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

TL;DR

This paper introduces a new generalized convolution operator involving trigonometric weights and special transforms, analyzes its properties in Lebesgue spaces, and discusses applications to convolution integro-differential equations.

Contribution

It develops a novel convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transforms, providing boundedness, norm estimates, and solvability conditions.

Findings

Boundedness in Lebesgue spaces established

Norm estimates in weighted $L_p$ spaces derived

Solvability conditions for convolution equations in $L_1$ space identified

Abstract

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.