Suppose B(z) is an infinite
Blaschke product with zeros {zk}. It is known that B′∉A2,0 (or D1∕2B∉H2). We
extend this to get B′∉Ap,p−2(p > 1) (or DβB∉H1∕β, β > 0) and apply this to the
Taylor coefficients of an infinite Blaschke product. We also present extended
versions of the Hardy-Littlewood theorem on fractional integrals and the
Hardy-Littlewood embedding theorem with simple proofs. These extensions show
that the above theorem becomes stronger as p ↑∞ (or β ↓ 0, respectively).
Finally, we give sufficient conditions on {zk} in order that DβB ∈ Ap,α
or ∈ Hp, which shows that the above result is best possible in a certain
sense.
