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September  2021, 20(9): 3161-3192. doi: 10.3934/cpaa.2021101

Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings

Beatrice Abbondanza 1, and  Stefano Biagi 2,,

1. 

Free researcher
2. 

Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

* Corresponding author

Received  February 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: The second author is member of INdAM and is partially supported by the INdAM-GNAMPA project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali

In this paper we use a potential-theoretic approach to establish various representation theorems and Poisson-Jensen-type formulas for subharmonic functions in sub-Riemannian settings. We also characterize the Radon measures in $ \mathbb{R}^N $ which are the Riesz-measures of bounded-above subharmonic functions in the whole space $ \mathbb{R}^N $.

Keywords: Representation theorems for subharmonic functions, Poisson-Jensen-type formulas, bounded-above subharmomnic functions, Green functions and potentials.
Mathematics Subject Classification: Primary: 35C15, 35H10, 31B05.
Citation: Beatrice Abbondanza, Stefano Biagi. Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3161-3192. doi: 10.3934/cpaa.2021101
References:
[1]

D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monogr. Math., Springer, London, 2001. doi: 10.1007/978-1-4471-0233-5.  Google Scholar

[2]

E. Battaglia and S. Biagi, Superharmonic functions associated with hypoelliptic non-Hörmander operators, Commun. Contemp. Math., 22 (2020), 32 pp. doi: 10.1142/S0219199718500712.  Google Scholar

[3]

E. Battaglia, S. Biagi and A. Bonfiglioli, The strong maximum principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators, Ann. Inst. Fourier (Grenoble), 66 (2016), 589-631.   Google Scholar

[4]

H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics 22 Springer-Verlag, Berlin-New York, 1966.  Google Scholar

[5]

S. Biagi, On the Gibbons conjecture for homogeneous Hörmander operators, Nonlinear Differ. Equ. Appl., 26 (2019), 26-49.  doi: 10.1007/s00030-019-0594-2.  Google Scholar

[6]

S. Biagi and A. Bonfiglioli, The existence of a global fundamental solution for homogeneous Hörmander operators via a global Lifting method, Proc. Lond. Math. Soc., 114 (2017), 855-889.  doi: 10.1112/plms.12024.  Google Scholar

[7]

S. Biagi and A. Bonfiglioli, An introduction to the Geometrical Analysis of Vector Fields. With Applications To Maximum Principles And Lie Groups, World Scientific Publishing Company, 2018.  Google Scholar

[8]

S. Biagi and A. Bonfiglioli, Global Heat kernels for parabolic homogeneous Hörmander operators, preprint, arXiv: 1910.09907 Google Scholar

[9]

S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares, J. Math. Anal. Appl., 498 (2021). doi: 10.1016/j.jmaa.2021.124935.  Google Scholar

[10]

S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates for the fundamental solution of homogeneous Hörmander sums of squares, arXiv: 1906.07836. Google Scholar

[11]

S. Biagi and M. Bramanti, Global Gaussian estimates for the heat kernel of homogeneous sums of squares, to appear in Potential Anal. Google Scholar

[12]

S. Biagi and M. Bramanti, Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds, arXiv: 2011.09322. Google Scholar

[13]

S. Biagi and E. Lanconelli, Large sets at infinity and Maximum Pinciple on unbounded domains for a class of sub-elliptic operators, J. Differ. Equ., 269 (2020), 9680-9719.  doi: 10.1016/j.jde.2020.06.060.  Google Scholar

[14]

S. Biagi, A. Pinamonti and E. Vecchi, Pohozaev-type identities for differential operators driven by homogeneous vector fields, Nonlinear Differ. Equ. Appl., 28 (2021). doi: 10.1007/s00030-020-00664-6.  Google Scholar

[15]

A. Bonfiglioli and C. Cinti, A Poisson-Jensen type representation formula for subharmonic functions on stratified Lie groups, Potential Anal., 22 (2005), 151-169.  doi: 10.1007/s11118-004-0588-4.  Google Scholar

[16]

A. Bonfiglioli and C. Cinti, The theory of energy for sub-Laplacians with an application to quasi-continuity, Manuscripta Math., 118 (2005), 289-309.  doi: 10.1007/s00229-005-0579-9.  Google Scholar

[17]

A. Bonfiglioli and E. Lanconelli, Subharmonic functions in sub-Riemannian settings, J. Eur. Math. Soc., 15 (2013), 387-441.  doi: 10.4171/JEMS/364.  Google Scholar

[18]

A. Bonfiglioli, E. Lanconelli and A. Tommasoli, Convexity of average operators for subsolutions to subelliptic equations, Anal. Partial Differ. Equ., 7 (2014), 345-373.  doi: 10.2140/apde.2014.7.345.  Google Scholar

[19]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, New York, N.Y., 2007.  Google Scholar

[20]

J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.   Google Scholar

[21]

M. Brelot, Axiomatique des Fonctions Harmoniques, Les Presses de l'Université de Montréal, Montréal, 1969.  Google Scholar

[22]

M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1960.  Google Scholar

[23]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111.  doi: 10.1134/S0081543808010070.  Google Scholar

[24]

C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, 1972.  Google Scholar

[25]

L. D'Ambrosio and E. Mitidieri, Representation formulae of solutions of second order elliptic inequalities, Nonlinear Anal., 178 (2019), 310-336.  doi: 10.1016/j.na.2018.08.014.  Google Scholar

[26]

L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[27]

L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications, SB Math., 201 (2010), 855-871.  doi: 10.1070/SM2010v201n06ABEH004094.  Google Scholar

[28]

L. D'Ambrosio and E. Mitidieri, Liouville Theorems of some second order elliptic inequalities, Preprint, 2018, 40 pp. doi: 10.1016/j.na.2018.08.014.  Google Scholar

[29]

L. D'Ambrosio, E. Mitidieri and S. I. Pohozaev, Representation formulae and inequalities for solutions of a class of second order partial differential equations, Trans. Amer. Math. Soc., 358 (2006), 893-910.  doi: 10.1090/S0002-9947-05-03717-7.  Google Scholar

[30]

N. du Plessis, An Introduction to Potential Theory, Oliver and Boyd, Edinburgh, 1970.  Google Scholar

[31]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.  Google Scholar

[32]

R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571.   Google Scholar

[33]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[34]

E. Mitidieri and S. I. Pohozaev, Positivity property of solutions of some elliptic inequalities on $\mathbb{R}^n$, Dokl. Math., 68 (2003), 339-344.   Google Scholar

[35]

A. Parmeggiani, A remark on the stability of $C^\infty $-hypoellipticity under lower-order perturbations, J. Pseudo-Differ. Oper. Appl., 6 (2015), 227-235.  doi: 10.1007/s11868-015-0118-8.  Google Scholar

[36]

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.  Google Scholar

[37] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, London, 1967.   Google Scholar

show all references

References:
[1]

D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monogr. Math., Springer, London, 2001. doi: 10.1007/978-1-4471-0233-5.  Google Scholar

[2]

E. Battaglia and S. Biagi, Superharmonic functions associated with hypoelliptic non-Hörmander operators, Commun. Contemp. Math., 22 (2020), 32 pp. doi: 10.1142/S0219199718500712.  Google Scholar

[3]

E. Battaglia, S. Biagi and A. Bonfiglioli, The strong maximum principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators, Ann. Inst. Fourier (Grenoble), 66 (2016), 589-631.   Google Scholar

[4]

H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics 22 Springer-Verlag, Berlin-New York, 1966.  Google Scholar

[5]

S. Biagi, On the Gibbons conjecture for homogeneous Hörmander operators, Nonlinear Differ. Equ. Appl., 26 (2019), 26-49.  doi: 10.1007/s00030-019-0594-2.  Google Scholar

[6]

S. Biagi and A. Bonfiglioli, The existence of a global fundamental solution for homogeneous Hörmander operators via a global Lifting method, Proc. Lond. Math. Soc., 114 (2017), 855-889.  doi: 10.1112/plms.12024.  Google Scholar

[7]

S. Biagi and A. Bonfiglioli, An introduction to the Geometrical Analysis of Vector Fields. With Applications To Maximum Principles And Lie Groups, World Scientific Publishing Company, 2018.  Google Scholar

[8]

S. Biagi and A. Bonfiglioli, Global Heat kernels for parabolic homogeneous Hörmander operators, preprint, arXiv: 1910.09907 Google Scholar

[9]

S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares, J. Math. Anal. Appl., 498 (2021). doi: 10.1016/j.jmaa.2021.124935.  Google Scholar

[10]

S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates for the fundamental solution of homogeneous Hörmander sums of squares, arXiv: 1906.07836. Google Scholar

[11]

S. Biagi and M. Bramanti, Global Gaussian estimates for the heat kernel of homogeneous sums of squares, to appear in Potential Anal. Google Scholar

[12]

S. Biagi and M. Bramanti, Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds, arXiv: 2011.09322. Google Scholar

[13]

S. Biagi and E. Lanconelli, Large sets at infinity and Maximum Pinciple on unbounded domains for a class of sub-elliptic operators, J. Differ. Equ., 269 (2020), 9680-9719.  doi: 10.1016/j.jde.2020.06.060.  Google Scholar

[14]

S. Biagi, A. Pinamonti and E. Vecchi, Pohozaev-type identities for differential operators driven by homogeneous vector fields, Nonlinear Differ. Equ. Appl., 28 (2021). doi: 10.1007/s00030-020-00664-6.  Google Scholar

[15]

A. Bonfiglioli and C. Cinti, A Poisson-Jensen type representation formula for subharmonic functions on stratified Lie groups, Potential Anal., 22 (2005), 151-169.  doi: 10.1007/s11118-004-0588-4.  Google Scholar

[16]

A. Bonfiglioli and C. Cinti, The theory of energy for sub-Laplacians with an application to quasi-continuity, Manuscripta Math., 118 (2005), 289-309.  doi: 10.1007/s00229-005-0579-9.  Google Scholar

[17]

A. Bonfiglioli and E. Lanconelli, Subharmonic functions in sub-Riemannian settings, J. Eur. Math. Soc., 15 (2013), 387-441.  doi: 10.4171/JEMS/364.  Google Scholar

[18]

A. Bonfiglioli, E. Lanconelli and A. Tommasoli, Convexity of average operators for subsolutions to subelliptic equations, Anal. Partial Differ. Equ., 7 (2014), 345-373.  doi: 10.2140/apde.2014.7.345.  Google Scholar

[19]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, New York, N.Y., 2007.  Google Scholar

[20]

J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.   Google Scholar

[21]

M. Brelot, Axiomatique des Fonctions Harmoniques, Les Presses de l'Université de Montréal, Montréal, 1969.  Google Scholar

[22]

M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1960.  Google Scholar

[23]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111.  doi: 10.1134/S0081543808010070.  Google Scholar

[24]

C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, 1972.  Google Scholar

[25]

L. D'Ambrosio and E. Mitidieri, Representation formulae of solutions of second order elliptic inequalities, Nonlinear Anal., 178 (2019), 310-336.  doi: 10.1016/j.na.2018.08.014.  Google Scholar

[26]

L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[27]

L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications, SB Math., 201 (2010), 855-871.  doi: 10.1070/SM2010v201n06ABEH004094.  Google Scholar

[28]

L. D'Ambrosio and E. Mitidieri, Liouville Theorems of some second order elliptic inequalities, Preprint, 2018, 40 pp. doi: 10.1016/j.na.2018.08.014.  Google Scholar

[29]

L. D'Ambrosio, E. Mitidieri and S. I. Pohozaev, Representation formulae and inequalities for solutions of a class of second order partial differential equations, Trans. Amer. Math. Soc., 358 (2006), 893-910.  doi: 10.1090/S0002-9947-05-03717-7.  Google Scholar

[30]

N. du Plessis, An Introduction to Potential Theory, Oliver and Boyd, Edinburgh, 1970.  Google Scholar

[31]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.  Google Scholar

[32]

R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571.   Google Scholar

[33]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[34]

E. Mitidieri and S. I. Pohozaev, Positivity property of solutions of some elliptic inequalities on $\mathbb{R}^n$, Dokl. Math., 68 (2003), 339-344.   Google Scholar

[35]

A. Parmeggiani, A remark on the stability of $C^\infty $-hypoellipticity under lower-order perturbations, J. Pseudo-Differ. Oper. Appl., 6 (2015), 227-235.  doi: 10.1007/s11868-015-0118-8.  Google Scholar

[36]

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.  Google Scholar

[37] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, London, 1967.   Google Scholar
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