### 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 |

### Fingerprint

### Keywords

- Boundedness
- Compactness
- Sarason's composition operator

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Sarason's composition operator over the half-plane.** / Choe, Boo Rim; Koo, Hyung Woon; Smith, Wayne.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Sarason's composition operator over the half-plane

AU - Choe, Boo Rim

AU - Koo, Hyung Woon

AU - Smith, Wayne

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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

AB - 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

KW - Boundedness

KW - Compactness

KW - Sarason's composition operator

UR - http://www.scopus.com/inward/record.url?scp=85042357604&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042357604&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2018.02.046

DO - 10.1016/j.jmaa.2018.02.046

M3 - Article

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

ER -