Maximum principles and direct methods for tempered fractional operators

In this paper, we are concerned with the tempered fractional operator \(-(\Delta+\lambda)^{\alpha\over{2}}\) with α ∈ (0, 2) and λ is a sufficiently small positive constant. We first establish various maximum principle principles and develop the direct moving planes and sliding methods for anti-symmetric functions involving tempered fractional operators. And then we consider tempered fractional problems. As applications, we extend the direct method of moving planes and sliding methods for the tempered fractional problem, and discuss how they can be used to establish symmetry, monotonicity, Liouville-type results and uniqueness results for solutions in various domains. We believe that our theory and methods can be conveniently applied to study other problems involving tempered fractional operators.