Boundedness of functional calculi of Schrödinger operators on generalized Lebesgue spaces $${L^{p(\cdot)}(\mathbb{R}^n)}$$

Abstract

Let L = −Δ + V be a Schrodinger operator on \({\mathbb{R}^n}\) , where \({V\in L^1_{\rm loc}({\mathbb R}^n)}\) is a nonnegative function on \({{\mathbb{R}^n}}\) . Let \({L^{p(\cdot)}(\mathbb{R}^n)}\) be the generalized Lebesgue spaces. In this article we use a technical atomic decomposition of Hardy space associated with L to show that the Lp(·)-norm of f can be controlled by the sum of the Lp(·)-norms of two variants of sharp maximal functions of f. As a result, we obtain boundedness of functional calculi of Schrodinger operators on generalized Lebesgue spaces \({L^{p(\cdot)}(\mathbb{R}^n)}\) .