Convexity of average operators for subsolutions to subelliptic equations

Bonfiglioli, Andrea; Lanconelli, Ermanno; Tommasoli, Andrea

doi:10.2140/apde.2014.7.345

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Convexity of average operators for subsolutions to subelliptic equations

Andrea Bonfiglioli, Ermanno Lanconelli and Andrea Tommasoli

Vol. 7 (2014), No. 2, 345–373

DOI: 10.2140/apde.2014.7.345

Abstract

We study convexity properties of the average integral operators naturally associated with divergence-form second-order subelliptic operators $ℒ$ with nonnegative characteristic form. When $ℒ$ is the classical Laplace operator, these average operators are the usual average integrals over Euclidean spheres. In our subelliptic setting, the average operators are (weighted) integrals over the level sets

\partial Ω_{r} (x) = {y : Γ (x, y) = 1 ∕ r}

of the fundamental solution $Γ (x, y)$ of $ℒ$ . We shall obtain characterizations of the $ℒ$ -subharmonic functions $u$ (that is, the weak solutions to $- ℒ u \leq 0$ ) in terms of the convexity (w.r.t. a power of $r$ ) of the average of $u$ over $\partial Ω_{r} (x)$ , as a function of the radius $r$ . Solid average operators will be considered as well. Our main tools are representation formulae of the (weak) derivatives of the average operators w.r.t. the radius. As applications, we shall obtain Poisson–Jensen and Bôcher type results for $ℒ$ .

Keywords

subharmonic functions, hypoelliptic operator, convex functions, average integral operator, divergence-form operator.

Mathematical Subject Classification 2010

Primary: 26A51, 31B05, 35H10

Secondary: 31B10, 35J70

Milestones

Received: 11 December 2012

Accepted: 21 May 2013

Published: 30 May 2014

Authors

Andrea Bonfiglioli
Dipartimento di Matematica Università degli Studi di Bologna Piazza di Porta San Donato, 5 I-40126 Bologna Italy

Ermanno Lanconelli
Dipartimento di Matematica Università degli Studi di Bologna Piazza di Porta San Donato, 5 I-40126 Bologna Italy

Andrea Tommasoli
Dipartimento di Matematica Università degli Studi di Bologna Piazza di Porta San Donato, 5 I-40126 Bologna Italy

Vol. 7, No. 2, 2014