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Degenerate Diffusion Operators Arising in Population Biology (AM-185)$
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Charles L. Epstein and Rafe Mazzeo

Print publication date: 2013

Print ISBN-13: 9780691157122

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691157122.001.0001

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Holder Estimates for the 1-dimensional Model Problems

Holder Estimates for the 1-dimensional Model Problems

Chapter:
(p.78) Chapter SixHölder Estimates for the 1-dimensional Model Problems
Source:
Degenerate Diffusion Operators Arising in Population Biology (AM-185)
Author(s):

Charles L. Epstein

Rafe1 Mazzeo

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691157122.003.0006

This chapter establishes Hölder space estimates for the 1-dimensional model problems. It gives a detailed treatment of the 1-dimensional case, in part because all of the higher dimensional estimates are reduced to estimates on heat kernels for the 1-dimensional model problems. It also presents the proof of parabolic Schauder estimates for the generalized Kimura diffusion operator using the explicit formula for the heat kernel, along with standard tools of analysis. Finally, it considers kernel estimates for degenerate model problems, explains how Hölder estimates are obtained for the 1-dimensional model problems, and describes the properties of the resolvent operator.

Keywords:   model problem, heat kernel, Schauder estimate, generalized Kimura diffusion, Kimura diffusion operator, Hölder space, resolvent operator

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