Some integral identities and inequalities for entire functions and their application to the coherent state transform

Eric A. Carlen

doi:https://doi.org/10.1016/0022-1236(91)90022-W

Some integral identities and inequalities for entire functions and their application to the coherent state transform

Eric A. Carlen

Research output: Contribution to journal › Article › peer-review

47 Scopus citations

Abstract

Let Φ be an entire function on Cⁿ, 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 Cⁿ. ∝ |▽F_r^{s 2}|² dμ_h = nπs h ∝ F_r^sdμ_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)|^pe^{-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)
Pages (from-to)	231-249
Number of pages	19
Journal	Journal of Functional Analysis
Volume	97
Issue number	1
DOIs	https://doi.org/10.1016/0022-1236(91)90022-W
State	Published - Apr 1991
Externally published	Yes

ASJC Scopus subject areas

Analysis

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Cite this

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title = "Some integral identities and inequalities for entire functions and their application to the coherent state transform",

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

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T1 - Some integral identities and inequalities for entire functions and their application to the coherent state transform

AU - Carlen, Eric A.

PY - 1991/4

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N2 - Let Φ be an entire function on Cn, and for any h > 0 and r > 0 define Fr = |Φ(z)|r e -2π|z|2 h. Let dμh denote h-n times Lebesgue measure on Cn. ∝ |▽Fr s 2|2 dμh = nπs h ∝ Frsdμh. From this and a logarithmic Sobolev inequality we easily deduce q n q∥Fr∥q ≤ p n p∥Fr∥p for all 0 < p ≤ q < t8 where the Lp norms are taken with respect to the measure dμh above. We apply these results to the study of the spaces Ap consisting of all entire functions Φ satisfying ∝ |Φ(z)|pe -2π|z|2 h dμh < ∞ obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in Ap for 1 ≤ p < ∞. We then apply our results to the coherent state transform, extending and simplifying some previously known results.

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

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Some integral identities and inequalities for entire functions and their application to the coherent state transform

Abstract

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