Weak type estimates for certain Calderón--Zygmund singular integral operators (Q2717715)

Let \(\Omega : S^{n-1} \rightarrow C\) be a complex-valued integrable function with mean zero. We consider the Calderón-Zygmund operator defined by NEWLINE\[NEWLINE T_{\Omega}f(x) = \int_{R^n}\frac{\Omega(y/ |y |)}{|y |^n} f(x-y)dy. NEWLINE\]NEWLINE Under the size condition NEWLINE\[NEWLINE \int_{S^{n-1}} |\Omega (\theta) |\log^{+} |\Omega (\theta) |d \theta < \infty , NEWLINE\]NEWLINE \textit{A. P. Calderón} and \textit{A. Zygmund} [Am. J. Math. 78, 289-309 (1956; Zbl 0072.11501)] proved that \(T_\Omega\) is bounded on \(L^p\), and \textit{A. Seeger} [J. Am. Math. Soc. 9, No.~1, 95-105 (1996; Zbl 0858.42008)] proved that \(T_\Omega\) is of weak type (1,1). NEWLINENEWLINENEWLINE\textit{W. C. Connett} [Harmonic analysis in Euclidean spaces, Part I, Proc. Symp. Pure Math. 35, 163-165 (1979; Zbl 0431.42008)] proved that if \(\Omega\in H^1(S^{n-1})\) then \(T_\Omega\) is bounded on \(L^p\). NEWLINENEWLINENEWLINEThe author considers the following problem: If \(\Omega \in H^1(S^{n-1})\) then \(T_{\Omega}\) is of weak type (1,1)? He proves that \(T_{\Omega}\) is of weak type \((1,1)\) when \(\Omega\) belongs to a special class of \(H^1(S^{n-1})\).