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

Jun Cao, Xiaoshen Dou, Mengyao Gao and Yongyang Jin

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4533-4558

Abstract: 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.

Date: 2023
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