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  • Bulletin of the Australian Mathematical Society, Volume 75, Issue 3
  • June 2007, pp. 431-446

Integration Operator Acting on Hardy and Weighted Bergman Spaces

  • Jouni Rättyä (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700039356
  • Published online: 01 April 2009
Abstract

Questions related to the operator Jg(f)(z):= ∫xof (ζ)g′(ζ) , induced by an analytic function g in the unit disc, are studied. It is shown that a function G is the derivative of a function in the Hardy space Hp if and only if it is of the form G = Fψ′ where FHq, ψ ∈ H3 and 1/s = 1/p − 1/q. Moreover, a complete characterisation of when Jg is bounded or compact from one weighted Bergman space into another is established, and an asymptotic formula for the essential norm of Jg, the distance from compact operators in the operator norm, is given. As an immediate consequence it is obtained that if p < 2 + α and α > −1, then any primitive of belongs to where q = ((2 + α) p)/(2 + α − p). For α = −1 this is a sharp result by Hardy and Littlewood on primitives of functions in Hardy space , 0 < p < 1.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
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