Integral transformations by the index of Lommel's function (Q1878599)

Given the integral operator \[ (S_\mu f) (x)=2^{1-\mu}\int^\infty_0 f(\tau)\left| \Gamma\left( \frac{1-\mu+i\tau}{2}\right)\right| ^2 S_{\mu,i\tau}(x) \,d\tau,\;x>0,\;\mu\in \mathbb R,\;| \mu| <1, \] where \(f\) belongs to the Lebesgue space of summable functions, \(\Gamma(z)\) is the Euler gamma-function, \(S_{\mu,i\tau}(x)\) is the kernel Lommel function (a special Bessel type function), where \(\mu\) is a fixed number, one studies its boundedness and inversion properties by using the methods of Fourier, Laplace and Kontorovich-Lebedev transforms theory. An inversion formula is finally proved.