Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

Zoltán M. Balogh, Andrea Calogero, A. Kristály

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H-convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H-convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Original languageEnglish
Pages (from-to)2669-2708
Number of pages40
JournalJournal of Functional Analysis
Volume269
Issue number9
DOIs
Publication statusPublished - Nov 1 2015

Fingerprint

Comparison Principle
Heisenberg Group
Maximum Principle
Convex function
Horizontal
Degree Theory
Harnack Inequality
Boundary Behavior
Set-valued Map
Sharpness

Keywords

  • Aleksandrov-type maximum principle
  • Comparison principle
  • H-convex functions
  • Heisenberg group

ASJC Scopus subject areas

  • Analysis

Cite this

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group. / Balogh, Zoltán M.; Calogero, Andrea; Kristály, A.

In: Journal of Functional Analysis, Vol. 269, No. 9, 01.11.2015, p. 2669-2708.

Research output: Contribution to journalArticle

@article{8fd6bafa1d0847a99faee46f137b80a4,
title = "Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group",
abstract = "In this paper we solve a problem raised by Guti{\'e}rrez and Montanari about comparison principles for H-convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H-convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.",
keywords = "Aleksandrov-type maximum principle, Comparison principle, H-convex functions, Heisenberg group",
author = "Balogh, {Zolt{\'a}n M.} and Andrea Calogero and A. Krist{\'a}ly",
year = "2015",
month = "11",
day = "1",
doi = "10.1016/j.jfa.2015.08.014",
language = "English",
volume = "269",
pages = "2669--2708",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "9",

}

TY - JOUR

T1 - Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

AU - Balogh, Zoltán M.

AU - Calogero, Andrea

AU - Kristály, A.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H-convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H-convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

AB - In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H-convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H-convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

KW - Aleksandrov-type maximum principle

KW - Comparison principle

KW - H-convex functions

KW - Heisenberg group

UR - http://www.scopus.com/inward/record.url?scp=84941261327&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941261327&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2015.08.014

DO - 10.1016/j.jfa.2015.08.014

M3 - Article

VL - 269

SP - 2669

EP - 2708

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 9

ER -