Weighted hardy spaces associated with operators satisfying reinforced off-diagonal estimates

Abstract

Let L be a nonnegative self-adjoint operator on L2(ℝn) satisfying the reinforced (pl,p'l) off-diagonal estimates, where pl [1,2) and p'L denotes its conjugate exponent. Assume that p (0,1] and the weight w satisfies the reverse Hölder inequality of order (p'L/p)'. In particular, if the heat kernels of the semigroups {e~tL}t>0 satisfy the Gaussian upper bounds, thenpL = 1 and hence w G A00(ℝn). In this paper, the authors introduce the weighted Hardy spaces HpL w(ℝn) associated with the operator L, via the Lusin area function associated with the heat semigroup generated by L. Characterizations of HpL w (ℝn), in terms of the atom and the molecule, are obtained. As applications, the bounded-ness of singular integrals such as spectral multipliers, square functions and Riesz transforms on weighted Hardy spaces HLpw(ℝn) are investigated. Even for the Schrödinger operator - Δ + V with 0 < V e L1oc(ℝn), the obtained results in this paper essentially improve the known results by extending the narrow range of the weights into the whole A∞(Rn) weights.