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Estimates of the derivatives for parabolic operators with unbounded coefficients

Authors:	Marcello Bertoldi Luca Lorenzi

Institution:	Applied Mathematical Analysis, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands ; Dipartimento di Matematica, Università di Parma, Via M. D'Azeglio 85/A, 43100 Parma, Italy

Abstract:	We consider a class of second-order uniformly elliptic operators $\mathcal{A}$ with unbounded coefficients in $\mathbb{R}^N$ . Using a Bernstein approach we provide several uniform estimates for the semigroup generated by the realization of the operator $\mathcal{A}$ in the space of all bounded and continuous or Hölder continuous functions in $\mathbb{R}^N$ . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation $\lambda u-\mathcal{A}u=f$ ( $\lambda>0$ ) and the nonhomogeneous Dirichlet Cauchy problem $D_tu=\mathcal{A}u+g$ . Then, we prove two different kinds of pointwise estimates of that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup in weighted -spaces related to the invariant measure associated with the semigroup.

Keywords:	Elliptic and parabolic operators with unbounded coefficients in ${\mathbb R}^N$ Markov semigroups uniform and pointwise estimates optimal Schauder estimates

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