A convolution operator related to the generalized Mehler-Fock and Kontorovich-Lebedev transforms
The authors consider the following generalization of the index integral \[ e^{-\gamma z}J_m(yR)= \sqrt{{2\over\pi y}} \int_{R_+} {\tau\tanh(\pi\tau)\over Z^m_\tau} K_{i\tau}(y)P^m_{-{1\over 2}+ i\tau} (\mu)P^m_{-{1\over 2}+ i\tau}(\eta)\,d\tau, \] where \(J_m(\omega)\), \(K_{i\tau}(y)\) are Bessel functions, \(P^m_{-{1\over 2}+i\tau}(\omega)\), \(\text{Re}(\omega)> 0\), \(m\in\mathbb{N}_0\), is the associated Legendre function and \(y> 0\); \(R=\sqrt{(\eta^2- 1)(1-\mu^2)}\), \(\mu\in [0,1]\), \(\eta\in [1,\infty)\), \[ Z^m_\tau= {\Gamma({1\over 2}+ m+i\tau)\over \Gamma({1\over 2}- m+i\tau)}. \] The aim of this paper is to study the convergence properties of the index integral. The special convolution construction associated with this integral is related the Kontorovich-Lebedev and generalized Mehler-Fock transforms. The investigations give norm estimates in weighted \(L_p\)-spaces, \(p\in[1,2]\). Some convolution integral equations are solved in \(L_2\) as applications.
scientific article; zbMATH DE number 6135263