Projection Operators in Analysis and Geometry of classical Banach Spaces

This project is a detailed investigation of projection opertators arising in classical analysis and the geometry of Banach spaces. We study - as an isometric problem - the asymptotic behaviour of Lebesgue constants of orthogonal projecton operators associated to generalized Franklin and spline bases. Here we continue research initiated by Z. Ciesielski, A. Kamont, A. Yu. Shadrin and C. de Boor. The anisotropic 2D Haar system is best adapted for studying isomorphic Banach space problems of the mixed norm Lebesgue spaces Lp(Lq). Orthogonal projections built from block-bases of the 2D Haar system play a fundamental role in the problems under investigation: The Gamlen-Gaudet problem in Lp(Lq), the subspace problem, and Andrew`s stability for the 2D Haar system in Lp(Lq).

In this project, we obtained new results about the convergence of spline series. They built the theoretical framework for the fast convergence of adaptive spline algorithms and therefore they have applications in the numerical solution of certain physical problems.By means of orthogonal projections onto block bases of the bi-variate Haarsystem we obtained factorization theorems for operators on biparameter-BMO and mixed norm Hardy spaces. We thus obtained extesions for Andrews theorem in the one variable case.Additionally, we proved results that connect Orlicz norms with simpler norms of certain random variables. Those findings have applications on the one hand in Banach space theory considering embedding problems and, on the other hand in Probability theory, especially in the area of order statistics of random variables. On spaces of homogeneous type rearrangement operators were estimated using combinatorics of deterministic dyadic grids.With conditioned Brownian motion and Doobs projection we obtained martingale decompositions that preserve holomorphy. Consequences thereof are novel Davis and Garsia inequalities for Hardy martingales and their dyadic perturbations.