Weighted-Hardy functions with Hadamard gaps on the unit ball

Abstract

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, f (z) = ∑k = 1∞ Pnk (z) (the homogeneous polynomial expansion of f) satisfying nk + 1 / nk ≥ λ > 1 for all k ∈ N, belongs to the space Bpα (B) = fenced(f | sup0 < r < 1 (1 - r2)α {norm of matrix} R fr {norm of matrix}p < ∞, f ∈ H (B)), α, p > 0 if and only if limsupk → ∞ {norm of matrix} Pnk {norm of matrix}p nk1 - α < ∞. Moreover, we show that the following asymptotic relation holds {norm of matrix} f {norm of matrix}Bpα {equivalent to} supk ∈ N {norm of matrix} Pnk {norm of matrix}p nk1 - α. Also we prove that limr → 1 (1 - r2)α {norm of matrix} R fr {norm of matrix}p = 0 if and only if limk → ∞ {norm of matrix} Pnk {norm of matrix}p nk1 - α = 0. These results confirm two conjectures from the following recent paper [S. Stević, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].