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

Abstract

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.

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators

“…Moreover, Bui et al [13,14,12,11] obtained various maximal function characterizations of some new (local) Hardy spaces associated with operators on X . Furthermore, S. Yang and D. Yang [95] established atomic and maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators on X . Unlike the case of the aforementioned Hardy spaces in the sense of Coifman and Weiss on X , real-variable characterizations of these Hardy-type spaces on X associated with operators mainly depend on the functional calculus of the semigroup generated by the relevant operator to obtain the related Calderón reproducing formulae so as to get rid of the dependence on the reverse doubling assumption of the measure µ under study of X .…”

Characterizations of vanishing Campanato classes related to Schrödinger operators via fractional semigroups on stratified groups

“…At the same time, localized Morrey–Campanato spaces related with Schrödinger operators on metric measure spaces were introduced by Yang et al 5 The localized Hardy spaces H 1 related to admissible functions on RD spaces were investigated by Yang and Zhou 6 . In the setting of nilpotent Lie groups, Jiang et al 7 obtained maximal function characterizations of Hardy spaces associated with Schrödinger operators (see also Yang and Yang 8 for the atomic and maximal characterizations of Musielak–Orlicz–Hardy spaces associated to nonnegative self‐adjoint operators on spaces of homogeneous type).…”

Difference Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

“…In recent years, function spaces and their applications on spaces of homogeneous type, with some additional assumptions, have been extensively investigated; see, for instance, [2,12,26,49,53] for the Ahlfors d-regular space case, and [24,25,55,57,58] for the RD-space case. Recall that an RD-space is a doubling metric measure space satisfying the following reverse doubling condition: there exist positive constants C (µ) ∈ (0, 1] and κ ∈ (0, ω] such that, for any ball B(x, r) with r ∈ (0, diam X/2) and λ ∈ [1, diam X/(2r)), C (µ) λ κ µ(B(x, r)) ≤ µ(B(x, λr)); here and thereafter, diam X := sup{d(x, y) : x, y ∈ X}.…”

“…Moreover, Bui et al [8,9,7,6] obtained various maximal function characterizations of a new local-type Hardy spaces associated with operators. Besides, S. Yang and D. Yang [53] established atomic and maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type.…”