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

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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 and 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.
[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.
[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.
[4]	H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics 22 Springer-Verlag, Berlin-New York, 1966.
[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.
[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.
[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.
[8]	S. Biagi and A. Bonfiglioli, Global Heat kernels for parabolic homogeneous Hörmander operators, preprint, arXiv: 1910.09907
[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.
[10]	S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates for the fundamental solution of homogeneous Hörmander sums of squares, arXiv: 1906.07836.
[11]	S. Biagi and M. Bramanti, Global Gaussian estimates for the heat kernel of homogeneous sums of squares, to appear in Potential Anal.
[12]	S. Biagi and M. Bramanti, Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds, arXiv: 2011.09322.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, New York, N.Y., 2007.
[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.
[21]	M. Brelot, Axiomatique des Fonctions Harmoniques, Les Presses de l'Université de Montréal, Montréal, 1969.
[22]	M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1960.
[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.
[24]	C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, 1972.
[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.
[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.
[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.
[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.
[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.
[30]	N. du Plessis, An Introduction to Potential Theory, Oliver and Boyd, Edinburgh, 1970.
[31]	G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.
[32]	R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571.
[33]	L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.
[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.
[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.
[36]	W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.
[37]	F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, London, 1967.

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.
[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.
[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.
[4]	H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics 22 Springer-Verlag, Berlin-New York, 1966.
[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.
[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.
[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.
[8]	S. Biagi and A. Bonfiglioli, Global Heat kernels for parabolic homogeneous Hörmander operators, preprint, arXiv: 1910.09907
[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.
[10]	S. Biagi, A. Bonfiglioli and M. Bramanti, Global estimates for the fundamental solution of homogeneous Hörmander sums of squares, arXiv: 1906.07836.
[11]	S. Biagi and M. Bramanti, Global Gaussian estimates for the heat kernel of homogeneous sums of squares, to appear in Potential Anal.
[12]	S. Biagi and M. Bramanti, Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds, arXiv: 2011.09322.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, New York, N.Y., 2007.
[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.
[21]	M. Brelot, Axiomatique des Fonctions Harmoniques, Les Presses de l'Université de Montréal, Montréal, 1969.
[22]	M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1960.
[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.
[24]	C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, 1972.
[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.
[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.
[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.
[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.
[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.
[30]	N. du Plessis, An Introduction to Potential Theory, Oliver and Boyd, Edinburgh, 1970.
[31]	G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.
[32]	R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571.
[33]	L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.
[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.
[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.
[36]	W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.
[37]	F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, London, 1967.

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