A new Kontorovich–Lebedev-type transform and its associated Gabor transform

In this paper, we introduce a modified Kontorovich–Lebedev transform associated with a normalized K-Bessel kernel. We establish the fundamental results of the corresponding harmonic analysis, including the inversion formula, the Plancherel theorem, and a Hausdorff–Young inequality. A translation operator and the related convolution product are then defined and their main properties are investigated. As an application to time–frequency analysis, we introduce a Gabor transform associated with the MKL-transform. We establish its fundamental properties, including orthogonality relations, Plancherel and inversion formulas, and characterize its range as a reproducing kernel Hilbert space.