International Mathematics Research Notices

Yoshino, Taro

doi: 10.1155/S1073792804142499pmid: N/A

We consider the affine transformation of a nilpotent Lie group on ℝn. Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is compact isotropy (CI) in the sense of Kobayashi. This paper gives an affirmative solution to Lipsman's conjecture in the four-dimensional case.

Bombieri, E.; Bourgain, J.

doi: 10.1155/S1073792804143444pmid: N/A

This paper studies the classical problem of determining the maximum growth of the Taylor series majorant of holomorphic functions with bound 1 in the unit disk, as a function of the radius of the Taylor series. We determine for the first time the asymptotic behavior of this maximum as the radius tends to 1, with a very small error term. The proof uses full asymptotic expansions of certain oscillatory integrals and the discrete Hilbert transform.

Pestov, Leonid; Uhlmann, Gunther

doi: 10.1155/S1073792804142116pmid: N/A

We describe a relation between the scattering relation, the Hilbert transform in frequency space, and the geodesic ray transform for simple, two-dimensional compact Riemannian manifolds with boundary. We use this relation to give a characterization of the range of the geodesic X-ray transform acting on scalar functions and vector fields in terms of the scattering relation which involves only measurements at the boundary. We also give inversion formulas for the geodesic X-ray transform acting on scalar functions and vector fields.