The Bargmann transform on Lp(R)

We study the mapping properties of the Bargmann transform B on L-p spaces of the real line. It is well known that B maps L-2 (R) isometrically onto the Fock space F-2. When 2 < p <= infinity, we show that B maps L-p(R) boundedly into the Fock space F-p and that the mapping is not onto. When 1 < p < 2, we show that B maps L-p(R) boundedly into the Fock space F-q, where 1/p + 1/q = 1, and that B does not map L-p(R) into F-p. There is no reasonable way to define the Bargmann transform on L-p (R) when 0 < p < 1. (C) 2018 Elsevier Inc. All rights reserved.