Composition structure of polyconvolution associated with index Kontorovich–Lebedev transform and Fourier integrals

Using Kakichev's classical concept and extending Yakubovich-Britvina's approach (\textit{Results. Math.} 55(1-2):175-197, 2009) and (\textit{Integral Transforms Spec. Funct.} 21(4):259--276, 2010) for setting up Kontorovich-Lebedev convolution operators, this paper proposes a new polyconvolution structure associated with the KL-transform and Fourier integrals. Our main contributions include demonstrating a one-dimensional Watson-type transform, providing necessary and sufficient conditions for this transform to serve as unitary on $L_2(\mathbb{R}_+)$, and inferring its inverse operator in symmetric form. The existence of this structure over specific function spaces and its connection with previously known convolutions are pointed out. Establish the Plancherel-type theorem, prove the convergence in the mean-square sense in $L_2(\mathbb{R}_+)$, and prove the boundedness of dual spaces via Riesz-Thorin's theorem. Derives new weighted $L_p$-norm inequalities and boundedness in a three-parametric family of Lebesgue spaces. These theoretical findings are applied to solve specific classes of the Toeplitz-Hankel equation, providing a priori estimations based on the established conditions for $L_1$ solvability.