Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

Athreya, Siva; Bass, Richard; Perkins, Edwin

doi:10.1090/S0002-9947-05-03638-X

Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
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by Siva R. Athreya, Richard F. Bass and Edwin A. Perkins PDF

Trans. Amer. Math. Soc. 357 (2005), 5001-5029 Request permission

Abstract:

We introduce a new method for proving the estimate \[ \left \Vert \frac {\partial ^2 u}{\partial x_i \partial x_j} \right \Vert _{C^\alpha }\leq c\|f\|_{C^\alpha },\] where $u$ solves the equation $\Delta u-\lambda u=f$. The method can be applied to the Laplacian on $\mathbb {R}^\infty$. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

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