On sharp constants for dual Segal–Bargmann LpLp spaces

We study dilated holomorphic LpLp space of Gaussian measures over CnCn, denoted Hp,αn with variance scaling parameter α>0α>0. The duality relations (Hp,αn)⁎≅Hp′,α hold with 1p+1p′=1, but not isometrically. We identify the sharp lower constant comparing the norms on Hp′,αHp′,α and (Hp,αn)⁎, 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=1n=1.