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Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further obtains the...

Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{\Phi }\in (0,1]$ and $\omega \in A_{\infty }^{loc}\left(ℝⁿ\right)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ via the local grand maximal function. Let $\rho \left(t\right)\equiv t^{-1}/{\Phi }^{-1}\left(t^{-1}\right)$ for all t ∈ (0,∞). We also introduce the BMO-type space $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$ and establish the duality between $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ and $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$. Characterizations of $h_{\omega }^{\Phi }\left(ℝⁿ\right)$, including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented....

Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order...