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Sharp weighted bounds for the q-variation of singular integrals. (English) Zbl 1271.42021

T. P. Hytönen [Ann. Math. (2) 175, No. 3, 1473–1506 (2012; Zbl 1250.42036)] proved the following sharp weighted inequality for Calderón-Zygmund operators, known as the \(A_2\) theorem: \[ \| Tf \|_{L^2(w)} \lesssim [w]_{A_2} \| f \|_{L^2(w)}. \] It remains valid when the linear Calderón-Zygmund operator is replaced by the pointwise larger, and nonlinear, maximal truncated singular integral \(T_{*}\). The authors replace \(T\) by the \(q\)-variation operator \(V_q^{\phi}T, q>2\), which measures the pointwise rate of convergence of the truncated version of \(T\): \[ V_q^{\phi}Tf(x) := \sup_{ \{ \varepsilon_i \}_{i \in {\mathbb Z} } } \left( \sum_{i \in {\mathbb Z}} | T_{\varepsilon_i}^{\phi}f(x) - T_{\varepsilon_{i+1}}^{\phi}f(x) |^q \right)^{1/q}, \] where \(T_{\varepsilon_i}^{\phi}\) is a smooth truncation of \(T\), and the supremum is taken over all increasing sequences of positive numbers. Unweighted norm inequalities for these operators have been studied by J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl [Trans. Am. Math. Soc. 355, No. 5, 2115–2137 (2003; Zbl 1022.42012)]. To prove the sharp weighted bounds, they show that the maximal truncation of a Calderón-Zygmund operator is dominated by the maximal operators and positive dyadic shifts by a non-probabilistic method which is different from the probabilistic one by T. P. Hytönen and M. T. Lacey.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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