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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 $B(x,r) = \{y \in \mathbb{R}^{d} : \vert y-x\vert < r\}$, $n$ 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 $x \neq y$, $$\vert K(x,y)\vert \leq C \vert x-y\vert ^{-n},$$ and for $\vert x-y\vert \geq 2\vert x-x'\vert $, $$\vert K(x,y) - K(x',y)\vert + \vert K(y,x) - K(y,x')\vert \leq C \, \frac{\vert x-x'\vert ^{\delta}}{\vert x-y\vert ^{n+\delta}}, $$ 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 {\it X. Tolsa} in [Trans. Am. Math. Soc. 355, No. 1, 315--348 (2003; Zbl 1021.42010)] and their new atomic characterization.

42B20Singular and oscillatory integrals, several variables
42B30$H^p$-spaces (Fourier analysis)
42B25Maximal functions, Littlewood-Paley theory
43A99Miscellaneous topics in harmonic analysis
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