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$ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients
Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings
1. | Free researcher |
2. | Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy |
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 $.
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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