Noncommutative maximal operators with rough kernels

Abstract

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 (∞, ∞) estimate, we obtain its Lp boundedness for all 1 < p < ∞. The proof of weak type (1,1) estimate is based on the noncommutative Calderón-Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both bad and good functions.

Maximal singular integral operators acting on noncommutative $L_p$-spaces

“…Motivated by Cuculescu's maximal weak (1,1) estimate for noncommutative martingales [8], Parcet [36] formulated a kind of noncommutative Calderón-Zygmund decomposition and established the weak type (1,1) estimate for the operator-valued Calderón-Zygmund singular integrals, which finds its unexpected application in the complete resolution of the Nazarov-Peller conjecture arising from the perturbation theory [3]. For more related results on weak type (1,1) estimates and noncommutative Calderón-Zygmund decomposition we refer the reader to [1,17,19,30,33].…”

“…Therefore, from this perspective Cadilhac's decomposition seems to be more efficient; in particular, in the present setting for maximal estimates, the pseudolocalization principle is still unavailable, and thus Parcet's decomposition may not work. However, it is worthy to point out that Parcet's decomposition or his pseudo-localization might be particularly useful for kernels without regularity where the argument based on the T T * -estimate plays the key role, see for instance [6,42] in the commutative case and [19,30] in the noncommutative setting.…”

“…As a consequence, we can finish the proof of the weak type (1, 1) estimate in the similar way as in the proof of Theorem 1.1. The only difference is that when reducing to the lacunary subsequence in (3.1), Mei's Hardy-Littlewood maximal inequalities should be replaced by the generalized ones for rough kernels These maximal inequalities (6.12) have been established by the second author in [30] for the rough kernels which include the L 2 -Dini assumption in Theorem 1.4 (ii). Thus using the same argument as for Theorem 1.1, we complete the proof of Theorem 1.4 (ii).…”

Noncommutative analysis of Hermite expansions

“…As a consequence, Parcet obtained the noncommutative weak type (1, 1) estimates of Calderón-Zygmund operators associated with the standard kernels acting on operator-valued functions, which found an unexpected application in solving the Nazarov-Peller conjecture in the perturbation theory [5]. The reader is referred to [3,19,21,33,37,6,14,16,20,58,59,21,17] for more related works.…”