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 $$T_Nf=\sum_{k=N_1}^{N_2}ν_{k}(M_{k}-\mathsf{E}_k)f,$$ 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 $(ν_{k})\in\ell_{\infty}$. 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.