### Abstract

Let Φ be an entire function on C^{n}, and for any h > 0 and r > 0 define F_{r} = |Φ(z)|^{r} e^{ -2π|z|2 h}. Let dμ_{h} denote h^{-n} times Lebesgue measure on C^{n}. ∝ |▽F_{r}^{ s 2}|^{2} dμ_{h} = nπs h ∝ F_{r}^{s}dμ_{h}. From this and a logarithmic Sobolev inequality we easily deduce q^{ n q}∥F_{r}∥_{q} ≤ p^{ n p}∥F_{r}∥_{p} for all 0 < p ≤ q < t8 where the L^{p} norms are taken with respect to the measure dμ_{h} above. We apply these results to the study of the spaces A^{p} consisting of all entire functions Φ satisfying ∝ |Φ(z)|^{p}e^{ -2π|z|2 h} dμ_{h} < ∞ obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in A^{p} for 1 ≤ p < ∞. We then apply our results to the coherent state transform, extending and simplifying some previously known results.

Original language | English (US) |
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Pages (from-to) | 231-249 |

Number of pages | 19 |

Journal | Journal of Functional Analysis |

Volume | 97 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1991 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis