Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type

Abstract

Let (, d, ) be a metric measure space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors prove that bilinear operators, which are finite combinations of compositions of commutators and Calderón-Zygmund operators, are bounded from H 1 ()×BMO() to L 1 (). The authors also prove that the commutator, generated by any b ∈ BMO() and Calderón-Zygmund operator, is bounded from the Hardy-type spacethat ensures the boundedness of the commutators from  to L 1 (). The novelties appearing in these approaches exist in applications of the multiresolution analysis of the wavelets on metric measure spaces of homogeneous type, the bilinear decomposition of the product space H 1 () × BMO(), the (sub)bilinear decomposition of commutators, the proof of off-diagonal estimates of the action of Calderón-Zygmund operators on the wavelet functions, and the boundedness of the almost diagonal matrix on the spaces H 1 () and BMO(). Notably, throughout this article, is not assumed to satisfy the reverse doubling condition. KEYWORDSbilinear decomposition, BMO space, Calderón-Zygmund operator, commutator, Hardy space, space of homogeneous type, wavelet for the theory of singular integrals and function spaces. A metric space (, d) is a set  equipped with a metric d such that, for any x, , z ∈ , (i) d(x, y) = d(y, x);

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

“…The space H log (X ) is also important to the local version of the above bilinear decomposition in [16]. Moreover, the bilinear decomposition in [19] is also useful to the endpoint boundedness of the (sub-)linear commutator [b, T ] of a sublinear operator T and b ∈ BMO(X ) on Hardy spaces in [46,47]; see the survey [15] for more details.…”

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

“…This motivates us to establish the related theory of local Hardy spaces on spaces of homogeneous type. Moreover, it should be mentioned that Hardy spaces on spaces of homogeneous type can be used in endpoint estimates on commutators and bilinear decompositions; see, for instance, [15,30,31].…”

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

“…This motivates us to establish the related theory of local Hardy spaces on spaces of homogeneous type. Moreover, it should be mentioned that Hardy spaces on spaces of homogeneous type can be used in endpoint estimates on commutators and bilinear decompositions; see, for instance, [15, 30, 31].…”