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Weighted norm inequalities for a class of rough singular integrals. (English) Zbl 1081.42005

Let \(K\) be a kernel of Calderón-Zygmund-type on \(\mathbb R^n\) given by \(K(x)=\Omega(x)| x| ^{-n}\), where \(\Omega\) is a homogeneous function of degree zero, and \[ \int_{S^{n-1}}\Omega(u)\,d\sigma(u)=0. \] For \(\Gamma\in C^1(\mathbb R_+)\), define the singular integral operator \(T_{\Gamma,\Omega}\) and its maximal truncated singular integral operator \(T_{\Gamma,\Omega}^\ast\) by \[ T_{\Gamma,\Omega}f(x)={\text{p.\,v.}}\int_{\mathbb R^n}f(x-\Gamma(| y| )y')K(y)\,dy, \] and \[ T_{\Gamma,\Omega}^\ast f(x) =\sup_{\varepsilon>0}\left| \int_{| y| >\varepsilon}f(x-\Gamma(| y| )y')K(y)\,dy\right| , \] where \(y'=y/| y| \in S^{n-1}\). For \(\alpha>0\), let \(F_\alpha(S^{n-1})\) denote the set of \(\Omega\) satisfying \[ \sup_{\xi\in S^{n-1}}\int_{S^{n-1}}| \Omega(y)| \big(\log| \xi\cdot y| ^{-1}\big)^{1+\alpha}\,d\sigma(y)<\infty. \] In this paper, under some hypothesis on \(\Gamma\), the author obtains some weighted norm inequalities for \(T_{\Gamma,\Omega}\) when \(\Omega\in F_\alpha(S^{n-1})\) with \(\alpha>0\) and for \(T_{\Gamma,\Omega}^\ast\) when \(\Omega\in F_\alpha(S^{n-1})\) with \(\alpha>1/2\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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