Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f   on the unit ball BB with Hadamard gaps, that is, f(z)=∑k=1∞Pnk(z) (the homogeneous polynomial expansion of f  ) satisfying nk+1/nk⩾λ>1nk+1/nk⩾λ>1 for all k∈Nk∈N, belongs to the space Bpα(B)=f|sup00 if and only if limsupk→∞‖Pnk‖pnk1-α<∞. Moreover, we show that the following asymptotic relation holds ‖f‖Bpα≍supk∈N‖Pnk‖pnk1-α. Also we prove that limr→1(1-r2)α‖Rfr‖p=0limr→1(1-r2)α‖Rfr‖p=0 if and only if limk→∞‖Pnk‖pnk1-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)].