Distributions of anisotropic order and applications to H-distributions

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.

Extension theorem of Whitney type for S(Rd+) by use of the Kernel theorem

We study the expansions of the elements in S(Rd +) and S?(Rd +) with respect to the Laguerre orthonormal basis, extending the result of M. Guillemot-Teissier in the one dimensional case. As a consequence, we obtain Kernel theorem for S(Rd +) and S?(Rd +) and an extension theorem of Whitney type for S(Rd +).

Regular nonlinear generalized functions and applications

We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G8-regularity introduced by M. Oberguggenberger. As a first application we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to micro local analysis of singularities of generalized functions, with respect to these regularities. AMS Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.