Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications

A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

“…In 2014, Bui and Duong [4] introduced Hardy spaces associated with discrete Laplacian on doubling graphs which can be seen as special case of spaces of homogeneous type, via atomic decomposition. In 2018, Bui et al [5] obtained the maximal function characterizations of local Hardy spaces associated with some non-negative self-adjoint operator L. Recently, Bui et al [6] showed that, when µ(X) < ∞, for any operator L with its heat kernels having good decay properties, then the Hardy spaces associated with L coincide with the Hardy spaces of Coifman and Weiss.…”

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

“…The Musielak-Orlicz Hardy space on R n has proved useful in harmonic analysis (see, for example, [56,18,57,79]) and, especially, naturally appears in the endpoint estimate for both the div-curl lemma and the commutator of Caldrón-Zygmund operators (see [9,10,79]). Moreover, the real-variable theory of Hardy-type spaces associated with operators on the space of homogeneous type was also widely studied (see, for example, [12,13,14,15,29,41,45,74,84,51,87]).…”

“…Recently, under the assumption that X is a space of homogeneous type with µ(X) = ∞, the non-tangential maximal function and the radial maximal function characterizations, associated with L, of the Hardy space H p L (X) (p ∈ (0, 1]) were established in [74], which improves the results obtained in [73] even when X := R n . Moreover, the equivalent characterizations of the Hardy space H p L (X) (p ∈ (0, 1]) associated with L, including the atom, the non-tangential maximal function and the radial maximal function associated with L, were established in [14] via a different method from that in [74], which improve the results in [74] by removing the assumption that µ(X) = ∞. Recently, the atomic and the maximal function characterizations of the weighted Hardy space H p L, ω (X), with any p ∈ (0, 1], associated with L, were obtained in [13] via applying the method used in [14], without assuming µ(X) = ∞, where ω ∈ A ∞ (X).…”

“…Moreover, the equivalent characterizations of the Hardy space H p L (X) (p ∈ (0, 1]) associated with L, including the atom, the non-tangential maximal function and the radial maximal function associated with L, were established in [14] via a different method from that in [74], which improve the results in [74] by removing the assumption that µ(X) = ∞. Recently, the atomic and the maximal function characterizations of the weighted Hardy space H p L, ω (X), with any p ∈ (0, 1], associated with L, were obtained in [13] via applying the method used in [14], without assuming µ(X) = ∞, where ω ∈ A ∞ (X). Here and hereafter, A q (X) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [39]).…”

“…Let X be a space of homogeneous type and L a non-negative self-adjoint operator on L 2 (X) whose heat kernels satisfy the Gaussian upper bound estimates. Motivated by [14,73,74,85], in this article, we establish the equivalent characterizations of the Musielak-Orlicz-Hardy space H ϕ, L (X), in terms of the atom, the non-tangential maximal function and the radial maximal function associated with L, which improve the corresponding results for the space H ϕ, L (X) established in [12,51,84] via removing the assumption that µ(X) = ∞. In particular, when µ(X) < ∞, the local non-tangential and radial maximal function characterizations of H ϕ, L (X) are obtained, which imply that the global Musielak-Orlicz-Hardy space H ϕ, L (X) and the local Musielak-Orlicz-Hardy space h ϕ, L (X) are equivalent when µ(X) < ∞.…”