Hardy spaces H ^p _L (ℝⁿ) associated with operators satisfying k-Davies-Gaffney estimates

Abstract

Let L be a one-to-one operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k ∈ ℕ. In this paper, the authors introduce the Hardy space H ^p _L (ℝⁿ) with p ∈ (0, 1] associated with L in terms of square functions defined via $\left\{ {e^{ - t^{2k} L} } \right\}_{t > 0}$ and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L ₁ with complex bounded measurable coefficients and the 2k-order Schrödinger type operator L ²:= (−Δ)^k + V ^k, where Δ is the Laplacian and 0 ⩽ V ∈ L ^k_loc (ℝⁿ). Moreover, as an application, for i ∈ {1, 2}, the authors prove that the associated Riesz transform ▿ ^k(L ^−1/2 _i ) is bounded from $H_{L_i }^p \left( {\mathbb{R}^n } \right)$ to H ^p(ℝⁿ) for p ∈ (n/(n + k), 1] and establish the Riesz transform characterizations of $H_{L_1 }^p \left( {\mathbb{R}^n } \right)$ for p ∈ (rn/(n + kr), 1] if $\left\{ {e^{ - tL_1 } } \right\}_{t > 0}$ satisfies the L ^r − L ² k-off-diagonal estimates with r ∈ (1, 2]. These results when k:= 1 and L:= L ₁ are known.