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Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications. (English) Zbl 1133.42020

Let \((X, d, \mu)\) be a homogeneous-type space introduced by R. R. Coifman and G. Weiss [Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. Lect. Notes Math. 242 (Berlin-Heidelberg-New York): Springer-Verlag. (1971; Zbl 0224.43006); Bull. Am. Math. Soc. 83, 569–645 (1977; Zbl 0358.30023)]. Here \(X\) is a set, \(d\) is a quasi-metric and \(\mu\) is a positive Borel regular measure with the doubling property. Let \(K\) be a locally integrable function on \(X \times X \setminus \{(x,y) : x=y\}\) satisfying the following size condition and the standard Hörmander condition, that is, there exists a constant \(C > 0\) such that for all \(R > 0\), and all \(y, y' \in X\), \[ \int_{R < d(x,y) \leq 2R} [| K(x,y)| + | K(y,x)| ] \, d\mu(x) \leq C \tag{1} \] and \[ \int_{ d(x,y) \geq 2d(y,y')} [| K(x,y)-K(x,y')| + | K(y,x)-K(y,x')| ]\, d\mu(x) \leq C.\tag{2} \] Then define the truncated operator \(T_{\epsilon}\) for any \(\epsilon > 0\) by \[ T_{\epsilon}f(x) = \int_{d(x,y) > \epsilon} K(x,y)\, f(y) \, d\mu(y) \] and the maximal operator \(T^{*}\) by \[ T^{*}f(x) = \sup_{\epsilon > 0}| T_{\epsilon}f(x)| ,\tag{3} \] where \(x \in X\), \(f \in L_{c}^{\infty}(X)\), and \(\mu\)– \(a.e.\) \(x \notin \text{supp} f\). In this paper if \(T^{*}\) is the the maximal operator as in (ref {3}) with \(K\) satisfying (ref {1}) and (ref {2}), some equivalent characterizations are proved via certain norm inequalities on John-Strömberg sharp maximal functions without using the boundedness of these operators themselves. They also generalized the results of L. Grafakos [Colloq. Math. 96, 167–177 (2003; Zbl 1036.42008)]; on Euclidean spaces to homogeneous-type spaces. In addition, they present two applications of their main results to maximal Monge-Ampère singular integral operators in [C. E. D’Attellis, E. M. Fernández-Berdaguer, Wavelet Theory and Harmonic Analysis in Applied Sciences. Applied and Numerical Harmonic Analysis. (Boston), MA: Birkhäuser. (pp.3–13) (1997; Zbl 0868.00052)] and maximal A. Nagel-E. M. Stein singular integral operators on certain specific smooth manifolds in [Rev. Mat. Iberoam. 20, 531–561 (2004; Zbl 1057.42016)], respectively.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47A20 Dilations, extensions, compressions of linear operators
43A99 Abstract harmonic analysis
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