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Calderón-Zygmund operators on Hardy spaces without the doubling condition. (English) Zbl 1113.42008

In this paper the authors establish the boundedness of Calderón-Zygmund operator associated to a non-negative Radon measure μ without the doubling condition on the Hardy space H 1 (μ). More precisely, the Euclidean space d is endowed with a non-negative Radon measure μ which only satisfies the following growth condition that there exists C>0 such that

μ(B(x,r))Cr n

for all x d and r>0, where B(x,r)={y d :|y-x|<r}, n is a fixed number and 0<nd. For such measure μ it is not necessary to be doubling. Let K be a function on d ×{(x,y):x=y} satisfying for xy,

|K(x,y)|C|x-y| -n ,

and for |x-y|2|x-x ' |,

|K(x,y)-K(x ' ,y)|+|K(y,x)-K(y,x ' )|C|x-x ' | δ |x-y| n+δ ,

where δ(0,1] and C>0 is a constant. The Calderón-Zygmund operator associated to the above kernel K and μ is defined by

Tf(x)= d K(x,y)f(y)dμ(y)·

For ε>0 we denote T ε by the truncated operators of T. If the operators {T ε } ε>0 are bounded on L 2 (μ) uniformly on ε>0, T is bounded on L 2 (μ). In this case there is an operator

T ˜f(x)= d K(x,y)f(y)dμ(y),x d supp(f)

which is the weak limit as ε0 of some subsequences of operators {T ε } ε>0 . In the main theorem the authors prove that if T ˜ is bounded on L 2 (μ) and T ˜ * 1=0, then T ˜ is bounded on H 1 (μ). Here, T ˜ * 1=0 implies that for any bounded function b with compact support and d dμ=0, d T ˜b(x)dμ(x)=0. They adapt the Hardy space H 1 (μ) as the characterization of a grand maximal function developed by X. Tolsa in [Trans. Am. Math. Soc. 355, No. 1, 315–348 (2003; Zbl 1021.42010)] and their new atomic characterization.

MSC:
42B20Singular and oscillatory integrals, several variables
42B30H p -spaces (Fourier analysis)
42B25Maximal functions, Littlewood-Paley theory
43A99Miscellaneous topics in harmonic analysis