An Example of an Unbounded Maximal Singular Operator

Suppose that an even integrable function Ω on the unit sphere S ¹ in R ² with mean value zero satisfies

$mathop{mathrm{essup}}limits_{xiin mathbf{S}^{1}}biggl|int_{mathbf{S}^{1}}Omega(theta)logfrac{1}{|thetacdotxi|},dthetabiggr|<+infty,$

then the singular integral operator T _Ω given by convolution with the distribution p.v.?Ω(x/|x|)|x|^?2, initially defined on Schwartz functions, extends to an L ²-bounded operator. We construct examples of a function Ω satisfying the above conditions and of a continuous bounded integrable function f such that

$limsup_{epsilonto 0^+}biggl|int_{epsilon<|y|}Omega(y/|y|)|y|^{-2}f(x-y)dybiggr|=inftyquad hbox{a. e.}$