Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

In this paper we deal with Banach spaces of analytic functions X   defined on the unit disk satisfying that Rtf∈XRtf∈X for any t>0t>0 and f∈Xf∈X, where Rtf(z)=f(eitz)Rtf(z)=f(eitz). We study the space of functions in X   such that ‖Pr(Df)‖X=O(ω(1−r)1−r), r→1−r→1− where Df(z)=∑n=0∞(n+1)anzn and ω is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X   satisfying any of the following conditions: (a) ‖Rtf−f‖X=O(ω(t))‖Rtf−f‖X=O(ω(t)), (b) ‖Prf−f‖X=O(ω(1−r))‖Prf−f‖X=O(ω(1−r)), (c) ‖Δnf‖X=O(ω(2−n))‖Δnf‖X=O(ω(2−n)), or (d) ‖f−snf‖X=O(ω(n−1))‖f−snf‖X=O(ω(n−1)), where Prf(z)=f(rz)Prf(z)=f(rz), snf(z)=∑k=0nakzk and Δnf=s2nf−s2n−1fΔnf=s2nf−s2n−1f. Our results extend those known for Hardy or Bergman spaces and power weights ω(t)=tαω(t)=tα.