Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

Abstract

Let L ≔ −Δ + V be a Schrödinger operator on ℝⁿ with n ⩾ 3 and V ⩾ 0 satisfying Δ⁻¹ V ∈ L ^∞(ℝⁿ). Assume that φ: ℝⁿ × [0,∞) → [0,∞) is a function such that φ(x, ·) is an Orlicz function, φ(·, t) ∈ A _∞(ℝⁿ) (the class of uniformly Muckenhoupt weights). Let w be an L-harmonic function on ℝⁿ with 0 < C ₁ ⩽ w ⩽ C ₂, where C ₁ and C ₂ are positive constants. In this article, the author proves that the mapping $H_{\phi ,L} (\mathbb{R}^n ) \mathrel\backepsilon f \mapsto wf \in H_\phi (\mathbb{R}^n )$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L,H_{\phi ,L} (\mathbb{R}^n )$, to the Musielak-Orlicz-Hardy space $H_\phi (\mathbb{R}^n )$ under some assumptions on φ. As applications, the author further obtains the atomic and molecular characterizations of the space $H_{\phi ,L} (\mathbb{R}^n )$ associated with w, and proves that the operator ${( - \Delta )^{ - 1/2}}{L^{1/2}}$ is an isomorphism of the spaces $H_{\phi ,L} (\mathbb{R}^n )$ and $H_\phi (\mathbb{R}^n )$. All these results are new even when φ(x, t) ≔ t ^p, for all x ∈ ℝⁿ and t ∈ [0,∞), with p ∞ (n/(n + μ₀), 1) and some μ₀ ∈ (0, 1].