Noncommutative maximal differential transforms associated to averaging operators

Abstract

In this paper, we establish the noncommutative maximal weak type $(1,1)$ and strong type $(p,p)$ estimates for the family of operators $(T_N{)}_N$, defined by $$T_{N}f=\sum _{k=N_1}^{N_2}{\nu }_k(M_k-{\mathsf{E}}_k)f,$$ where $M_k$ denotes the dyadic Hardy--Littlewood average operator, ${\mathsf{E}}_k$ is the conditional expectation with respect to the dyadic cubes of side-length $2^{-k}$, $N=(N_1,N_2)$ with $N_1<N_2$ and $({\nu }_k)\in {\ell }_{\infty }$. The main novelty of our approach is the development of a noncommutative Cotlar-type inequality for non-smooth kernels, a result that is new even in classical harmonic analysis. As an application, we obtain the boundedness theory of the noncommutative maximal differential transforms for averaging operators.

22 pages