Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
type: Item
instance of: scholarly article
title: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
zbMATH Open document ID: 1175.35081
DOI: 10.4171/RMI/565
author: Sergio Polidoro
author: Maria Alessandra Ragusa
published in: Revista Matemática Iberoamericana
publication date: 2009-02-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1228834303
full work available at URL: https://eudml.org/doc/42405
review text: This article deals with regularity results for solutions of ultraparabolic equations. More precisely the following equation in \(\mathbb{R}^{N+1}\) is considered: \[ {\mathcal L}_vu:-{\mathcal L}_0u+{\mathcal V}u= 0\tag{1} \] where \({\mathcal L}_0\) is a linear second order operator of the form \[ {\mathcal L}_0=\sum^m_{k=1}X^2_k+X_0-\partial_t \tag{2} \] the terms \(X_k\) denote smooth vector fields on \(\mathbb{R}^N\) i.e. \[ X_k(x)= \sum^N_{j-1}a^k_j(x)\partial_{c_j}\;k=0,1,\dots,m.\tag{3} \] Finally \({\mathcal V}\) is a singular potential belonging to a Stummel-Kato class defined by the fundamental solution \(\Gamma_0\) of \({\mathcal L}_0\). The main result is an invariant Harnack inequality for positive solutions of (1) that extends previous results of A. Kogoj and E. Lanconelli for the solutions of the equation \({\mathcal L}_0u=0\). The proof, inspired by some arguments by Safonov, is based on pointwise lower bounds for the Green function \(G_0\) of \({\mathcal L}_0\) related to suitable ``cylindrical'' open sets. The Green function for the operator \({\mathcal L}_\nu\) is also constructed by the Levi parametrix method and \(L^p\) estimates and pointwise lower bounds are proved. An uniqueness result for the Cauchy-Dirichlet problem is also obtained.
reviewed by: Maria Agostina Vivaldi
Mathematics Subject Classification ID: 35K70
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5509272
zbMATH Keywords: hypoelliptic operator
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Green function
MaRDI profile type: Publication
cites work: Sub-Laplacians with drift on Lie groups of polynomial volume growth
cites work: Stratified Lie Groups and Potential Theory for their Sub-Laplacians
cites work: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées
cites work: Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs
cites work: Harnack's Inequality for Sum of Squares of Vector Fields Plus a Potential
cites work: The \(L^ p\)-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations
cites work: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
cites work: Subelliptic estimates and function spaces on nilpotent Lie groups
cites work: Level Sets of the Fundamental Solution and Harnack Inequality for Degenerate Equations of Kolmogorov Type
cites work: Hypoelliptic second order differential equations
cites work: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations
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cites work: Link of groups and homogeneous Hörmander operators
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cites work: On the fundamental solution for hypoelliptic second order partial differential equations with non-negative characteristic form
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cites work: On Harnack's inequality for a class of strongly degenerate Schrödinger operators formed by vector fields
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description: scientific article; zbMATH DE number 5509272