Real inversion formulas of the Laplace transform on weighted function spaces (Q841665)

Let \({\mathcal L}\) be the Laplace transform. In this article, the authors study the following weighted norm inequalities for the transform \(L\): \[ \int^\infty_0 |(Lf)(x)|^2 u(x)\,dx\preceq \int^\infty_0|f'(x)|^2 w(x)\,dx, \] where \((Lf)(x)= x({\mathcal L}f)(x)\), \(u\), \(v\) are positive continuous functions and \(f\) is an absolutely continuous function with \(f(0)= 0\). The following is the main result of the paper: Theorem 2.1. Assume that for a given pair \(w\) and \(u\) of weights, \[ M= \int\int_{\mathbb{R}^+\times \mathbb{R}^+} e^{-2tx}{u(x)\over w(t)}\,dx\,dt< \infty. \] Then the operator \(L\) is a compact operator from \(L^2_1(\mathbb{R}^+, w)\) or \(L^2(\mathbb{R}^+,u)\) with norm less than or equal to \(\sqrt{M}\).