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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731492

Digital Object Identifier
doi:10.2140/apde.2014.7.345

Mathematical Reviews number (MathSciNet)
MR3218812

Zentralblatt MATH identifier
1302.35133

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.apde/1513731492


Export citation

References

  • B. Abbondanza and A. Bonfiglioli, “The Dirichlet problem and the inverse mean-value theorem for a class of divergence form operators”, J. Lond. Math. Soc. $(2)$ 87:2 (2013), 321–346.
  • D. H. Armitage and S. J. Gardiner, Classical potential theory, Springer, London, 2001.
  • S. Axler, P. Bourdon, and W. Ramey, “Bôcher's theorem”, Amer. Math. Monthly 99:1 (1992), 51–55.
  • A. Bonfiglioli and E. Lanconelli, “Gauge functions, eikonal equations and Bôcher's theorem on stratified Lie groups”, Calc. Var. Partial Differential Equations 30:3 (2007), 277–291.
  • A. Bonfiglioli and E. Lanconelli, “A new characterization of convexity in free Carnot groups”, Proc. Amer. Math. Soc. 140:9 (2012), 3263–3273.
  • A. Bonfiglioli and E. Lanconelli, “Subharmonic functions in sub-Riemannian settings”, J. Eur. Math. Soc. $($JEMS$)$ 15:2 (2013), 387–441.
  • A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer, Berlin, 2007.
  • J.-M. Bony, “Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés”, Ann. Inst. Fourier $($Grenoble$)$ 19:1 (1969), 277–304.
  • C. Constantinescu and A. Cornea, Potential theory on harmonic spaces, Grundlehren der Mathematischen Wissenschaften 158, Springer, New York, 1972.
  • D. Danielli, N. Garofalo, and D.-M. Nhieu, “Notions of convexity in Carnot groups”, Comm. Anal. Geom. 11:2 (2003), 263–341.
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions, vol. 1, London Mathematical Society Monographs 9, Academic Press, London, 1976.
  • L. H örmander, Notions of convexity, Progress in Mathematics 127, Birkhäuser, Boston, 1994.
  • P. Juutinen, G. Lu, J. J. Manfredi, and B. Stroffolini, “Convex functions on Carnot groups”, Rev. Mat. Iberoam. 23:1 (2007), 191–200.
  • G. Lu, J. J. Manfredi, and B. Stroffolini, “Convex functions on the Heisenberg group”, Calc. Var. Partial Differential Equations 19:1 (2004), 1–22.
  • V. Magnani and M. Scienza, “Regularity estimates for convex functions in Carnot–Carathéodory spaces”, preprint, 2012.
  • P. Negrini and V. Scornazzani, “Wiener criterion for a class of degenerate elliptic operators”, J. Differential Equations 66:2 (1987), 151–164.