Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations

The main objective of this work is to introduce the generalized convolution with trigonometric weighted $\gamma =\sin y$ involving the Fourier cosine-sine and Kontorovich-Lebedev transforms, and to study its fundamental results. We establish boundedness properties in a two-parametric family of Lebesgue spaces for this convolution operator. Norm estimation in the weighted $L_p$ space is obtained and applications of the corresponding class of convolution integro-differential equations are discussed. The conditions for the solvability of these equations in $L_1$ space are also founded.