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

Abstract

The local potential operator with integral kernel restricted in a ball of radius less than some fixed number 𝑟 ∈ (0, ∞) 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 𝑟 → ∞. As an application, we establish uniform estimate for the resolvent (𝜆 − ) −1 of some generalized Schrödinger operator  ∶=  0 + 𝑉 on the Orlicz space. An explicit representation in its operator norm on the dependence of 𝜆 ∈ (0, ∞) is also given.