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Weighted inequalities for commutators of one-sided singular integrals. (English) Zbl 1069.42014

The classical result of R. R. Coifman [Proc. Natl. Acad. Sci. USA 69, 2838–2839 (1972; Zbl 0243.44006)] states that a Calderón-Zygmund integral operator \(T\) is controlled by the Hardy-Littlewood maximal operator \(M\) in the sense \(\int _{\mathbb R^ n}| Tf| ^ pw\leq C\int _{\mathbb R^ n} (Mf)^ pw\) whenever \(0<p<\infty \) and \(w\) satisfies the \(A_\infty \) condition of Muckenhoupt. C. Pérez [J. Lond. Math. Soc., II. Ser. 49, No. 2, 296–308 (1994; Zbl 0797.42010)] showed that \(\int _{\mathbb R^ n}| Tf| ^ pw\leq C\int _{\mathbb R^ n} | f| ^ pM^ {[p]+1}w\) with no assumption on \(w\) and later [J. Fourier Anal. Appl. 3, No. 6, 743–756 (1997; Zbl 0894.42006)] generalized both the above inequalities to the context of the higher-order commutators \[ (T^ k_bf)(x):=\int _{\mathbb R^ n}(b(x)-b(y))^ kK(x-y)f(y)\, dy, \] where \(k=0,1,2,\dots \) Namely, he proved that \[ \int _{\mathbb R^ n}| T^ k_bf| ^ pw\leq C \| b\| _{BMO}^ {kp} \int _{\mathbb R^ n}(M^ {k+1}f)^ pw \] for \(0<p<\infty \) and \(w\in A_\infty \) and \[ \int _{\mathbb R^ n}| T^ k_bf| ^ pw\leq C \| b\| _{BMO}^ {kp} \int _{\mathbb R^ n}| f| ^ pM^ {[(k+1)p]+1}w \] for \(1<p<\infty \) and any weight \(w\).
The authors of the paper under review obtain analogous results for the one-sided version of Calderón singular integral operators, namely those with kernel supported either in \((0,\infty )\) or \((-\infty ,0)\) [cf. H. Aimar, L. Forzani and F. J. Martín-Reyes [Proc. Am. Math. Soc. 125, No. 7, 2057–2064 (1997; Zbl 0868.42007)]. Here, naturally, the controlling operator is the appropriate one-sided maximal operator, and the condition on weight is \(A^ +_\infty \), introduced by F. J. Martín-Reyes, the reviewer and A. de la Torre [Can. J. Math. 45, No. 6, 1231–1244 (1993; Zbl 0797.42012)].

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B15 Multipliers for harmonic analysis in several variables
47G10 Integral operators
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