Weak $(1,1)$ boundedness of singular integrals with nonsmooth kernel

Abstract:

If $\Omega \in {L^q}\left ( {{S^1}} \right )$ for some $q > 1,\int _{{S^1}} {\Omega = 0}$, and $\Omega$ is homogeneous of degree 0, then the operator defined in two dimensions by ${T_\varepsilon }f\left ( x \right ) = \int _{\left | y \right | > \varepsilon } {f\left ( {x - y} \right )\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}dy}$ is of weak-type $(1,1)$ with bound independent of $\varepsilon > 0$.