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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Siva R. Athreya, Richard F. Bass and Edwin A. Perkins
Journal: Trans. Amer. Math. Soc. 357 (2005), 5001-5029
MSC (2000): Primary 35J15; Secondary 35R15, 47D07, 60J35
Published electronically: March 10, 2005
MathSciNet review: 2165395
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Abstract: We introduce a new method for proving the estimate

\begin{displaymath}\left\Vert\frac{\partial^2 u}{\partial x_i \partial x_j} \right\Vert_{C^\alpha}\leq c\Vert f\Vert _{C^\alpha},\end{displaymath}

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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Additional Information

Siva R. Athreya
Affiliation: Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Edwin A. Perkins
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03638-X
PII: S 0002-9947(05)03638-X
Keywords: Semigroups, Schauder estimates, H\"older spaces, perturbations, resolvents, elliptic operators, Laplacian, Ornstein-Uhlenbeck processes, infinite-dimensional stochastic differential equations
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: February 13, 2004
Published electronically: March 10, 2005
Additional Notes: The first author’s research was supported in part by an NBHM travel grant.
The second author’s research was supported in part by NSF grant DMS0244737.
The third author’s research was supported in part by an NSERC Research Grant
Article copyright: © Copyright 2005 American Mathematical Society