An extension of the Córdoba-Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators
Let the geometric maximal operator \(M_\theta f(x)=\sup_{x\in R}\frac{1}{|R|}\int_R|f(y)|\,dy\) and the multiplier operator \(T_\theta\), \(\widehat{T_\theta f}(\xi)=\chi_{P_\theta}(\xi)\widehat f(\xi)\), where \(\theta_1>\theta_1>\dots\) is a decreasing sequence of angles between 0 and \(\pi/2\), and the supremum in the first formula extends over the collection of rectangle in the plane of arbitrary eccenfrity oriented in one of the directions \(\theta_i\). Extending a result of \textit{A. Córdoba} and \textit{R. Fefferman} [Proc. Natl. Acad. Sci. USA 74, 423--425 (1977; Zbl 0342.42003)], there is proved the following theorem: (a) if \(M_\theta\) is bounded on \(L^p(\mathbb{R}^2)\) for some \(1<p<\infty\), then \(M_\theta\) and \(T_\theta\) are bounded on \(L^q(\mathbb{R}^2)\) for all \(1<q<\infty\), (b) if \(M_\theta\) satisfies the tauberian condition \(|\{x:M_\theta\chi_E(x)>1/2\}\leq C|E|\), then \(M_\theta\) and \(T_\theta\) are bounded on \(L^q(\mathbb{R}^2)\) for all \(1<q<\infty\).
scientific article; zbMATH DE number 5362117