On a testing-function space for distributions associated with the Kontorovich--Lebedev transform
Let \(A_x\) denote the differential operator \(A_x=x^2-x\frac{d}{dx}\left[x\frac{d}{dx}\right]\) and \(A_x^k\) its \(k\)-th iterate. For any \(2\leq p < \infty\) and \(\nu > 0\), the author considers the space \({\mathcal A}_{\nu,p}\) consisting of those complex-valued smooth function on \((0,+\infty)\) such that \[ \alpha_{k,\nu,p}(\varphi):=\left(\int_0^{\infty}| A_x^k\varphi| ^px^{\nu p -1}\,dx\right)^{\frac{1}{p}}<\infty \] for every \(k\in{\mathbb N}\). Then the Kontorovich-Lebedev transform of \(f\in {\mathcal A}^\prime_{\nu,p}\) is defined as \[ {\mathcal KL}[f](s):=\left<f, K_s(\cdot)\right>,\;s\in{\mathbb C}, \] where \(K_s\) is the modified Bessel function of complex index \(s\). This extends the classical Kantorovich--Lebedev transform, defined only for appropriate functions and for a purely imaginary index. The author proves that \({\mathcal KL}[f]\) is an analytic function on a vertical strip and obtains an inversion formula for the above transformation. As an application, a Dirichlet problem for a wedge is considered.
scientific article; zbMATH DE number 5079948