Self Improving Sobolev-Poincaré Inequalities, Truncation and Symmetrization

Abstract

Abstract. In [12] we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in W 1,1 0 (Ω). In this paper we extend our method to Sobolev functions that do not vanish at the boundary.

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

“…[29,31] and the references therein), the new Gaussian counterpart (1.11) we obtain here leads to corresponding optimal Gaussian Sobolev-Poincaré inequalities as well. The corresponding analog of (1.6) is: given any rearrangement-invariant space X on the interval (0, 1), we have the optimal inequality, valid for Lip functions (cf.…”

Pointwise symmetrization inequalities for Sobolev functions and applications

“…Therefore, by Lemma 2 of[85], we see that (3.16) holds.Remark 3. Suppose that there exists α > 1, such that the isoperimetric estimator I α is concave.…”

Isoperimetry and symmetrization for Sobolev spaces on metric spaces

“…[4]), and our methods can be readily adapted to provide an almost painless extension. In particular, the results of this note, when combined with the method developed 1 in [10], produce concentration inequalities in metric spaces, as well as a Sobolev metric space version of the Pólya-Szegö principle; while our results combined with the method of [ [8], Theorem 3] imply metric Faber-Krahn inequalities.…”