Rough maximal functions and rough singular integral operators applied to integrable radial functions
Consider the maximal operator \[ M_\Omega f(x)= \sup_{R>0} R^{-n}\int|\Omega(y') f(x-y)|dy \] and the singular integral operator \[ T_\Omega f(x)= \int\Omega(y')|y|^{-n} f(x-y)dy, \] where \(\Omega\in L^1(S^{n-1})\) is extended to a function homogeneous of degree 0; in the definition of \(T_\Omega\) one assumes cancellation: \(\int\Omega(y') d\sigma(y')=0\). It is an open question whether \(M_\Omega\) (or \(T_\Omega\) for odd \(\Omega\)) is of weak type (1,1) although this has been proved under the stronger assumption \(\Omega\in L\log L\). In this interesting paper, the authors prove that \(M_\Omega\) and \(T_\Omega\) are of weak (1,1) on \textit{radial} functions in \(\mathbb{R}^n\), where \(\Omega\) is merely assumed to be in \(L^1(S^{n-1})\). The assumption that \(\Omega\) be odd is not needed in the result for radial functions. The authors also prove the almost everywhere existence of the principal value integral, for radial \(f\in L^1\) by establishing the weak-type (1,1) estimate for the corresponding maximal singular integral (the corresponding question for general \(f\) seems to be open even under the assumption \(\Omega\in L^\infty\)). The proof is quite delicate; it partially relies on methods developed by the authors in a previous paper [Isr. J. Math. 95, 211-229 (1996)].
scientific article; zbMATH DE number 1052954