On sharp constants for dual Segal–Bargmann Lp spaces

Abstract

We study dilated holomorphic L p space of Gaussian measures over C n , denoted H n p,α with variance scaling parameter α > 0. The duality relations (H n p,α ) * ∼ = H p ′ ,α hold with 1 p + 1 p ′ = 1, but not isometrically. We identify the sharp lower constant comparing the norms on H p ′ ,α and (H n p,α ) * , and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp bound on each Taylor coefficient of a function in the Fock space for n = 1.

Vanishing Bergman kernels on the disk

“…This theorem can also be used to obtain the estimate (1.6) of Theorem 1.2 of [8], which deals with a similar question for the Segal-Bargmann space. Note that the dependence of p is suppressed by the way the Segal-Bargmann norm depends on p. We will address this topic in more detail in the final section of the paper.…”

“…To adapt Theorem 2.2 to the Segal-Bargmann case, note that for 0 < p < ∞ the L p -norm of F p γ actually comes from F 2 γp/2 . Thus, since the case p = ∞ is obvious, we can easily obtain the sharp bound |f (z)| ≤ e (γ/2)|z| 2 f F p γ , which is the formula (1.6) of [8], see also Theorem 2.8 [21]. To be completely honest, the case 0 < p < ∞ also needs part of the argument from [17].…”