Applications of the spaces of homogeneous polynomials to some problems on the ball algebra

Abstract

Denote by B 2 {B_2} the unit ball in C 2 {{\mathbf {C}}^2} . The existence is shown of a uniformly bounded orthonormal basis in H 2 ( B 2 ) {H^2}({B_2}) , by constructing such systems in the spaces of homogeneous polynomials. In the second part of the paper, those spaces of homogeneous polynomials are exploited to disprove the existence of generalized analytic projections, the so-called ( i p − π p ) ({i_p} - {\pi _p}) property, for the ball algebra.