An Example of an Unbounded Maximal Singular Operator

Abstract

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

$$\mathop{\mathrm{essup}}\limits_{\xi\in \mathbf{S}^{1}}\biggl|\int_{\mathbf{S}^{1}}\Omega(\theta)\log\frac{1}{|\theta\cdot\xi|}\,d\theta\biggr|<+\infty,$$

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

$$\limsup_{\epsilon\to 0^+}\biggl|\int_{\epsilon<|y|}\Omega(y/|y|)|y|^{-2}f(x-y)dy\biggr|=\infty\quad \hbox{a. e.}$$