Abstract
We prove some asymptotic characterizations for the subsolutions to a class of diffusion equations on homogeneous Lie groups. These results are the diffusion counterpart of the classical Blaschke, Privaloff, Reade and Saks Theorems for harmonic functions.
Similar content being viewed by others
References
Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Am. Math. Soc. 35, 662–674 (1933)
Blaschke, W.: Ein Mittelwersatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials. Leipz. Ber. 68, 3–8 (1916)
Bonfiglioli, A., Lanconelli, E.: Subharmonic functions in sub-Riemannian settings. J. Eur. Math. Soc. (2012, to appear)
Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin (2007)
Cinti, C.: Sub-solutions and mean value operators for ultraparabolic equations on Lie groups. Math. Scand. 101, 83–103 (2007)
Cinti, C., Lanconelli, E.: Riesz and Poisson–Jensen representation formulas for a class of ultraparabolic operators on Lie groups. Potential Anal. 30, 179–200 (2009)
Freitas, P., Matos, J.P.: On the characterization of harmonic and subharmonic functions via mean-value properties. Potential Anal. 32, 189–200 (2010)
Garofalo, N., Lanconelli, E.: Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients. Math. Ann. 283, 211–239 (1989)
Hörmander, L.: Notions of Convexity. Birkhäuser, Boston (2007)
Kogoj, A.E., Lanconelli, E.: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterr. J. Math. 1, 51–80 (2004)
Kogoj, A.E., Lanconelli, E.: Link of groups and homogeneous Hörmander operators. Proc. Am. Math. Soc. 135, 2019–2030 (2007)
Kozakiewicz, W.: Un théorème sur les opérateurs et son application à la théorie des Laplaciens généralisés. C. R. Soc. sc. Varsovie 26, 18–24 (1933)
Lanconelli, E., Pascucci, A.: Superparabolic functions related to second order hypoellitic operators. Potential Anal. 11, 303–323 (1999)
Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Pol. Torino 52(1), 29–63 (Partial Differ. Equ.) (1994)
Negrini, P., Scornazzani, V.: Superharmonic functions and regularity of boundary points for a class of elliptic-parabolic partial differential operators. Bollettino UMI An. Funz. Appl. Serie VI, Vol. III-C 1, 85–106 (1984)
Pini, B.: Maggioranti e minoranti delle soluzioni delle equazioni paraboliche. Ann. Math. Pures Appl. 37, 249–264 (1954)
Pini, B.: Su un integrale analogo al potenziale logaritmico. Boll. Unione Mat. Ital. 9(3), 244–250 (1954)
Potts, D.H.: A note on the operators of Blaschke and Privaloff. Bull. Am. Math. Soc. 54, 782–787 (1948)
Privaloff, I.: Sur les fonctions harmoniques. Rec. Math. Moscou (Mat. Sbornik) 32, 464–471 (1925)
Privaloff, I.: On a theorem of S. Saks. Rec. Math. Moscou (Mat. Sbornik) 9, 457–460 (1941)
Reade, M.: Some remarks on subharmonic functions. Duke Math. J. 10, 531–536 (1943)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
Saks, S.: On the operators of Blaschke and Privaloff for subharmonic functions. Rec. Math. Moscou (Mat. Sbornik) 9, 451–456 (1941)
Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1995)
Watson, N.A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. 26, 385–417 (1973)
Watson, N.A.: Nevanlinna’s first fundamental theorem for supertemperatures. Math. Scand. 73, 49–64 (1993)
Watson, N.A.: Volume mean values of subtemperatures. Colloq. Math. 86, 253–258 (2000)
Watson, N.A.: Elementary proofs of some basic subtemperatures theorems. Colloq. Math. 94, 111–140 (2002)
Watson, N.A.: A generalized Nevanlinna theorem for supertemperatures. Ann. Acad. Sci. Fenn. Math. 28, 35–54 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kogoj, A.E., Tralli, G. Blaschke, Privaloff, Reade and Saks Theorems for Diffusion Equations on Lie Groups. Potential Anal 38, 1103–1122 (2013). https://doi.org/10.1007/s11118-012-9309-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-012-9309-6