Weighted weak-type $(1,1)$ inequalities for rough operators

Abstract:

Let $\Omega$ be homogeneous of degree 0, have mean value 0 on the circle, and belong to ${L^q}\left ( {{S^1}} \right ),1 < q \leq \infty$. Then the two-dimensional operator defined by \[ Tf\left ( x \right ) = ''{\text {pv''}}\int {\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}} f\left ( {x - y} \right )dy\] is shown to be of weak-type $(1,1)$ with respect to the weighted measures ${\left | x \right |^\alpha }dx$, if $- 2 + 1/q < \alpha < 0$. Under the weaker assumption that $\Omega$ belongs to $L {\log ^ + }L\left ( {{S^1}} \right )$, the same result holds if $- 1 < \alpha < 0$. Similar results are also obtained for the related maximal operator \[ {M_\Omega }f\left ( x \right ) = \sup \limits _{r > 0} {r^{ - 2}}\int _{|y| < r} {\left | {\Omega \left ( y \right )f\left ( {x - y} \right )} \right |dy.} \]