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## Degenerate Holder Spaces

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0005
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces the degenerate Hölder spaces on the spaces Sₙ,ₘ, and their heat-space counterparts on Sₙ,ₘ x [0, T]. It also establishes the basic properties of these spaces. The objective is ... More

## Introduction

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0001
- Subject:
- Mathematics, Probability / Statistics

This book proves the existence, uniqueness and regularity results for a class of degenerate elliptic operators known as generalized Kimura diffusions, which act on functions defined on manifolds with ... More

## Holder Estimates for Euclidean Models

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0008
- Subject:
- Mathematics, Probability / Statistics

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator ... More

## Degenerate Diffusion Operators Arising in Population Biology (AM-185)

*Charles L. Epstein and Rafe Mazzeo*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as ... More

## Holder Estimates for General Models

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining ... More

## Holder Estimates for the 1-dimensional Model Problems

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0006
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Existence of Solutions

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0010
- Subject:
- Mathematics, Probability / Statistics

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The ... More

## Holder Estimates for Higher Dimensional Corner Models

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0007
- Subject:
- Mathematics, Probability / Statistics

This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous ... More

## The Model Solution Operators

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods ... More

## The Resolvent Operator

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0011
- Subject:
- Mathematics, Probability / Statistics

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ-L) R(μ) f = f, R(μ) ... More

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