Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces

The Hardy Space H1 with Non-doubling Measures and Their Applications

“…In [88], Lin and Yang proved that the L p .X ; / boundedness with p 2 .1; 1/ of the Marcinkiewicz integral is equivalent to either of its boundedness from L 1 .X ; / into L 1; 1 .X ; / or from the atomic Hardy space H 1 .X ; / into L 1 .X ; /. The equivalence of these statements is still unknown when 1Cı.B; S/ is replaced by Q ı .…”

Herz-Type Hardy Spaces Associated with Operators

“…It is known that many classical function spaces and the Hardy type spaces associated with operators have the atomic decompositions and the molecular decompositions, and the atomic and molecular decompositions of function spaces make the linear operators acting on spaces very simple; see [20][21][22][23][24][25][26][27][28][29], etc. In fact, the characterizations of spaces of functions or distributions, including the atomic and molecular characterizations, have many important applications in harmonic analysis.…”

Boundedness of certain commutators over non-homogeneous metric measure spaces

“…Sawano et al [33] presented an example showing that, if (X , d, μ) is not geometrically doubling, then Morrey spaces depend on the auxiliary parameters. More research on function spaces and the boundedness of various operators on metric measure spaces of non-homogeneous type can be found in [1,3,4,17,21,[24][25][26][27][28]. We refer the reader to the survey [45] and the monograph [46] for more developments on harmonic analysis in this setting.…”