Abstract We study the boundary behavior of some open, light, and unclosed ring mappings. The mappings are considered on general metric measure spaces, and our results are new in the class of quasiregular mappings or in some important class of mappings of finite dilatation.
Abstract In this article, we prove that a smooth projective variety (defined over a field k ) of the nonnegative Kodaira dimension that has a k -rational point, a separable polarized self-map, and a virtually abelian geometric fundamental group must admit an abelian variety as a finite étale cover.
Abstract The classical Hilbert transform of a function on the real line arises as the boundary value of the function’s conjugate-harmonic extension. In this article, harmonic extensions to the upper half-integer lattice Z × N $\mathbb {Z} \times \mathbb {N}$ double struck upper Z times double struck upper N are constructed for a given boundary sequence on the integers; these serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by: (i) discrete harmonicity with respect to a two-dimensional Laplacian, (ii) a Cauchy–Riemann system, and (iii) boundary values involving a discrete Hilbert transform. This new discrete Hilbert transform is compared to the Riesz–Titchmarsh transform, and Lebesgue bounds are proved. The construction is also extended to the higher dimensional case. These results provide a discrete harmonic-analytic model analogous to the classical theory.
