Narrow Results

In this paper, the authors establish the inhomogeneous Plancherel–Pôlya inequalities on spaces of homogeneous type by use of the inhomogeneous discrete Calderón reproducing formulas. As an application, the authors prove that the Lebesgue norms of the inhomogeneous Littlewood–Paley g-function and S-function on spaces of homogeneous type are equivalent. All results are new even for ℝⁿ.

Let be a metric space with doubling measure, L a nonnegative self-adjoint operator in satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p_ω∈(0, 1] and ρ(t) = t^-1/ω^-1(t^-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space via the Lusin area function associated to the heat semigroup, and the BMO-type space . The authors then establish the duality between and ; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space . Characterizations of , including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let and L = -Δ+V be a Schrödinger operator, where is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L^-1/2 is bounded from H_{ω, L}(ℝⁿ) to L(ω). Moreover, if there exist q₁, q₂∈(0, ∞) such that q₁<1<q₂ and [ω(t^q₂)]^q₁ is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space H_{ω, L}(ℝⁿ), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t^p for all t ∈ (0, ∞) and p ∈ (0, 1).

Let φ : ℝⁿ × [0,∞) → [0,∞) be a growth function such that φ(x, ⋅) is nondecreasing, φ(x, 0) = 0, φ(x, t) > 0 when t > 0, lim_t→∞φ(x, t) = ∞, and φ(⋅, t) is a Muckenhoupt A_∞(ℝⁿ) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak–Orlicz Hardy space H_φ(ℝⁿ) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the φ-Carleson measure characterization of the Musielak–Orlicz BMO-type space BMO_φ(ℝⁿ), which was proved to be the dual space of H_φ(ℝⁿ) by Luong Dang Ky.

Let $L$ be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all $p\in (0,\infty )$ and $L$ satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces $h_L^p({ℝ}^n)$ associated with $L$, which coincide with Goldberg’s local Hardy spaces $h^p({ℝ}^n)$ for all $p\in (0,\infty )$ when $L\equiv -\mathrm{\Delta }$ (the Laplace operator). The authors also establish a real-variable theory of $h_L^p({ℝ}^n)$, which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when $L\equiv -\mathrm{\text{div}}(A\nabla )$ (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that $h_L^p({ℝ}^n)$ coincides with the Hardy space $H_{L+\delta }^p({ℝ}^n)$ associated with the operator $L+\delta$ for all $p\in (0,\infty )$, where $\delta$ is some positive constant depending on the ellipticity and the off-diagonal estimates of $L$. As an application, the authors establish some mapping properties for the local Riesz transforms ${\nabla }^k{(L+\delta )}^{-1/2}$ on $H_{L+\delta }^p({ℝ}^n)$, where $k\in \{0,\dots ,m\}$ and $p\in (0,2]$.