Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Abstract

Let \(\mathcal{X}\) be a metric space with doubling measure and L a nonnegative self-adjoint operator in \(L^{2}(\mathcal{X})\) satisfying the Davies–Gaffney estimates. Let \(\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)\) be a function such that φ(x,⋅) is an Orlicz function, \(\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})\) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2−I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space \(H_{\varphi,L}(\mathcal{X})\), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space \(\mathrm{BMO}_{\varphi,L}(\mathcal{X})\), which is further proved to be the dual space of \(H_{\varphi,L}(\mathcal{X})\) and hence whose φ-Carleson measure characterization is deduced. Characterizations of \(H_{\varphi,L}(\mathcal{X})\), including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize \(H_{\varphi,L}(\mathcal{X})\) in terms of the Littlewood–Paley \(g^{\ast}_{\lambda}\)-function \(g^{\ast}_{\lambda,L}\) and establish a Hörmander-type spectral multiplier theorem for L on \(H_{\varphi,L}(\mathcal{X})\). Moreover, for the Musielak–Orlicz–Hardy space H φ,L (ℝn) associated with the Schrödinger operator L:=−Δ+V, where \(0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H φ,L (ℝn) to the Musielak–Orlicz space L φ(ℝn) when i(φ)∈(0,1], and from H φ,L (ℝn) to the Musielak–Orlicz–Hardy space H φ (ℝn) when \(i(\varphi)\in(\frac{n}{n+1},1]\), where i(φ) denotes the uniformly critical lower type index of φ.