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2014 Convexity of average operators for subsolutions to subelliptic equations
Andrea Bonfiglioli, Ermanno Lanconelli, Andrea Tommasoli
Anal. PDE 7(2): 345-373 (2014). DOI: 10.2140/apde.2014.7.345


We study convexity properties of the average integral operators naturally associated with divergence-form second-order subelliptic operators with nonnegative characteristic form. When 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

Ω r ( x ) = { y : Γ ( x , y ) = 1 r }

of the fundamental solution Γ(x,y) of . We shall obtain characterizations of the -subharmonic functions u (that is, the weak solutions to u0) in terms of the convexity (w.r.t. a power of r) of the average of u over Ωr(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 .


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Andrea Bonfiglioli. Ermanno Lanconelli. Andrea Tommasoli. "Convexity of average operators for subsolutions to subelliptic equations." Anal. PDE 7 (2) 345 - 373, 2014.


Received: 11 December 2012; Accepted: 21 May 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1302.35133
MathSciNet: MR3218812
Digital Object Identifier: 10.2140/apde.2014.7.345

Primary: 26A51‎ , 31B05 , 35H10
Secondary: 31B10 , 35J70

Keywords: average integral operator , Convex functions , divergence-form operator. , hypoelliptic operator , subharmonic functions

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 2 • 2014
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