LAUSR

Abstract 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 ≤ ∞ , 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 .