Differentiability of generalized Fourier transforms associated with Schrödinger operators (Q1088052)

In a previous paper [J. Reine Angew. Math. 337, 18-67 (1982; Zbl 0486.35026)] the author developed a theory of generalized Fourier transform \({\mathcal F}\) associated with the Schrödinger operator \(H=- \Delta +V(x)\) in \(L^ 2({\mathbb{R}}^ n)\) (n\(\geq 2)\) with a long-range potential V(x). The main purpose of this paper is to prove that for \(f\in L^{2,N+r}\) \((r>\), \(N\in {\mathbb{N}})\), (\({\mathcal F}f)(\xi)\) is N-times differentiable with respect to \(\xi\neq 0\).