Weighted Hardy Spaces Associated With Operators Satisfying Reinforced Off-Diagonal Estimates

Let L be a nonnegative self-adjoint operator on L²(Rⁿ) satisfying the reinforced (pL, pL') off-diagonal estimates, where pL∈[1,2) and pL' 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 -tL> 0 satisfy the Gaussian upper bounds, then pL=1 and hence w∈ A_∞( 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 boundedness of singular integrals such as spectral multipliers, square functions and Riesz transforms on weighted Hardy spaces HᵖL,\,w(Rⁿ) are investigated. Even for the Schrödinger operator -Δ+V with 0≤ V∈ Lloc¹ (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.