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- W1837994127 abstract "Given a complex Banach space E , a semigroup of analytic functions ( φ t ) and an analytic function F : D → E we introduce the modulus w φ ( F , t ) = sup | z | < 1 ‖ F ( φ t ( z ) ) − F ( z ) ‖ . We show that if 0 < α ≤ 1 and F belongs to the vector-valued disc algebra A ( D , E ) , the Lipschitz condition M ∞ ( F ′ , r ) = O ( ( 1 − r ) 1 − α ) as r → 1 is equivalent to w φ ( F , t ) = O ( t α ) as t → 0 for any semigroup of analytic functions ( φ t ) , with φ t ( 0 ) = 0 and infinitesimal generator G , satisfying that φ t ′ and G belong to H ∞ ( D ) with sup 0 ≤ t ≤ 1 ‖ φ ′ ‖ ∞ < ∞ , and in particular is equivalent to the condition ‖ F − F r ‖ A ( D , E ) = O ( ( 1 − r ) α ) as r → 1 . We apply this result to particular semigroups ( φ t ) and particular spaces of analytic functions E , such as Hardy or Bergman spaces, to recover several known results about Lipschitz type functions." @default.
- W1837994127 created "2016-06-24" @default.
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- W1837994127 date "2016-03-01" @default.
- W1837994127 modified "2023-09-29" @default.
- W1837994127 title "Modulus of continuity with respect to semigroups of analytic functions and applications" @default.
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- W1837994127 doi "https://doi.org/10.1016/j.jmaa.2015.10.060" @default.
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