Local potential operator and uniform resolvent estimate for generalized Schrödinger operator in Orlicz spaces

Abstract

AbstractThe local potential operator with integral kernel restricted in a ball of radius less than some fixed number has appeared frequently in the spectral estimates of the Schrödinger operator. In this paper, we establish a good‐λ inequality for this operator and characterize its uniform boundedness on the weighted Orlicz space in both strong and weak senses. The uniformity in r of this boundedness enables us to recover the classical boundedness of “global” potential operator, by letting . As an application, we establish uniform estimate for the resolvent of some generalized Schrödinger operator on the Orlicz space. An explicit representation in its operator norm on the dependence of is also given.