity 11

Around the study of Vladimir Ma'z'ya I

Function Spaces

Edited via Ari Laptev

Professor Maz'ya is likely one of the most popular 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).

Professor Maz'ya is without doubt one of the premiere gurus in a variety of fields of sensible 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 themes are mentioned during this quantity: Orlicz-Sobolev areas, weighted Sobolev areas, Besov areas with destructive exponents, Dirichlet areas and comparable 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 house surroundings, Hardy sort, Sobolev, Poincare, and pseudo-Poincare inequalities in numerous 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 potential 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 collage London (UK) and Royal Institute of know-how (Sweden). Ari Laptev is a world-recognized professional in Spectral conception 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 fresh advances in a number of fields of arithmetic via the sequence Editor and a founding father of the IMS Tamara Rozhkovskaya.

Cover photo: Vladimir Maz'ya

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5. Any κ-concave probability measure µ on Rn , −∞ < κ < 0, such that α |x| exp dµ(x) 2, α, λ > 0, λ satisfies the weak Poincar´e type inequality with rate function C(p) = Cλ (1 − κ) 3 2−p 1/α , where C is a universal constant. 7 Examples. Perturbation Given a spherically invariant, absolutely continuous probability measure µ on Rn , we write its density in the form p(x) = 1 −V (|x|) e , Z x ∈ Rn , Distributions with Slow Tails 37 where V = V (t) is defined and finite for t > 0 and Z is a normalizing factor.

As we know, the (unique) point 2 of maximum of ϕ on [0, +∞) is ε0 = Qα log 1 and, at this point, s ϕ(ε0 ) = 2 Qα log 2/α 1 s e−2/α = 2 Qαe 2/α 1 log2/α 1s . 44 S. Bobkov and B. , s 2/α a2 b2/α Qαe = a2 b2/α ϕ(ε0 ) 2 log2/α 1 s 2a2 (Qbe)2/α log2/α 1 , s where we used ( α2 )2/α e1/e < 2. 12) is proved. Now, we assume that s e−2/Qα . Then ϕ is increasing and is maximized a2 b2/α s−Q . So, we need a bound of on [0,1] at ε = 1, which gives βq (s) the form s−Q A log2/α (2/s) or, equivalently, s log2/Qα (2/s) A−1/Q in the interval e−2/Qα s 1.

Indeed, then we may use f p C γ+1 , so E |f |p C p(γ+1) . 5) would follow from 1 2pp , which is true since, on the positive half-axis, the function 2pp is minimized at p = 1e and has the minimum value 2e−1/e > 1. 5) holds in the range p > 0. Now, by the Chebyshev inequality, for any t > 0 µ{|f | t} E |f |p tp 2γ+1 (Cp)p(γ+1) = 2γ+1 (Dq)q , tp where q = p (γ + 1) and D = (γ+1) tC1/(γ+1) . The quantity (Dq)q is minimized, when q = 1/(De), and the minimum is e−1/(De) = exp − (γ + 1) t1/(γ+1) Ce = exp − 4 (γ + 1) t1/(γ+1) .

### Around the Research of Vladimir Maz'ya I: Function Spaces by Farit Avkhadiev, Ari Laptev (auth.), Ari Laptev (eds.)

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