Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators

Abstract

Let φ be a growth function, and let A:= −(∇−ia)·(∇−ia)+V be a magnetic Schrödinger operator on L ²(ℝⁿ), n ⩾ 2, where a:= (a ₁, a ₂, …, a _n ) ∈ L ²_loc (ℝⁿ,ℝⁿ) and 0 ⩽ V ∈ L ¹_loc (ℝⁿ). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space H _{A, φ} (ℝⁿ), defined by the Lusin area function associated with $\{ e^{ - t^2 A} \} _{t > 0}$, in terms of the Lusin area function associated with $\{ e^{ - t\sqrt A } \} _{t > 0}$, the radial maximal functions and the non-tangential maximal functions associated with $\{ e^{ - t^2 A} \} _{t > 0}$ and $\{ e^{ - t\sqrt A } \} _{t > 0}$, respectively. The boundedness of the Riesz transforms L _k A ^−1/2, k ∈ {1, 2, …, n}, from H _{A, φ} (ℝⁿ) to H ^φ(ℝⁿ) is also presented, where L _k is the closure of $\frac{\partial } {{\partial x_k }}$ - ia _k in L ²(ℝⁿ). These results are new even when φ(x, t):= ω(x)t ^p for all x ∈ ℝⁿ and t ∈ (0,+∞) with p ∈ (0, 1] and ω ∈ A _∞(ℝⁿ) (the class of Muckenhoupt weights on ℝⁿ).