The book focuses on the basic theory of nonlocal and fractional operators and it is divided into two main parts covering respectively the basic theory of the "singular-integral" fractional Laplacian operator, which also includes a number of explicit examples of its evaluations, and the regularity theory in L^p spaces, by going through Bessel and Besov spaces. A number of appendices then go over additional material in order to make the exposition as self-contained as possible.

The book is suitable for a PhD course but can be also read by advanced undergraduates. It contains didactic elements, but it is also useful for researchers, since it collects essential material in a concise and self-constrained way.

Flexible and easy to subdivide Rigorous, but intuitive. Intuitive, but rigorous Clear, explicit, simple to read

Autorentext

Nicola Abatangelo is an associate professor of Mathematical Analysis at the University of Bologna (Italy) since September 2024. Previously he has been assistant professor in the same university, a Humboldt post-doctoral fellow at the University of Frankfurt (Germany), and a post-doc at the Universities of Zurich (Switzerland) and Brussels (Belgium). He has obtained his PhD from the Universities of Amiens (France) and Milan (Italy) in 2015. His research focuses on the analysis of fractional Laplacian type operators, with particular emphasis on nonlocal elliptic PDEs and the qualitative properties of the associated solutions. Serena Dipierro is a Professor of Mathematics, Fellow of the Australian Mathematical Society, and Australian Mathematical Society Hanna Neumann Lecturer. She carried out her academic career in Santiago de Chile, Edinburgh, Magdeburg, Berlin, Melbourne, Milan, and Perth. She has obtained prestigious awards, including the Australian Mathematical Society Medal, the Mahony-Neumann-Room Prize, the Bartolozzi Prize of the Italian Mathematical Union, the Christopher Heyde Medal of the Australian Academy of Science, the Book Prize of the Italian Mathematical Union, and the Prize of the Bulletin of the Brazilian Mathematical Society. Enrico Valdinoci is a Professor of Mathematics and Australian Laureate Fellow. He carried out his academic career in Pisa, Rome, Milan, Berlin, Melbourne, and Perth. He is a highly cited researcher, Ladyzhenskaya Lecturer and has been awarded the James S. W. Wong Prize, the Mahony-Neumann-Room Prize, the Orazio Arena Prize, the Book Prize of the Unione Matematica Italiana, the Prize of the Bulletin of the Brazilian Mathematical Society, and the Amerio Gold Medal Prize.

Inhalt

Chapter 1. Equivalent definitions of the fractional Laplacian.- Chapter 2. Some explicit calculations of fractional Laplacians.- Chapter 3. Liouville-type results.- Chapter 4. Regularity theory in Lebesgue spaces for global solutions.- Chapter 5. Interior regularity theory in Lebesgue spaces.