Tatyana Shaposhnikova

A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions

Presentation of new results on the latest topics of the theory of Sobolev spaces, partial differential equations, analysis and mathematical physicsThe authors and editors are world-renowned specialists, working in different countriesPublication on the centenary of Sobolev’s birth with two short biographical articles and unique archive photos of S. Sobolev which have not yet been published in the English-language literatureThis volume is dedicated to the centenary of the outstanding mathematician of the XXth century Sergey Sobolev and, in a sense, to his celebrated work On a theorem of functional analysis published in 1938, exactly 70 years ago, where the original Sobolev inequality was proved. This double event is a good case to gather experts for presenting the latest results on the study of Sobolev inequalities which play a fundamental role in analysis, the theory of partial differential equations, mathematical physics, and differential geometry. In particular, the following topics are discussed: Sobolev type inequalities on manifolds and metric measure spaces, traces, inequalities with weights, unfamiliar settings of Sobolev type inequalities, Sobolev mappings between manifolds and vector spaces, properties of maximal functions in Sobolev spaces, the sharpness of constants in inequalities, etc. The volume opens with a nice survey reminiscence My Love Affair with the Sobolev Inequality by David R. Adams.

On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms

Abstract We prove the Gagliardo–Nirenberg type inequality ‖u‖ W θs,p/θ ⩽c(n) p p−1 θ 1−s 1−θ θ/p ‖u‖ θ W s,p ‖u‖ 1−θ L ∞ , where 0 W s,p ( R n ) . The dependence of the constant factor in the right-hand side on each of the parameters s, θ, and p is precise in a sense.

Brezis–Gallouet–Wainger type inequality for irregular domains†

A Brezis–Gallouet–Wainger logarithmic interpolation-embedding inequality is proved for various classes of irregular domains, in particular, for power cusps and λ-John domains. †Dedicated to Viktor Burenkov on the occasion of his 70th birthday.

Traces of multipliers in pairs of weighted Sobolev spaces

We prove that the pointwise multipliers acting in a pair of fractional Sobolev spaces form the space of boundary traces of multipliers in a pair of weighted Sobolev space of functions in a domain.