Some global Sobolev inequalities related to Kolmogorov-type operators
DOI:
https://doi.org/10.6092/issn.2240-2829/10584Keywords:
Global a priori estimates, Kolmogorov-Fokker-Planck diffusion, fractional powers of hypoelliptic operatorsAbstract
In this note we review a recent result in [17] in collaboration with N. Garofalo, where we establish global versions of Hardy-Littlewood-Sobolev inequalities attached to hypoelliptic equations of Kolmogorov type. The relevant Sobolev spaces are defined through the fractional powers of the operator under consideration. We outline the main steps of the semigroup approach we adopt.
References
F. Abedin, G. Tralli. Harnack inequality for a class of Kolmogorov-Fokker-Planck equations in non-divergence form. Arch. Rational Mech. Anal., 233 (2019) 867-900.
F. Anceschi, M. Eleuteri, S. Polidoro. A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients. Commun. Contemp. Math., 21 (2019) 1850057.
F. Anceschi, S. Polidoro, M.A. Ragusa. Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients. Nonlinear Anal., 189 (2019) 111568, doi.org/10.1016/j.na.2019.07.001.
F. Anceschi, S. Polidoro. A survey on the classical theory for Kolmogorov equation. ArXiv preprint 1907.05155.
A.V. Balakrishnan. Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math., 10 (1960) 419-437.
V.I. Bogachev. Ornstein-Uhlenbeck operators and semigroups. Russian Math. Surveys, 73 (2018) 191-260.
M. Bramanti, G. Cupini, E. Lanconelli, E. Priola. Global L^p estimates for degenerate Ornstein-Uhlenbeck operators. Math. Z., 266 (2010) 789-816.
F. Buseghin, N. Garofalo. A chain rule for a class of evolutive nonlocal hypoelliptic equations. ArXiv preprint 1910.08803.
S. Chandrasekhar. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys., 15 (1943) 1-89.
C. Cinti, A. Pascucci, S. Polidoro. Pointwise estimates for a class of non-homogeneous Kolmogorov equations. Math. Ann., 340 (2008) 237-264.
G. Citti, A. Pascucci, S. Polidoro. Regularity properties of viscosity solutions of a non-H ̈ormander degenerate equation. J. Math. Pures Appl. (9), 80 (2001) 901-918.
F. Ferrari, B. Franchi. Harnack inequality for fractional sub-Laplacians in Carnot groups. Math. Z., 279 (2015) 435-458.
N. Garofalo, E. Lanconelli. Level sets of the fundamental solution and Harnack inequality for de- generate equations of Kolmogorov type. Trans. Amer. Math. Soc., 321 (1990) 775-792.
N. Garofalo. Some properties of sub-Laplacians. Proceedings of the Conference “Two nonlinear days in Urbino 2017”, Elect. J. Diff. Equations, 25 (2018) 103-131.
N. Garofalo, G. Tralli. A class of nonlocal hypoelliptic operators and their extensions. To appear in Indiana Univ. Math. J. (preprint at www.iumj.indiana.edu/IUMJ/forthcoming.php).
N. Garofalo, G. Tralli. Functional inequalities for class of nonlocal hypoelliptic equations of H ̈ormander type. To appear in Nonlinear Anal., doi.org/10.1016/j.na.2019.06.026, Special Issue ‘Fractional and Nonlocal equations’.
N. Garofalo, G. Tralli. Hardy-Littlewood-Sobolev inequalities for a class of non-symmetric and non- doubling hypoelliptic semigroups. ArXiv preprint 1904.12982.
N. Garofalo, G. Tralli. Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators. ArXiv preprint 1907.02281.
F. Golse, C. Imbert, C. Mouhot, A.F. Vasseur. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19 (2019) 253-295.
L. H ̈ormander. Hypoelliptic second order differential equations. Acta Math., 119 (1967) 147-171.
A.E. Kogoj, G. Tralli. Blaschke, Privaloff, Reade and Saks theorems for diffusion equations on Lie groups. Potential Anal., 38 (2013) 1103-1122.
A.E. Kogoj, E. Lanconelli, G. Tralli. An inverse mean value property for evolution equations. Adv. Differential Equations, 19 (2014) 783-804.
A.E. Kogoj, E. Lanconelli, G. Tralli. Wiener-Landis criterion for Kolmogorov-type operators. Dis- crete Contin. Dyn. Syst., 38 (2018) 2467-2485.
A.N. Kolmogorov. Zuf ̈allige Bewegungen (Zur Theorie der Brownschen Bewegung). Ann. of Math. (2), 35 (1934) 116-117.
E. Lanconelli, S. Polidoro. On a class of hypoelliptic evolution operators. Partial differential equa- tions, II (Turin, 1993). Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994) 29-63.
E. Lanconelli, F. Lascialfari. A boundary value problem for a class of quasilinear operators of Fokker- Planck type. Proceedings of the Conference “Differential Equations” (Ferrara, 1996), Ann. Univ. Ferrara Sez. VII (N.S.), 41 (1996) 65-84.
E. Lanconelli, A. Pascucci, S. Polidoro. Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. Nonlinear problems in mathematical physics and related topics, II, 243265, Int. Math. Ser. (N. Y.) 2, Kluwer/Plenum, New York, 2002.
A. Lanconelli, A. Pascucci, S. Polidoro. Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients. To appear in J. Evol. Equ. (ArXiv preprint 1704.07307).
F. Lascialfari, D. Morbidelli. A boundary value problem for a class of quasilinear ultraparabolic equations. Comm. Partial Differential Equations, 23 (1998) 847-868.
N. Lohou ́e. Puissances complexes de l’op ́erateur de Laplace-Beltrami. C. R. Acad. Sci. Paris S ́er. A-B, 290 (1980) 605-608.
M. Manfredini. The Dirichlet problem for a class of ultraparabolic equations. Adv. Differential Equa- tions, 2 (1997) 831-866.
A. Pascucci, S. Polidoro. The Moser’s iterative method for a class of ultraparabolic equations. Commun. Contemp. Math., 6 (2004) 395-417.
E. Priola, J. Zabczyk. Liouville theorems for non-local operators. J. Funct. Anal., 216 (2004) 455- 490.
E.M. Stein. Topics in harmonic analysis related to the Littlewood-Paley Theory, Annals of Mathematics Studies no. 63, Princeton University Press, Princeton (1970).
N.Th. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal., 63 (1985) 240-260.
W. Wang, L. Zhang. The Cα regularity of weak solutions of ultraparabolic equations. Discrete Contin. Dyn. Syst., 29 (2011) 1261-1275.
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