6533b7ddfe1ef96bd12751c7

RESEARCH PRODUCT

Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

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Abstract In this paper we deal with Banach spaces of analytic functions X defined on the unit disk satisfying that R t f ∈ X for any t > 0 and f ∈ X , where R t f ( z ) = f ( e i t z ) . We study the space of functions in X such that ‖ P r ( D f ) ‖ X = O ( ω ( 1 − r ) 1 − r ) , r → 1 − where D f ( z ) = ∑ n = 0 ∞ ( n + 1 ) a n z n 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) ‖ R t f − f ‖ X = O ( ω ( t ) ) , (b) ‖ P r f − f ‖ X = O ( ω ( 1 − r ) ) , (c) ‖ Δ n f ‖ X = O ( ω ( 2 − n ) ) , or (d) ‖ f − s n f ‖ X = O ( ω ( n − 1 ) ) , where P r f ( z ) = f ( r z ) , s n f ( z ) = ∑ k = 0 n a k z k and Δ n f = s 2 n f − s 2 n − 1 f . Our results extend those known for Hardy or Bergman spaces and power weights ω ( t ) = t α .