Composition as an integral operator

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Abstract

Let S be the unit sphere and B the unit ball in Cn, and denote by L1(S) the usual Lebesgue space of integrable functions on S. We define four "composition operators" acting on L1(S) and associated with a Borel function ϕ:S→B-, by first taking one of four natural extensions of f∈L1(S) to a function on B-, then composing with ϕ and taking radial limits. Classical composition operators acting on Hardy spaces of holomorphic functions correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on Lt(S), 1≤t<∞. Dependence on t and relations between the characterizations for the different operators are also studied.

Original languageEnglish
Pages (from-to)149-187
Number of pages39
JournalAdvances in Mathematics
Volume273
DOIs
Publication statusPublished - 2015 Mar 9

Keywords

  • Boundedness
  • Compactness
  • Composition operator

ASJC Scopus subject areas

  • Mathematics(all)

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