Weighted Duality and Mixed Operational Calculus of the Two-Sided Fourier–Kontorovich–Lebedev Transform

Abstract

This paper constructs a mixed testing calculus in which a two-sided Fourier factor and an imaginary-order Macdonald factor act jointly on the half-strip. A countable family of seminorms generated by Cartesian derivatives and even radial operators produces a Fréchet space of rapidly controlled amplitudes. The product kernel is shown to be a holomorphic section of that space on every compact frequency sub-strip and every compact index interval. Dual pairings therefore define a distributional transform attached to an open spectral region whose width is fixed by the admissible weight. The transform is jointly continuous on compact spectral sets, holomorphic in the frequency variable, and of exponential type in the index whenever the dual is of finite order. Translation, dilation, and mixed Helmholtz generators pass to explicit multipliers. A cutoff inversion integral recovers every finite-order dual in the weak-star topology, uniqueness holds on connected spectral regions, and the construction tensorizes to several Cartesian and radial factors. Growth dictionaries relate dual order to strip width and to polynomial degree in the Kontorovich–Lebedev index, so mixed partial operators on the half-strip become multiplication operators on the spectral side. Keywords: two-sided Fourier–Kontorovich–Lebedev kernel, weighted Fréchet testing space, dual pairing on the half-strip, Macdonald bounds, strip holomorphy, time-shift multiplier, radial dilation, weak-star inversion, mixed Helmholtz symbol, sequential multinorm completeness