We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.
The Schwartz kernel theorem for the tempered distributions on the Heisenberg group
