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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