A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

Let { K t } t >0 be the semigroup of linear operators generated by a Schrödinger operator − L = Δ − V ( x ) on ℝ d , d ≥ 3, where V ( x ) ≥ 0 satisfies Δ −1 V ∈ L ∞ . We say that an L 1 -function f belongs to the Hardy space H L 1 if the maximal function ℳ L f ( x ) = sup t >0 | K t f ( x )| belongs to L 1 (ℝ d ). We prove that the operator (− Δ ) 1/2 L −1/2 is an isomorphism of the space H L 1 with the classical Hardy space H 1 (ℝ d ) whose inverse is L 1/2 (− Δ ) −1/2 . As a corollary we obtain that the space H L 1 is characterized by the Riesz transforms R j = ∂ ∂ x j L − 1 / 2 .