Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics

**Code:** J1-1690

**Project:** p-Ellipticity in Harmonic Analysis and Partial Differential Equations

**Period:** 1. 7. 2019 - 30. 6. 2023

**Range per year:** 1,45 FTE, **category:** C

**Head:** Oliver Dragičević

**Research activity**: Natural sciences and mathematics

**Citations for bibliographic records**

**Project description:**

The project primarily concerns the area of harmonic analysis and partial differential equations (PDE). Thus it lies at the junction of two very active fields of contemporary mathematics. The core of our interest is a concept named *p*‐ellipticity, that was introduced by the principal investigator and his collaborator from the University of Genova, A. Carbonaro, in a joint paper in J. Eur. Math. Soc. (accepted for publication). In the paper they, among the rest, establish connections between this condition and several phenomena concerning the *Lp* theory of elliptic differential operators in divergence form. Independently of them, M. Dindoš in J. Pipher found a condition that permits extending a variant of reverse Hölder inequalities, which are a key step in Moser's proof of the celebrated De Giorgi ‐ Nash ‐ Moser theorem, to operators with complex coefficients. It turned out that their condition was equivalent to *p*‐ellipticity, i.e., it is precisely its reformulation. Likewise, it emerged that by means of *p*‐ellipticity one may reformulate a very similar (yet not identical) condition, considered in 2005 by A. Cialdea and V. Maz'ya in the context of *Lp* dissipativity of sesquilinear forms which give rise to elliptic operators with smooth complex coefficients, and *Lp* contractivity of the corresponding operator semigroups. All this clearly testifies about the significance of *p*‐ellipticity. The primary aim of the project is to further investigate the reach of this condition in harmonic analysis and PDE.

Apart from *p*‐ellipticity, attention within the project itself will be devoted to other mathematical problems, as well (dispersive estimates for Schrödinger operators and the analysis of Lie grupoids).