Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Luciana Angiuli[a] and Luca Lorenzi[b]

On the estimates of the derivatives of solutions to nonautonomous Kolmogorov equations and their consequences

Pages: 421-471
Accepted in revised form: 1 February 2017
Mathematics Subject Classification (2010): 35K10, 35K15, 35B40.
Keywords: Elliptic operators with unbounded coefficients, estimates of the spatial derivatives, evolution systems of measures, logarithmic Sobolev inequalities, Poincaré inequality, asymptotic behaviour, summability improving properties.
[a]: University of Salento, Department of Mathematics and Physics , via per Arnesano, s.n.c. Lecce, 73100, Italy
[b]: University of Parma, Department of Mathematical, Physical and Computer Sciences Mathematical and Computer Sciences Building, Parco Area delle Scienze 53/A Parma, 43124, Italy

Abstract: We consider evolution operators G(t; s) associated to a class of nonautonomous elliptic operators with unbounded coefficients, in the space of bounded and continuous functions over $$\mathbb{R}^d$$. We prove some new pointwise estimates for the spatial derivatives of the function G(t; s)f, when f is bounded and continuous or much smoother. We then use these estimates to prove smoothing effects of the evolution operator in $$L^p$$-spaces. Finally, we show how pointwise gradient estimates have been used in the literature to study the asymptotic behaviour of the evolution operator and to prove summability improving results in the $$L^p$$-spaces related to the so-called tight evolution system of measures.

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