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

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 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.