Convolution operators related to the Fourier cosine and Kontorovich-Lebedev transformations
The authors construct certain convolution operators, which are associated with the Fourier cosine transformation \[ (F_cf)(x)= \lim_{N\to\infty} \sqrt{{\pi\over 2}} \int^N_{1/N} f(t)\cos xt\,dt \] and the Kontorovich-Lebedev transformation \[ K_{ix}[f]= \lim_{N\to\infty} \int^N_{1/N} f(t) K_{ix}(t)\,dt, \] where the modified Bessel function \(K_{ix}(t)\) can be represented as \[ K_{ix}(t)= \int^\infty_0 e^{-t\cosh u}\cos x\,u\,du,\quad t> 0 \] and it also can be extended to the strip \(\delta\in[0,\pi/2]\) in the upper half-plane by \[ K_{ix}(t)={1\over 2} \int^{i\delta+\infty}_{i\delta-\infty} e^{-t\cosh z+ ixz}dz. \] The authors establish boundedness in a two-parametric family of Lebesgue spaces for these convolution operators related to the transformations \(F_cf\) and \(K_{ix}[f]\) and obtain some norm estimates on the weighted \(L^p\) spaces. Further more, they demonstrate some applications to the corresponding class of convolution integral equations and find necessary and sufficient conditions for the solvability of these equations in the weighted \(L^2\) spaces.
scientific article; zbMATH DE number 5627208