### Abstract

Let 1 < p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N ^{p} (μ) consists of all measurable functions h on D such that log^{+} |h| L ^{p} (μ), and the area Nevanlinna space N _{α} ^{p} is the subspace consisting of all holomorphic functions, in N ^{p} ((1-|z|^{2})^{α} dv(z)), where α > -1 and ν is area measure on D. We characterize Carleson measures for N _{α} ^{p} , defined to be those measures μ for which N _{α} ^{p} ⊂ N ^{p} (μ). As an application, we show that the spaces N _{α} ^{p} are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given.

Original language | English |
---|---|

Pages (from-to) | 207-233 |

Number of pages | 27 |

Journal | Journal d'Analyse Mathematique |

Volume | 104 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Journal d'Analyse Mathematique*,

*104*(1), 207-233. https://doi.org/10.1007/s11854-008-0022-8

**Carleson measures for the area Nevanlinna spaces and applications.** / Choe, Boo Rim; Koo, Hyung Woon; Smith, Wayne.

Research output: Contribution to journal › Article

*Journal d'Analyse Mathematique*, vol. 104, no. 1, pp. 207-233. https://doi.org/10.1007/s11854-008-0022-8

}

TY - JOUR

T1 - Carleson measures for the area Nevanlinna spaces and applications

AU - Choe, Boo Rim

AU - Koo, Hyung Woon

AU - Smith, Wayne

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Let 1 < p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N p (μ) consists of all measurable functions h on D such that log+ |h| L p (μ), and the area Nevanlinna space N α p is the subspace consisting of all holomorphic functions, in N p ((1-|z|2)α dv(z)), where α > -1 and ν is area measure on D. We characterize Carleson measures for N α p , defined to be those measures μ for which N α p ⊂ N p (μ). As an application, we show that the spaces N α p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given.

AB - Let 1 < p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N p (μ) consists of all measurable functions h on D such that log+ |h| L p (μ), and the area Nevanlinna space N α p is the subspace consisting of all holomorphic functions, in N p ((1-|z|2)α dv(z)), where α > -1 and ν is area measure on D. We characterize Carleson measures for N α p , defined to be those measures μ for which N α p ⊂ N p (μ). As an application, we show that the spaces N α p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given.

UR - http://www.scopus.com/inward/record.url?scp=58449084347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58449084347&partnerID=8YFLogxK

U2 - 10.1007/s11854-008-0022-8

DO - 10.1007/s11854-008-0022-8

M3 - Article

AN - SCOPUS:58449084347

VL - 104

SP - 207

EP - 233

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -