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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 \(\mu\) without the doubling condition on the Hardy space \(H^{1}(\mu)\). More precisely, the Euclidean space \(\mathbb{R}^{d}\) is endowed with a non-negative Radon measure \(\mu\) which only satisfies the following growth condition that there exists \(C > 0\) such that \[ \mu(B(x,r)) \leq C \, r^n \] for all \(x \in \mathbb{R}^{d}\) and \(r > 0\), where \(B(x,r) = \{y \in \mathbb{R}^{d} : | y-x| < r\}\), \(n\) is a fixed number and \(0 < n \leq d\). For such measure \(\mu\) it is not necessary to be doubling. Let \(K\) be a function on \(\mathbb{R}^{d} \times \{(x,y) : x=y\}\) satisfying for \(x \neq y\), \[ | K(x,y)| \leq C | x-y| ^{-n}, \] and for \(| x-y| \geq 2| x-x'| \), \[ | K(x,y) - K(x',y)| + | K(y,x) - K(y,x')| \leq C \, \frac{| x-x'| ^{\delta}}{| x-y| ^{n+\delta}}, \] where \(\delta \in (0,1]\) and \(C > 0\) is a constant. The Calderón-Zygmund operator associated to the above kernel \(K\) and \(\mu\) is defined by \[ Tf(x) = \int_{\mathbb{R}^{d}}K(x,y)f(y) \, d\mu(y). \] For \(\varepsilon > 0\) we denote \(T_{\varepsilon}\) by the truncated operators of \(T\). If the operators \(\{T_{\varepsilon}\}_{\varepsilon > 0}\) are bounded on \(L^{2}(\mu)\) uniformly on \(\varepsilon > 0\), \(T\) is bounded on \(L^{2}(\mu)\). In this case there is an operator \[ \widetilde{T}f(x) = \int_{\mathbb{R}^{d}}K(x,y)f(y) \, d\mu(y), \;\;x \in \mathbb{R}^{d} \setminus \text{supp} \, (f) \] which is the weak limit as \(\varepsilon \rightarrow 0\) of some subsequences of operators \(\{T_{\varepsilon}\}_{\varepsilon > 0}\). In the main theorem the authors prove that if \(\widetilde{T}\) is bounded on \(L^{2}(\mu)\) and \({\widetilde T}^{*}1 = 0\), then \(\widetilde{T}\) is bounded on \(H^{1}(\mu)\). Here, \({\widetilde T}^{*}1 = 0\) implies that for any bounded function \(b\) with compact support and \(\int_{\mathbb{R}^{d}} d\mu = 0\), \(\int_{\mathbb{R}^{d}} \widetilde{T}b(x) \, d\mu(x) = 0\). They adapt the Hardy space \(H^{1}(\mu)\) 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:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
43A99 Abstract harmonic analysis

Citations:

Zbl 1021.42010
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References:

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