The boundedness of Riesz transforms for Hermite expansions on the Hardy spaces

Let L = − Δ + | x | 2 be a Hermite operator, where Δ is the Laplacian on R d . In this paper, we first characterize the Hardy spaces H L 1 ( R d ) associated with L by a new version of area integral. Then, we use it to prove the boundedness of Riesz transforms R j L , j = 1 , 2 , … for L on H L 1 ( R d ) . Moreover, we characterize the Hardy space H L 1 ( R d ) by R j L . This gives a negative answer to a question asked by Thangavelu.