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The Hardy space H 1 on non-homogeneous metric spaces. (English) Zbl 1250.42076

According to T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)], a metric measure space (X,d,μ) is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ(x,r) and a positive constant C such that for each xX,rλ(x,r) is non-decreasing and, for all xX and r>0,


where B(x,r)={yX:d(x,y)<r}.

Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function λ(x,r)=μ(B(x,r)). Moreover, let μ be a non-negative Radon measure on n which only satisfies the polynomial growth condition

μ({y n :|x-y|<r})Cr a ·

By taking λ(x,r)=Cr a , we see that ( n ,|·|,μ) is also upper doubling measure space.

A metric space (X,d) is said to be geometrically doubling if there exists some N 0 such that for any ball B(x,r)X, there exists a finite ball covering {B(x i ,r/2)} i of B(x,r) such that the cardinality of this covering is at most N 0 .

Let (X,d,μ) be a metric space satisfying the upper doubling condition and the geometrical doubling condition. T. Hytönen introduced the regularized BMO space RBMO(μ). The authors introduce the atomic Hardy space H 1 (μ) and show that the dual space of H 1 (μ) is RBMO(μ). As an application they obtain the boundedness of Calderón–Zygmund operators from H 1 (μ) to L 1 (μ).

42B30H p -spaces (Fourier analysis)
42B35Function spaces arising in harmonic analysis