Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 345-373.

Convexity of average operators for subsolutions to subelliptic equations

Andrea Bonfiglioli, Ermanno Lanconelli, and Andrea Tommasoli

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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 .

Article information

Anal. PDE, Volume 7, Number 2 (2014), 345-373.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A51: Convexity, generalizations 31B05: Harmonic, subharmonic, superharmonic functions 35H10: Hypoelliptic equations
Secondary: 31B10: Integral representations, integral operators, integral equations methods 35J70: Degenerate elliptic equations

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


Bonfiglioli, Andrea; Lanconelli, Ermanno; Tommasoli, Andrea. Convexity of average operators for subsolutions to subelliptic equations. Anal. PDE 7 (2014), no. 2, 345--373. doi:10.2140/apde.2014.7.345.

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