On maximal functions with rough kernels in \(L(\log L)^{1/2} (\mathbb S^{n-1})\)
Let \(\Omega\) be a homogeneous function of degree zero with cancellation property \(\int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0\), and \(h\) be a suitable function on \((0,\infty)\). The author considers the following singular integrals \[ T_{\Omega,h}f(x)=\roman{p.v.}\int_{\mathbb R^n}f(x-y)| y| ^{-n}h(| y| )\Omega(y/| y| ) \,dy, \] and their maximal operator \(\mathcal M_{\Omega}(f)(x)=\sup_{h\in H; \| h\| _H\leq1}| T_{\Omega,h}f(x)| \), where \(H=L^2(\mathbb R^+, dr/r)\). \textit{L. K. Chen} and \textit{H. Lin} [Ill. J. Math. 34, No.~1, 120--126 (1990; Zbl 0682.42012)] gave that \(\mathcal M_{\Omega}\) is bounded on \(L^p\) for \(p>2n/(2n-1)\) under the assumption that \(\Omega\) is continuous. The author's main theorem is: \(\mathcal M_{\Omega}\) is bounded on \(L^p\) for \(p\geq2\) under the assumption \(\Omega\in L(\log L)^{1/2}\), and the exponent \(1/2\) is optimal. This improves Chen and Lin's result. He also gives related results on \(T_{\Omega,h}\) and the Marcinkiewicz integral.
scientific article; zbMATH DE number 2149099