Mean Lipschitz conditions and composition semigroups
A theorem of Hardy and Littlewood and its extension for Abel
means says that the following are equivalent for functions f in the Hardy space Hp of the disc:
(a) ‖f(eitz)−f(z)‖p=O(tα) as t→0, i.e. f∈Λ(p,α), the mean Lipschitz class.
(b) Mp(r,f′)=O((1−r)α−1) as r→1−,
(c) ‖f(rz)−f(z)‖p=O((1−r)α) as r→1−,
We will describe an "elementary" proof of the theorem, and will discuss an analogue on Bergman spaces.
Recall that in the abstract setting, the Favard class Fα, 0<α≤1, for a strongly continuous
operator semigroup (Tt) on a Banach space X is
Fα={f∈X:‖Tt(f)−f‖X=O(tα)}.
The above conditions then, interpreted in terms of the composition semigroups f→f(eitz) (rotations) and f→f(e−tz) (dilations) say that Fα=Λ(p,α) for both of these semigroups on Hp. The question arises, given a general composition semigroup Tt(f)=f∘ϕt acting on Hardy space (or on other spaces of analytic functions), to identify its Favard class Fα in terms of a condition analogous to (b), involving the derivative.