Isomorphisms of variable Hardy spaces associated with Schrödinger operators

Abstract

Let L:= −Δ + V be the Schrodinger operator on ℝn with n ≥ 3, where V is a non-negative potential satisfying Δ−1 (V) ∈ L∞(ℝn). Let w be an L-harmonic function, determined by V, satisfying that there exists a positive constant o such that, for any x ∈ ℝn, 0 < δ ≤ w(x) ≤ 1. Assume that p(·): ℝn → (0, 1] is a variable exponent satisfying the globally log-Holder continuous condition. In this article, the authors show that the mappings $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto wf \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ and $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto {\left( { - {\rm{\Delta }}} \right)^{1/2}}{L^{ - 1/2}}\left( f \right) \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ are isomorphisms between the variable Hardy spaces (ℝn), associated with L, and the variable Hardy spaces Hp(·)(ℝn).