Quasiconformal Harmonic Mappings and Steinhaus Lattice-Point Problem with Partitions of Unity on Banach Spaces

Abstract

The bi-Lipschitz type inqualities with the quasi – isometries of harmonic quasiconformal mappings and between smooth Jordan domains are studied. The quasiconformal maps with controlled Laplacian and coefficients estimates for harmonic υ – Bloch mappings and harmonic k- quasiconformal mappings with the curvature of the boundary are considered. We give the smooth functions, partitions of unity, lines and spaces and partitions of unity on certain Banach spaces. We show the dual locally uniformaly rotund and convex norms, the structure of WUR Banach spaces. Smooth norms and approximation in Banach spaces of type C(k). the three lattice – point problems of Steinhaus and Steinhaus tilling problem for Banach spaces are dealt with.