Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for ...

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the unif...

‎By considering a degenerate $p(x)-$Laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent Sobolev space is presented‎. ‎Multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding Nehari manifold‎. 

Let 0 < a < 1 and set Φ(t) = |t|, ∀ t ∈ R. Let 1 < p < ∞ and n ≥ 1. We prove that the superposition operator u 7→ Φ(u) maps the Sobolev space W (R) into the fractional Sobolev space W (R). We also investigate the case of more general nonlinearities. Résumé. Superposition avec des puissances sousunitaires dans les espaces de Sobolev. Pour 0 < a < 1, soit Φ(t) = |t|, ∀ t ∈ R. Soient 1 < p < ∞ et ...

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and FaberKrahn estimates for Hörmander vector fields.

In this paper we prove a sharp affine Lp Sobolev inequality for functions on R. The new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin [A2] and Talenti [T], even though it uses only the vector space structure and standard Lebesgue measure on R. For the new inequality, no inner product, norm, or conformal structure is needed at...

We deal with Orlicz-Sobolev embeddings in open subsets of R. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces ...

In view of the good properties of nonstationary wavelet frames and the better flexibility of wavelets in Sobolev spaces, the nonstationary dual wavelet frames in a pair of dual Sobolev spaces are studied in this paper. We mainly give the oblique extension principle and the mixed extension principle for nonstationary dual wavelet frames in a pair of dual Sobolev spaces H(R) and H−s(Rd). Keywords...

We investigate the steady transport equation λz + w · ∇z + az = f, λ > 0 in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions w, a are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular ...

In our recent paper [12] we developed a new principle of “symmetrization by truncation” to obtain symmetrization inequalities of Sobolev type via truncation. In this note we consider the corresponding results for Sobolev spaces on domains, without assuming that the Sobolev functions vanish at the boundary. The explicit connection between Sobolev-Poincaré inequalities and isoperimetric inequalit...