**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

**Received:** 11 November 2016

**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.

**Author address:**

[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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