×

On weighted weak type norm inequalities for one-sided oscillatory singular integrals. (English) Zbl 1257.42022

The authors consider one-sided weighted classes of Muckenhoupt type and study the weighted weak \((1,1)\) norm inequalities for certain one-sided oscillatory singular integrals with smooth kernel.
One-sided oscillatory singular integral operators \(T^+\) and \(T^-\) are defined by \[ \begin{aligned} T^+f(x) &= \lim_{\varepsilon\to 0+} \int^\infty_{x+\varepsilon} e^{iP(x, y)} K(x- y)f(y)\,dy,\\ T^- f(x) &= \lim_{\varepsilon\to 0+} \int^{x-\varepsilon}_{-\infty} e^{iP(x, y)} K(x- y)f(y)\,dy,\end{aligned} \] where \(P(x, y)\) is a real polynomial defined on \(\mathbb{R}\times\mathbb{R}\), and \(K\) is a one-sided Calderón-Zygmund kernel. The following theorem is the main result of the paper.
Let \(\overline\omega\in A^+_1\), the class of one-sided \(A_1\) weights. Then there exists a constant \(C\) depending on the degree of \(P\) and \(A^+_1(\overline\omega)\) such that \[ \sup_{\lambda> 0}\,\lambda\overline\omega(\{x\in \mathbb{R}:|T^+ f(x)|> \lambda\})\leq C\| f\|_{L^1(\overline\omega)} \] for all Schwartz functions \(f\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI arXiv Link