The degree of approximation by Hausdorff means of a conjugate Fourier series

The purpose of this paper is to analyze the degree of approximation of a function $\overline f$ that is a conjugate of a function $f$ belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series

Multivariate Davenport series

We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$ and $\{x\}$ is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.Comment: 43 page

Spectral Triples on Carnot Manifolds

We analyze whether one can construct a spectral triple for a Carnot manifold $M$, which detects its Carnot-Carath\'{e}odory metric and its graded dimension. Therefore we construct self-adjoint horizontal Dirac operators $D^H$ and show that each horizontal Dirac operator detects the metric via Connes' formula, but we also find that in no case these operators are hypoelliptic, which means they fail to have a compact resolvent. First we consider an example on compact Carnot nilmanifolds in detail, where we present a construction for a horizontal Dirac operator arising via pullback from the Dirac operator on the torus. Following an approach by Christian B\"ar to decompose the horizontal Clifford bundle, we detect that this operator has an infinite dimensional kernel. But in spite of this, in the case of Heisenberg nilmanifolds we will be able to discover the graded dimension from the asymptotic behavior of the eigenvalues of this horizontal Dirac operator. Afterwards we turn to the general case, showing that any horizontal Dirac operator fails to be hypoelliptic. Doing this, we develop a criterion from which hypoellipticity of certain graded differential operators can be excluded by considering the situation on a Heisenberg manifold, for which a complete characterization of hypoellipticity in known by the Rockland condition. Finally, we show how spectral triples can be constructed from horizontal Laplacians via the Heisenberg pseudodifferential calculus developed by Richard Beals and Peter Greiner. We suggest a few of these constructions, and discuss under which assumptions it may be possible to get an equivalent metric to the Carnot-Carath\'{e}odory metric from these operators. In addition, we mention a formula by which the Carnot-Carath\'{e}odory metric can be detected from arbitrary horizontal Laplacians.Comment: This is my PhD thesis, written at Leibniz Universitaet Hannover and submitted on May 14th, 2013. Email: [email protected]