User's guide to the fractional Laplacian and the method of semigroups

The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and Jos\'e L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-\Delta)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for $(-\Delta)^su(x)$. One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on $u$, the validity of the pointwise limits $$\lim_{s\to1^-}(-\Delta)^su(x)=-\Delta u(x)\quad\hbox{and}\quad\lim_{s\to0^+}(-\Delta)^su(x)=u(x).$$ The formula for the solution to the Poisson problem $(-\Delta)^su=f$ is found through the semigroup approach as the inverse of the fractional Laplacian $u(x)=(-\Delta)^{-s}f(x)$ (fundamental solution). We then present the Caffarelli--Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical H\"older and Schauder estimates $$(-\Delta)^{\pm s}:C^\alpha\to C^{\alpha\mp 2s}$$ are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of H\"older and Zygmund spaces with heat semigroups.