# fractional laplacian on $\mathbb{R}.$

Is there a book on fractional operator $(-\frac{d^2}{dx^2})^{s}$ with $s\in (0, \frac{1}{2}]$. The textbook should ideally contain estimates and properties of all kinds, and Schauder to Moser theory.

To my best knowledge, there is no complete book on this topic. Peter Michor already pointed out Triebel, which is a classical reference.

If you look for newer literature, there is e.g. The "Hitchhiker's guide to the fracional Sobolev spaces" https://arxiv.org/abs/1104.4345 . This is very basic and might not give you the complete answer you need.

Other, more specialized surveys where you might find some hints are e.g. the Abel lecture of J.L. Vazquez "Nonlinear diffusion with Fractional Laplacian Operators"

You may also give a look to the book authored by Molica Bisci - Radulescu and Servadei about fractional operators entitled Variational methods for nonlocal fractional problems.

A lot of information about the fractional Laplace operator can be found in a book Hypersingular Integrals and Their Applications by Stefan Samko. Another reference is "Fractional Integrals and Potentials" by Boris Rubin. There are plenty of regularity results there, but from a completely different perspective.

If you are more focused on operators with variable coefficients, you should look into some recent papers of PDE people, for example, Non-local diffusion and applications by Claudia Bucur and Enrico Valdinoci, or Nonlocal elliptic equations in bounded domains: A survey by Xavier Ros-Oton. You can find further references in my survey article Ten equivalent definitions of the fractional Laplace operator.

Not a book but a great resource on the fractional Laplacian etc. is the Nonlocal Equation Wiki maintained by Luis Silvestre (I think).