Product integrals II: Contour integrals

This paper continues the joint work of the authors begun in the article “On Strong Product Integration” [J. Functional Analysis, submitted]. We consider product integrals along contours; the point of view and development is analogous to the usual complex variable theory of ordinary contour integrals. Our main results are Theorem 2.3 (homotopy invariance of product integrals, an analog of Cauchy's integral theorem) and Theorem 3.4 (an analog of Cauchy's integral formula or the residue theorem).     

Singular integrals,analytic capacity and rectifiability

In this survey we study some interplay between classical complex analysis (removable sets for bounded analytic functions), harmonic analysis (singular integrals), and geometric measure theory (rectifiability).General expositions and surveys closely related to this paper are given in [6, 12, 18, 21, 23, 24, 40, 49, 59, 61, 62].     

Stochastic integrals for nonprevisible,multiparameter processes

We develop a general theory for stochastic integrals of generalized stochastic processesX(t), depending on multidimensional time, within the framework of the space of Wiener distributions (D ^*).     

Singular integrals on symmetric spaces, II

We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by -bi-invariant kernels with certain large scale cancellation properties. As an application we prove boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.

     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Master integrals, superintegrability and quadratic algebras

In this paper we use a generalization of Oevel's theorem about master symmetries to relate them with superintegrability and quadratic algebras.     

Uniform distribution,invariant means,and Riemann integrals

Translated from Matematicheskie Zametki, Vol. 56, No. 3, pp. 144–154, September, 1994.