### Abstract

Let H={z∈C:Imz>0} be the upper half plane, and denote by L^{p}(R), 1≤p<∞ the usual Lebesgue space of functions on the real line R. We define two “composition operators” acting on L^{p}(R) induced by a Borel function φ:R→H‾ by first taking either the Poisson or Borel extension of f∈L^{p}(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 H^{p}(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 L^{p}(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 language | English |
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Journal | Journal of Mathematical Analysis and Applications |

DOIs | |

Publication status | Accepted/In press - 2018 Jan 1 |

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### Keywords

- Boundedness
- Compactness
- Sarason's composition operator

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics