On improvement of summability properties in nonautonomous Kolmogorov equations

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May 2014, 13(3): 1237-1265. doi: 10.3934/cpaa.2014.13.1237

On improvement of summability properties in nonautonomous Kolmogorov equations

L. Angiuli ^1, and Luca Lorenzi ^2,

1.	Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
2.	Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma

Received June 2013 Revised September 2013 Published December 2013

Abstract
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Under suitable conditions, we obtain some characterizations of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided.

Keywords: unbounded coefficients, supercontractivity, ultracontractivity., Nonautonomous second order elliptic operators, evolution operators, ultraboundedness, Harnack type inequality, evolution systems of measures.

Mathematics Subject Classification: Primary: 35K10; Secondary: 35K15, 37L4.

Citation: L. Angiuli, Luca Lorenzi. On improvement of summability properties in nonautonomous Kolmogorov equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1237-1265. doi: 10.3934/cpaa.2014.13.1237

References:

[1]	L. Angiuli, Pointwise gradient estimates for evolution operators associated with Kolmogorov operators , Arch. Math. (Basel), 101 (2013), 159-170. doi: 10.1007/s00013-013-0542-z.
[2]	L. Angiuli and L. Lorenzi, Compactness and invariance properties of evolution operators associated to Kolmogorov operators with unbounded coefficients , J. Math. Anal. Appl., 379 (2011), 125-149. doi: 10.1016/j.jmaa.2010.12.029.
[3]	L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations , Comm. Partial Differential Equations, 38 (2013), 2049-2080.
[4]	E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions , Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245-287.
[5]	G. Da Prato and A. Lunardi, Ultraboundedness for parabolic equations in convex domains without boundary conditions , Phys. D, 239 (2010), 1453-1457. doi: 10.1016/j.physd.2009.02.004.
[6]	E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians , J. Funct. Anal., 59 (1984), 335-395. doi: 10.1016/0022-1236(84)90076-4.
[7]	E. B. Davies, Heat kernes and spectral theory, Cambridge tracts in Mathematics, 92, Cambridge Univ. Press, Cambridge, 1990. doi: ISBN: 0-521-40997-7.
[8]	M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations , J. Lond. Math. Soc. (2), 77 (2008), 719-740. doi: 10.1112/jlms/jdn009.
[9]	L. Gross, Logarithmic Sobolev inequalities , Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[10]	O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité, J. Funct. Anal., 11 (1993), 155-196. doi: 10.1006/jfan.1993.1008.
[11]	M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients , Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: 10.1090/S0002-9947-09-04738-2.
[12]	M. Ledoux, Remarks on Logarithmic Sobolev constants, exponential integrability, and bounds on the diameter , J. Math. Kyoto Univ., 35 (1995), 211-220.
[13]	A. Lunardi, Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in $\mathbb{R}^{d}$ , in Parabolic Problems. The Herbert Amann Festschrift, Progress in Nonlinear Differential Equations and Their Applications, 80, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0075-4_23.
[14]	P. Maheux, New proofs of Davies-Simon's theorems about ultracontractivity and Logarithmic Sobolev inequalities related to Nash type inequalities , Available on arXiv (http://arxiv.org/abs/0609124v1), (2006).
[15]	E. Nelson, The free Markoff field , J. Funct. Anal., 12 (1973), 211-227. doi: 10.1016/0022-1236(73)90025-6.
[16]	M. Röckner and F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds , Forum Math., 15 (2003), 893-921. doi: 10.1515/form.2003.044.
[17]	F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemann manifolds , Probab. Theory Relat. Fields, 109 (1997), 417-434. doi: 10.1007/s004400050137.
[18]	F.-Y. Wang, Functional inequalities for empty essential spectrum , J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.
[19]	F.-Y. Wang, A character of the gradient estimate for diffusion semigroups , Proc. Amer. Math. Soc., 133 (2004), 827-834. doi: 10.1090/S0002-9939-04-07625-7.
[20]	F.-Y. Wang, Functional inequalities in Markov Semigroups and Spectral Theory, Science Press, Beijing, 2004. doi: ISBN: 978-0-08-044942-5.

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References:

[1]	L. Angiuli, Pointwise gradient estimates for evolution operators associated with Kolmogorov operators , Arch. Math. (Basel), 101 (2013), 159-170. doi: 10.1007/s00013-013-0542-z.
[2]	L. Angiuli and L. Lorenzi, Compactness and invariance properties of evolution operators associated to Kolmogorov operators with unbounded coefficients , J. Math. Anal. Appl., 379 (2011), 125-149. doi: 10.1016/j.jmaa.2010.12.029.
[3]	L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations , Comm. Partial Differential Equations, 38 (2013), 2049-2080.
[4]	E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions , Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245-287.
[5]	G. Da Prato and A. Lunardi, Ultraboundedness for parabolic equations in convex domains without boundary conditions , Phys. D, 239 (2010), 1453-1457. doi: 10.1016/j.physd.2009.02.004.
[6]	E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians , J. Funct. Anal., 59 (1984), 335-395. doi: 10.1016/0022-1236(84)90076-4.
[7]	E. B. Davies, Heat kernes and spectral theory, Cambridge tracts in Mathematics, 92, Cambridge Univ. Press, Cambridge, 1990. doi: ISBN: 0-521-40997-7.
[8]	M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations , J. Lond. Math. Soc. (2), 77 (2008), 719-740. doi: 10.1112/jlms/jdn009.
[9]	L. Gross, Logarithmic Sobolev inequalities , Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[10]	O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité, J. Funct. Anal., 11 (1993), 155-196. doi: 10.1006/jfan.1993.1008.
[11]	M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients , Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: 10.1090/S0002-9947-09-04738-2.
[12]	M. Ledoux, Remarks on Logarithmic Sobolev constants, exponential integrability, and bounds on the diameter , J. Math. Kyoto Univ., 35 (1995), 211-220.
[13]	A. Lunardi, Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in $\mathbb{R}^{d}$ , in Parabolic Problems. The Herbert Amann Festschrift, Progress in Nonlinear Differential Equations and Their Applications, 80, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0075-4_23.
[14]	P. Maheux, New proofs of Davies-Simon's theorems about ultracontractivity and Logarithmic Sobolev inequalities related to Nash type inequalities , Available on arXiv (http://arxiv.org/abs/0609124v1), (2006).
[15]	E. Nelson, The free Markoff field , J. Funct. Anal., 12 (1973), 211-227. doi: 10.1016/0022-1236(73)90025-6.
[16]	M. Röckner and F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds , Forum Math., 15 (2003), 893-921. doi: 10.1515/form.2003.044.
[17]	F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemann manifolds , Probab. Theory Relat. Fields, 109 (1997), 417-434. doi: 10.1007/s004400050137.
[18]	F.-Y. Wang, Functional inequalities for empty essential spectrum , J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.
[19]	F.-Y. Wang, A character of the gradient estimate for diffusion semigroups , Proc. Amer. Math. Soc., 133 (2004), 827-834. doi: 10.1090/S0002-9939-04-07625-7.
[20]	F.-Y. Wang, Functional inequalities in Markov Semigroups and Spectral Theory, Science Press, Beijing, 2004. doi: ISBN: 978-0-08-044942-5.

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