Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type. (April 2020)

• Record Type: Journal Article
• Authors: Garofalo, Nicola; Tralli, Giulio
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: 35H10 -- 35R11 -- 26D10 -- 46E99; Hörmander operators -- Kolmogorov equation -- Fractional powers -- Besov spaces -- Poincaré inequalities; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.06.026
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a class of second-order partial differential operators A of Hörmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the '30s. We analyse some properties of the nonlocal operators driven by the fractional powers of A, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincaré inequalities adapted to the underlying geometry.