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

The local potential operator with integral kernel restricted in a ball of radius less than some fixed number r∈(0,∞)$r\in (0,\infty )$ 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→∞$r\rightarrow \infty$. As an application, we establish uniform estimate for the resolvent (λ−L)−1$(\lambda -\mathcal {L})^{-1}$ of some generalized Schrödinger operator L:=L0+V$\mathcal {L}:=\mathcal {L}_0+V$ on the Orlicz space. An explicit representation in its operator norm on the dependence of λ∈(0,∞)$\lambda \in (0,\infty )$ is also given.