Finite Atomic Decomposition Characterization of Variable Anisotropic Hardy Spaces

In 2011,Dekel et al.developed highly geometric Hardy spaces H^(p)(Θ),for the full range 0<p≤1,which are constructed by continuous multi-level ellipsoid coverΘof R^(n) with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level.The authors obtain the finite atomic decomposition characterization of Hardy spaces H^(p)(Θ)and as an application,the authors prove that given an admissible triplet(p,q,l)with 1≤q≤∞,if T is a sublinear operator and uniformly bounded elements of some quasi-Banach space B for maps all(p,q,l)-atoms with q<∞(or all continuous(p,q,l)-atoms with q=∞),then T uniquely extends to a bounded sublinear operator from H^(p)(Θ)to B.These results generalize the known results on the anisotropic Hardy spaces of Bownik et al.