Modulus of continuity with respect to semigroups of analytic functions and applications

Given a complex Banach space E  , a semigroup of analytic functions (φt)(φt) and an analytic function F:D→EF:D→E we introduce the modulus wφ(F,t)=sup|z|<1⁡‖F(φt(z))−F(z)‖wφ(F,t)=sup|z|<1⁡‖F(φt(z))−F(z)‖. We show that if 0<α≤10<α≤1 and F   belongs to the vector-valued disc algebra A(D,E)A(D,E), the Lipschitz condition M∞(F′,r)=O((1−r)1−α)M∞(F′,r)=O((1−r)1−α) as r→1r→1 is equivalent to wφ(F,t)=O(tα)wφ(F,t)=O(tα) as t→0t→0 for any semigroup of analytic functions (φt)(φt), with φt(0)=0φt(0)=0 and infinitesimal generator GG, satisfying that φt′ and GG belong to H∞(D)H∞(D) with sup0≤t≤1⁡‖φ′‖∞<∞sup0≤t≤1⁡‖φ′‖∞<∞, and in particular is equivalent to the condition ‖F−Fr‖A(D,E)=O((1−r)α)‖F−Fr‖A(D,E)=O((1−r)α) as r→1r→1. We apply this result to particular semigroups (φt)(φt) and particular spaces of analytic functions E, such as Hardy or Bergman spaces, to recover several known results about Lipschitz type functions.