A remark on singular Calderón-Zygmund theory

Abstract:

It is shown that in ${{\mathbf {R}}^n}$ the operator \[ Hf\left ( x \right ) = pv\int _{ - \infty }^{ + \infty } {f\left ( {{x_1} - t, \ldots {x_n} - {t^n}} \right ){t^{ - 1}}dt} \] maps $L\left ( {\log L} \right )$ to weak ${L^1}$ locally. A slight variant of the Calderón-Zygmund procedure provides a new approach to the previously known ${L^p}$ boundedness of $H,1 < p < \infty$. Relatively sharp bounds are obtained as $p \to {1^ + }$, and extrapolation produces the result for $L\left ( {\log L} \right )$.