Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Duong, Xuan Thinh; Ly, Fu Ken

doi:10.1090/tran/7289

Cited by 63 publications

(32 citation statements)

References 38 publications

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“…In the case that µ(X) < ∞, we have the following further conclusion. Similarly to the proof of Theorem 2.9 below, we find that, for any α ∈ (0, ∞), the local Musielak-Orlicz-Hardy spaces h φ, α ϕ, L, max (X) and h φ ϕ, L, rad (X) coincide with equivalent quasi-norms (see [15,Proposition 4.4] for the case that ϕ(x, t) := t p , with p ∈ (0, 1], for any x ∈ X and t ∈ [0, ∞)). Moreover, replacing the maximal function φ * L, α ( f ) by φ * L, α, loc ( f ) and repeating the proof of Theorem 1.8, we know that, for any q…”

Section: Introductionsupporting

confidence: 54%

“…Examples of operators satisfying Assumptions 1.1 and 1.2 include both second-order selfadjoint elliptic operators in divergence form with bounded measurable coefficients and (degenerate) Schrödinger operators with non-negative potentials or with magnetic fields on R n or strongly Lipschitz domains, and Laplace-Beltrami operators on Heisenberg groups, connected and simply connected nilpotent Lie groups or complete Riemannian manifolds (see, for example, [4,14,15,28,71,74]). Now we describe the Musielak-Orlicz function considered in this article as follows.…”

Section: Introductionmentioning

confidence: 99%

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Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Yang

2019

Collect. Math.

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Let X be a metric space with doubling measure and L a non-negative self-adjoint operator on L 2 (X) whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function ϕ : X×[0, ∞) → [0, ∞) satisfies that ϕ(x, ·) is an Orlicz function and ϕ(·, t) ∈ A ∞ (X) (the class of uniformly Muckenhoupt weights). Let H ϕ, L (X) be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of L. In this article, the authors characterize the space H ϕ, L (X) by means of atoms, non-tangential and radial maximal functions associated with L. In particular, when µ(X) < ∞, the local non-tangential and radial maximal function characterizations of H ϕ, L (X) are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the "geometric" Musielak-Orlicz-Hardy spaces H ϕ, r (Ω) and H ϕ, z (Ω) on the strongly Lipschitz domain Ω in R n associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when ϕ(x, t) := t for any x ∈ R n and t ∈ [0, ∞), the equivalent characterizations of H ϕ, z (Ω) given in this article improve the known results via removing the assumption that Ω is unbounded.(1.2)V(x, λr) ≤ Cλ n V (x, r) 2010 Mathematics Subject Classification. Primary 42B25; Secondary 42B35, 46E30, 30L99. Key words and phrases. Musielak-Orlicz-Hardy space, atom, maximal function, non-negative self-adjoint operator, Gaussian upper bound estimate, space of homogeneous type, strongly Lipschitz domain.

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Section: Introductionsupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Yang

2019

Collect. Math.

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“…Of course X k (M ) ⊇ X k (M ), for h 1 (M ) properly contains H 1 (M ). It may be worth recalling that h 1 (M ) is the analogue on M of the classical space h 1 (R n ) introduced by D. Goldberg in [Go] and further investigated on specific measure metric spaces in various papers, including [HMY,YZ1,YZ2,BDL] (see also the references therein).…”

Section: Introductionmentioning

confidence: 99%

A family of Hardy-type spaces on nondoubling manifolds

Martini

Meda

Vallarino

2020

Annali di Matematica

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We introduce a decreasing one-parameter family X γ (M ), γ > 0, of Banach subspaces of the Hardy-Goldberg space h 1 (M ) on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that X 1/2 (M ) agrees with the space of all functions in h 1 (M ) whose Riesz transform is in L 1 (M ), and we obtain the surprising result that this space does not admit an atomic decomposition.2010 Mathematics Subject Classification. 42B20, 42B30, 42B35, 58C99.

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“…In 2018, Bui et al. [6] obtained the maximal function characterizations of local Hardy spaces associated with some nonnegative self‐adjoint operator L . Recently, Bui et al.…”

Section: Introductionmentioning

confidence: 99%

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

Yang

Yuan

2021

Mathematische Nachrichten

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Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h∗,pfalse(Xfalse) via local grand maximal functions and also characterize h∗,pfalse(Xfalse) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,pfalse(Xfalse). As an application, the authors give the dual spaces of h∗,pfalse(Xfalse) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1false(Xfalse). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hnormalcw1false(Xfalse) when μ(X)<∞.

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Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Cited by 63 publications

References 38 publications

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

A family of Hardy-type spaces on nondoubling manifolds

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

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