It is shown that if the Boyd indices of the rearrangement invariant Banach function space Lρ(R) are strictly between 0 and 1, ρ is an absolutely continuous function norm, Ω is a domain from R satisfying the restricted cone condition, denoting by ω the restriction of ρ to Ω, there exists an extension operator for the abstract Sobolev space W Lω(Ω). This is a generalization to abstract Sobolev sp...

We prove some general results on the existence of partitions of unity in Sobolev type spaces on various innnite dimensional manifolds. As special cases we obtain in particular, (continuous) partitions of unity a) in the Malliavin test functions on an abstract Wiener space; b) in the rst order Sobolev space on pinned and free loop spaces; c) in the rst order Sobolev spaces associated with revers...

Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of Hamilton–Jacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton–Jacobi equations. We obtain also ultracontractive estimations under general ...

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz’ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also gi...

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L−Sobolev inequalities. The logarithmic version of affine L−Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasur...

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on Rn, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly conve...

This paper is concerned with the existence of two non-trivial weak solutions for a p(x)-Kirchho type problem by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle and the theory of the variable exponent Sobolev spaces.