**Contemporary Mathematics**

Volume: 723;
2019;
214 pp;
Softcover

MSC: Primary 35;
Secondary 11; 26; 60; 91

Print ISBN: 978-1-4704-4110-4

Product Code: CONM/723

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-5151-6
Product Code: CONM/723.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# New Developments in the Analysis of Nonlocal Operators

Share this page *Edited by *
*Donatella Danielli; Arshak Petrosyan; Camelia A. Pop*

This volume contains the proceedings of the
AMS Special Session on New Developments in the Analysis of Nonlocal
Operators, held from October 28–30, 2016, at the University of
St. Thomas, Minneapolis, Minnesota.

Over the last decade there has been a resurgence of interest in
problems involving nonlocal operators, motivated by applications in
many areas such as analysis, geometry, and stochastic processes.

Problems represented in this volume include uniqueness for weak
solutions to abstract parabolic equations with fractional time
derivatives, the behavior of the one-phase Bernoulli-type free
boundary near a fixed boundary and its relation to a Signorini-type
problem, connections between fractional powers of the spherical
Laplacian and zeta functions from the analytic number theory and
differential geometry, and obstacle problems for a class of not
stable-like nonlocal operators for asset price models widely used in
mathematical finance.

The volume also features a comprehensive introduction to various
aspects of the fractional Laplacian, with many historical remarks and
an extensive list of references, suitable for beginners and more
seasoned researchers alike.

#### Readership

Graduate students and research mathematicians interested in analytic, geometric, and probabilistic aspects of nonlocal equations.

# Table of Contents

## New Developments in the Analysis of Nonlocal Operators

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Fractional thoughts 110
- 1. Introduction 211
- 2. The fractional Laplacean 918
- 3. Maximum principle, Harnack inequality and Liouville theorem 1726
- 4. A brief interlude about very classical stuff 1928
- 5. Fourier transform, Bessel functions and (-Ξ)^{π } 2736
- 6. The fractional Laplacean and Riesz transforms 3342
- 7. The fractional Laplacean of a radial function 3544
- 8. The fundamental solution of (-Ξ)^{π } 3645
- 9. The nonlocal Yamabe equation 4251
- 10. Traces of Bessel processes: The extension problem 4453
- 11. Fractional Laplacean and subelliptic equations 5059
- 12. Hypoellipticity of (-Ξ)^{π } 5766
- 13. Regularity at the boundary 6978
- 14. Monotonicity formulas and unique continuation for (-Ξ)^{π } 7584
- 15. Nonlocal Poisson kernel and mean-value formulas 8695
- 16. The heat semigroup \pt=π^{π‘(-Ξ)^{π }} 95104
- 17. Bochnerβs subordination: from π_{π‘} to (-Ξ)^{π } 102111
- 18. More subordination: from π_{π‘} to π^{(π )}_{π‘} 104113
- 19. A chain rule for (-Ξ)^{π } 106115
- 20. The Gamma calculus for (-Ξ)^{π } 107116
- 21. Are there nonlocal Li-Yau inequalities? 112121
- 22. A Li-Yau inequality for Bessel operators 116125
- 23. The fractional π-Laplacean 120129
- Acknowledgments 122131
- References 122131

- Uniqueness for weak solutions of parabolic equations with a fractional time derivative 137146
- Boundary regularity for the free boundary in the one-phase problem 149158
- Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups 167176
- 1. Introduction 167176
- 2. Negative powers in the unit circle and the Hurwitz zeta function 170179
- 3. Positive powers in the unit circle and the Hurwitz zeta and Fine functions 172181
- 4. The semigroup generated by the Dirichlet-to-Neumann map 173182
- 5. Negative powers and the Minakshisundaram zeta function 176185
- 6. Positive powers and the Dirichlet-to-Neumann map 178187
- 7. Fractional Laplacians and the heat semigroup on the sphere. Extension problem and Harnack inequality 182191
- References 188197

- Obstacle problems for nonlocal operators 191200
- 1. Introduction 191200
- 2. Stationary obstacle problem 198207
- 3. Evolution obstacle problem 208217
- References 213222

- Back Cover Back Cover1226