Fractional powers of a canonical Fourier–Bessel differential operator

We develop a fractional functional calculus for the canonical Fourier–Bessel differential operator associated with a real unimodular matrix. The starting point is the unitary chirp conjugacy between the canonical operator and the classical Bessel operator, which yields a natural self-adjoint realization in the weighted Bessel space and a direct spectral diagonalization by the canonical Fourier–Bessel transform. On this basis, fractional powers of the associated non-negative operator are defined by spectral multipliers, and their domains are described through the corresponding Sobolev scale. We also study the canonical generalized translation and convolution, which provide the physical space structures compatible with the transform and with heat propagation. For fractional orders between zero and one, we derive the Balakrishnan semigroup representation, an explicit heat-kernel formula, and a Lévy-type singular-integral representation. The same spectral framework is used to treat the fractional heat equation and to obtain a Caffarelli–Silvestre type extension problem whose Dirichlet-to-Neumann map realizes the fractional operator. Hardy-type estimates are then deduced by transferring known Bessel inequalities through the canonical unitary conjugacy. Several matrix specializations are discussed to recover the ordinary, inverse, fractional, Fresnel, scaled, and chirp-modulated Fourier–Bessel transforms. The resulting theory gives a unified description of nonlocal Bessel operators under linear canonical deformation and connects spectral, semigroup, translation, and extension methods in a common framework.