The atomic Hardy space for a general Bessel operator

We study Hardy spaces associated with a general multidimensional Bessel operator $$\mathbb {B}_\nu $$ B ν . This operator depends on a multiparameter of type $$\nu $$ ν that is usually restricted to a product of half-lines. Here we deal with the Bessel operator in the general context, with no restrictions on the type parameter. We define the Hardy space $$H^1$$ H 1 for $$\mathbb {B}_\nu $$ B ν in terms of the maximal operator of the semigroup of operators $$\exp (-t\mathbb {B}_\nu )$$ exp ( - t B ν ) . Then we prove that, in general, $$H^1$$ H 1 admits an atomic decomposition of local type.

Maximal characterisation of local Hardy spaces on locally doubling manifolds

We prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.

Hausdorff operators on homogeneous spaces of locally compact groups

Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.

Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

LetΩ⊂ℝnbe a nonsmooth convex domain and letfbe a distribution in the atomic Hardy spaceHatp(Ω); we study the Schrödinger equations-div⁡(A∇u)+Vu=finΩwith the singular potentialVand the nonsmooth coefficient matrixA. We will show the existence of the Green function and establish theLpintegrability of the second-order derivative of the solution to the Schrödinger equation onΩwith the Dirichlet boundary condition forn/(n+1)<p≤2. Some fundamental pointwise estimates for the Green function are also given.

Discrete Hardy Spaces Related to Powers of the Poisson Kernel

Discrete Hardy spaces $H^{1}_{\alpha}(\partial{T})$, related to powers $\alpha \ge 1/2$ of the Poisson kernels on boundaries $\partial{T}$ of regular rooted trees, are studied. The spaces for $\alpha > 1/2$ coincide with the ordinary atomic Hardy space on $\partial{T}$, which in turn is strictly smaller than $H^{1}_{1/2}(\partial{T})$. The Orlicz space $L\log\log L(\partial{T})$ characterizes the positive and increasing functions in $H^{1}_{1/2}(\partial{T})$, but there is no such characterization for general positive functions.

The Hardy space H1 on non-homogeneous metric spaces

Let (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures

Let μ be a non-negative Radon measure on which satisfies only the polynomial growth condition. Let 𝒴 be a Banach space and H 1(μ) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H 1(μ) to 𝒴 if and only if T maps all (p, γ)-atomic blocks into uniformly bounded elements of 𝒴; moreover, the authors prove that for a sublinear operator T bounded from L 1(μ) to L 1, ∞(μ), if T maps all (p, γ)-atomic blocks with p ∈ (1, ∞) and γ ∈ ℕ into uniformly bounded elements of L 1(μ), then T extends to a bounded sublinear operator from H 1(μ) to L 1(μ). For the localized atomic Hardy space h 1(μ), the corresponding results are also presented. Finally, these results are applied to Calderón–Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón–Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.