Narrow Results

In this paper, we show the regularity of divergence form parabolic equations on time-dependent quasiconvex domains. The objective is to study the optimal parabolic boundary condition for the L^p estimates. The time-dependent quasiconvex domain is a generalization of the time-dependent Reifenberg flat domain, and assesses some properties analog to the convex domain. As to the a priori estimates near the boundary, we will apply the maximal function technique, Vitali covering lemma and the compactness method.

For two weighted local Morrey spaces and we obtain general type sufficient conditions and necessary conditions imposed on the functions φ and ψ and the weights u and v for the boundedness of the maximal operator from to , with some "logarithmic gap" between the sufficient and necessary conditions. Both the conditions formally coincide if we omit a certain logarithmic factor in these conditions.

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).

Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces H^p((0,∞),dm_λ) associated with △_λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dm_λ(x) ≡ x^2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

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 $p\in (0,2]$.

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of $\delta$-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of $\delta$-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

In this paper, the authors introduce a class of mixed-norm Herz spaces, ${Ė}_{\overrightarrow{q}}^{\overrightarrow{\alpha },\overrightarrow{p}}({ℝ}^n)$, which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on ${Ė}_{\overrightarrow{q}}^{\overrightarrow{\alpha },\overrightarrow{p}}({ℝ}^n)$. Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on ${Ė}_{\overrightarrow{q}}^{\overrightarrow{\alpha },\overrightarrow{p}}({ℝ}^n)$. As applications, the authors develop various real-variable theory of Hardy spaces associated with ${Ė}_{\overrightarrow{q}}^{\overrightarrow{\alpha },\overrightarrow{p}}({ℝ}^n)$ by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of ${Ė}_{\overrightarrow{q}}^{\overrightarrow{\alpha },\overrightarrow{p}}({ℝ}^n)$ and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem.

The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions Φ and Ψ, and obtain the equivalent conditions for Φ and Ψ such that this kind of inequalities holds. In the case Φ = Ψ ∈ Δ₂, it is proved that this necessary and sufficient condition is equivalent to q_Φ > 1.

The purpose of this paper is to articulate an observation that many interesting types of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible.