We introduce several fundamental and useful properties for integro-differential operators. These properties are useful not only for inverse problems but also for forward problems. First, we review the rigorous definition of the fractional Laplacian and fractional Sobolev spaces. Second, we recall the well-known Caffarelli–Silvestre-type extension result. Last but not least, we prove remarkable unique continuation property, maximum principle, and \(L^p\) estimates for the fractional Laplacian.