Sarason's composition operator over the half-plane

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Abstract

Let H={z∈C:Imz>0} be the upper half plane, and denote by Lp(R), 1≤p<∞ the usual Lebesgue space of functions on the real line R. We define two “composition operators” acting on Lp(R) induced by a Borel function φ:R→H‾ by first taking either the Poisson or Borel extension of f∈Lp(R) to a function on H‾ then composing with φ and taking vertical limits. Classical composition operators, induced by holomorphic functions and acting on the Hardy spaces Hp(H) of holomorphic functions, correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on Lp(R), 1≤p<∞. The characterization for the case 1<p<∞ is independent of p and the same for the Poisson and the Borel extensions. The case p=1 is quite different.

Original languageEnglish
Pages (from-to)1309-1341
Number of pages33
JournalJournal of Mathematical Analysis and Applications
Volume462
Issue number2
DOIs
Publication statusPublished - 2018 Jun 15

Keywords

  • Boundedness
  • Compactness
  • Sarason's composition operator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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