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.

A groupoid approach to pseudodifferential calculi

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural {\mathbb{R}^{\times}_{+}}-action. Specifically, a properly supported semiregular distribution on {M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the {\mathbb{R}^{\times}_{+}}-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

A Spectral Identity on Jacobi Polynomials and its Analytic Implications

The Jacobi coefficientsare linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomialsinto a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.

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.