Quantified versions of Ingham's theorem (Q2812006)

Let \(X\) be a Banach space and let \(f\) be a bounded and uniformly continuous function from \( {\mathbb R}_+ \) to \(X\). Suppose that there exists a function \(F\in L^1_{\mathrm{loc}}({\mathbb R};X)\) such that NEWLINE\[NEWLINE\lim_{\alpha\to0+}\int_{\mathbb R} \widehat{f}(\alpha+ is)\psi(s)\,ds=\int_{\mathbb R} F(s)\psi(s)\,ds NEWLINE\]NEWLINE for all \(\psi\in C_c({\mathbb R})\), where \(\widehat{f}\) is the Laplace transform of \(f\). According to Ingham's classical Tauberian theorem, in this case \(f\in C_0({\mathbb R}_+; X)\).NEWLINENEWLINEThe authors consider a generalization of this theorem with \(F\in L^1_{\mathrm{loc}}({\mathbb R}\backslash\Sigma;X)\) for all \(\psi\in C_c({\mathbb R}\backslash\Sigma)\), where \(\Sigma\subset {\mathbb R}\) is a singular set. The three cases of interest in the paper are when the boundary function \(F\) has a `singularity at infinity', when \(F\) has a `singularity at zero' and when \(F\) has `singularities at zero and infinity'. Quantified versions of Ingham's theorem are proved in the paper. These versions allow to estimate the rate of decay of \(f(t)\) as \(t\to\infty\) given information about how the function \(F\) behaves near the singular points in \(\Sigma\) and near infinity.NEWLINENEWLINEThe proofs are based on a variant of Ingham's original proof. As corollaries, quantified decay estimates for \(C_0\)-semigroups are obtained.