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Weighted weak type estimates for commutators of Littlewood-Paley operator. (English) Zbl 1046.42013

Let \(\psi\) be a fixed function satisfying \(\int \psi(x)\,dx=0\), \(| \psi(x)| \leq C(1+| x| )^{-(n+1)}\), \(| \psi(x+y)-\psi(x)| \leq C| y| ^\varepsilon (1+| x| )^{-(n+1+\varepsilon)}\) \((| y| <| x| /2)\) for some \(\varepsilon>0\). The commutator of the Littlewood-Paley operator is defined by \[ g_{\psi,b}^m(f)(x)=\left(\int_0^\infty | F_{b,t}^m(x)| ^2dt/t\right)^{1/2}, \] where \(F_{b,t}^m(x)=\int_{\mathbb R^n}\psi_t(x-y)f(y)(b(x)-b(y))^mdy\), and \(\psi_t(x)=\psi(x/t)/t^n\).
The main theorem in the paper is: Let \(b\in BMO\), \(m\) be a positive integer, and \(\omega\in A_1\) (Muckenhoupt’s weight class). Then, there exists \(C>0\) such that \[ \omega(\{x\in \mathbb R^n; g_{\psi,b}^m(f)(x)>\lambda\}) \leq C\lambda^{-1}\int_{\mathbb R^n}| f(x)| (1+\log^+(| f(x)| /\lambda))^m\omega(x)dx \] \((\lambda>0)\). The author also gives two-weights weighted weak type inequalities and weighted \(L^p\) inequalities for \(g_{\psi,b}^m(f)\). These correspond to those for singular integrals by C. Pérez and D. Cruz-Uribe.

MSC:

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