Some maximal operators related to families of singular integral operators (Q1882488)

The authors treat two kinds of maximal operators related to some families of singular integrals; \[ T(f)(x)= \sup_{h}\left| \int_{\mathbb R^n}h(| y| )\frac{\Omega(y')}{| y| ^n}f(x-y)\,dy\right|, \] where \(\Omega\in H^1(S^{n-1})\) and \(\int_{S^{n-1}}\Omega(x')d\sigma(x')=0\), and the supremum is taken over the set of all radial functions \(h\) satisfying \((\int_0^\infty | h(r)| ^2\,dr/r)^{1/2}\leq 1\), and \[ S(f)(x)= \sup_{\Omega}\left| \int_{\mathbb R^n}\frac{\Omega(y')}{| y| ^n}f(x-y) \,dy\right|, \] where the supremum is taken over the set of all \(\Omega\in L^q(S^{n-1})\) satisfying \(\| \Omega\| _{L^q(S^{n-1})}\leq 1\) and \(\int_{S^{n-1}}\Omega(x')d\sigma(x')=0\). One of their main results is: (i) \(\| Tf\| _{p}\leq C_{n,p,\Omega}\| f\| _p\) for \(2\leq p<\infty\). (ii) If \(\Omega\in L^q(S^{n-1})\) for some \(1<q\leq2\) and \(\int_{S^{n-1}}\Omega(x') \,d\sigma(x')=0\), then \(\| Tf\| _{p}\leq C_{n,p,\Omega}\| f\| _p\) for \(2nq'/(nq'+2n-2)<p<\infty\). This improves the result in the case \(\Omega\in C(S^{n-1})\) given by \textit{L.-K. Chen} and \textit{H. Lin} [Ill. J. Math., 34, No. 1, 120--126 (1990; Zbl 0682.42012)]. As for \(S(f)\) they give the \(L^p\) boundedness for \(q>1\) and \(q'\leq p<\infty\). They extend their results to the product case.