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

On improvement of summability properties in nonautonomous Kolmogorov equations

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

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.
Citation: L. Angiuli, Luca Lorenzi. On improvement of summability properties in nonautonomous Kolmogorov equations. Communications on Pure & 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 ,, \emph{Arch. Math. (Basel)}, 101 (2013), 159. doi: 10.1007/s00013-013-0542-z. Google Scholar

[2]

L. Angiuli and L. Lorenzi, Compactness and invariance properties of evolution operators associated to Kolmogorov operators with unbounded coefficients ,, \emph{J. Math. Anal. Appl.}, 379 (2011), 125. doi: 10.1016/j.jmaa.2010.12.029. Google Scholar

[3]

L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations ,, \emph{Comm. Partial Differential Equations}, 38 (2013), 2049. Google Scholar

[4]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions ,, \emph{Ann. Inst. H. Poincar\'e Probab. Statist.}, 23 (1987), 245. Google Scholar

[5]

G. Da Prato and A. Lunardi, Ultraboundedness for parabolic equations in convex domains without boundary conditions ,, \emph{Phys. D}, 239 (2010), 1453. doi: 10.1016/j.physd.2009.02.004. Google Scholar

[6]

E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians ,, \emph{J. Funct. Anal.}, 59 (1984), 335. doi: 10.1016/0022-1236(84)90076-4. Google Scholar

[7]

E. B. Davies, Heat kernes and spectral theory,, Cambridge tracts in Mathematics, 92 (1990). doi: ISBN: 0-521-40997-7. Google Scholar

[8]

M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations ,, \emph{J. Lond. Math. Soc. (2)}, 77 (2008), 719. doi: 10.1112/jlms/jdn009. Google Scholar

[9]

L. Gross, Logarithmic Sobolev inequalities ,, \emph{Amer. J. Math.}, 97 (1975), 1061. doi: 10.2307/2373688. Google Scholar

[10]

O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité,, \emph{J. Funct. Anal.}, 11 (1993), 155. doi: 10.1006/jfan.1993.1008. Google Scholar

[11]

M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients ,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 169. doi: 10.1090/S0002-9947-09-04738-2. Google Scholar

[12]

M. Ledoux, Remarks on Logarithmic Sobolev constants, exponential integrability, and bounds on the diameter ,, \emph{J. Math. Kyoto Univ.}, 35 (1995), 211. Google Scholar

[13]

A. Lunardi, Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in $\R^d$ ,, in \emph{Parabolic Problems. The Herbert Amann Festschrift}, 80 (2011). doi: 10.1007/978-3-0348-0075-4_23. Google Scholar

[14]

P. Maheux, New proofs of Davies-Simon's theorems about ultracontractivity and Logarithmic Sobolev inequalities related to Nash type inequalities ,, Available on arXiv (\url{http://arxiv.org/abs/0609124v1}), (2006). Google Scholar

[15]

E. Nelson, The free Markoff field ,, \emph{J. Funct. Anal.}, 12 (1973), 211. doi: 10.1016/0022-1236(73)90025-6. Google Scholar

[16]

M. Röckner and F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds ,, \emph{Forum Math.}, 15 (2003), 893. doi: 10.1515/form.2003.044. Google Scholar

[17]

F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemann manifolds ,, \emph{Probab. Theory Relat. Fields}, 109 (1997), 417. doi: 10.1007/s004400050137. Google Scholar

[18]

F.-Y. Wang, Functional inequalities for empty essential spectrum ,, \emph{J. Funct. Anal.}, 170 (2000), 219. doi: 10.1006/jfan.1999.3516. Google Scholar

[19]

F.-Y. Wang, A character of the gradient estimate for diffusion semigroups ,, \emph{Proc. Amer. Math. Soc.}, 133 (2004), 827. doi: 10.1090/S0002-9939-04-07625-7. Google Scholar

[20]

F.-Y. Wang, Functional inequalities in Markov Semigroups and Spectral Theory,, Science Press, (2004). doi: ISBN: 978-0-08-044942-5. Google Scholar

show all references

References:
[1]

L. Angiuli, Pointwise gradient estimates for evolution operators associated with Kolmogorov operators ,, \emph{Arch. Math. (Basel)}, 101 (2013), 159. doi: 10.1007/s00013-013-0542-z. Google Scholar

[2]

L. Angiuli and L. Lorenzi, Compactness and invariance properties of evolution operators associated to Kolmogorov operators with unbounded coefficients ,, \emph{J. Math. Anal. Appl.}, 379 (2011), 125. doi: 10.1016/j.jmaa.2010.12.029. Google Scholar

[3]

L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations ,, \emph{Comm. Partial Differential Equations}, 38 (2013), 2049. Google Scholar

[4]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions ,, \emph{Ann. Inst. H. Poincar\'e Probab. Statist.}, 23 (1987), 245. Google Scholar

[5]

G. Da Prato and A. Lunardi, Ultraboundedness for parabolic equations in convex domains without boundary conditions ,, \emph{Phys. D}, 239 (2010), 1453. doi: 10.1016/j.physd.2009.02.004. Google Scholar

[6]

E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians ,, \emph{J. Funct. Anal.}, 59 (1984), 335. doi: 10.1016/0022-1236(84)90076-4. Google Scholar

[7]

E. B. Davies, Heat kernes and spectral theory,, Cambridge tracts in Mathematics, 92 (1990). doi: ISBN: 0-521-40997-7. Google Scholar

[8]

M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations ,, \emph{J. Lond. Math. Soc. (2)}, 77 (2008), 719. doi: 10.1112/jlms/jdn009. Google Scholar

[9]

L. Gross, Logarithmic Sobolev inequalities ,, \emph{Amer. J. Math.}, 97 (1975), 1061. doi: 10.2307/2373688. Google Scholar

[10]

O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité,, \emph{J. Funct. Anal.}, 11 (1993), 155. doi: 10.1006/jfan.1993.1008. Google Scholar

[11]

M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients ,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 169. doi: 10.1090/S0002-9947-09-04738-2. Google Scholar

[12]

M. Ledoux, Remarks on Logarithmic Sobolev constants, exponential integrability, and bounds on the diameter ,, \emph{J. Math. Kyoto Univ.}, 35 (1995), 211. Google Scholar

[13]

A. Lunardi, Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in $\R^d$ ,, in \emph{Parabolic Problems. The Herbert Amann Festschrift}, 80 (2011). doi: 10.1007/978-3-0348-0075-4_23. Google Scholar

[14]

P. Maheux, New proofs of Davies-Simon's theorems about ultracontractivity and Logarithmic Sobolev inequalities related to Nash type inequalities ,, Available on arXiv (\url{http://arxiv.org/abs/0609124v1}), (2006). Google Scholar

[15]

E. Nelson, The free Markoff field ,, \emph{J. Funct. Anal.}, 12 (1973), 211. doi: 10.1016/0022-1236(73)90025-6. Google Scholar

[16]

M. Röckner and F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds ,, \emph{Forum Math.}, 15 (2003), 893. doi: 10.1515/form.2003.044. Google Scholar

[17]

F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemann manifolds ,, \emph{Probab. Theory Relat. Fields}, 109 (1997), 417. doi: 10.1007/s004400050137. Google Scholar

[18]

F.-Y. Wang, Functional inequalities for empty essential spectrum ,, \emph{J. Funct. Anal.}, 170 (2000), 219. doi: 10.1006/jfan.1999.3516. Google Scholar

[19]

F.-Y. Wang, A character of the gradient estimate for diffusion semigroups ,, \emph{Proc. Amer. Math. Soc.}, 133 (2004), 827. doi: 10.1090/S0002-9939-04-07625-7. Google Scholar

[20]

F.-Y. Wang, Functional inequalities in Markov Semigroups and Spectral Theory,, Science Press, (2004). doi: ISBN: 978-0-08-044942-5. Google Scholar

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