Noncommutative maximal operators with rough kernels

This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak type $(1,1)$ boundedness for noncommutative maximal operator with a rough kernel. Then, via the noncommutative Marcinkiewicz type interpolation theorem, together with a trivial $(\infty,\infty)$ estimate, we obtain its $L_p$ boundedness for all $1<p<\infty$. The proof of weak type (1,1) estimate is based on the noncommutative Calder\'on-Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both bad and good functions.Comment: 31 page