Maximal singular integral operators acting on noncommutative L_p -spaces

In this paper, we study the boundedness theory for maximal Calderón–Zygmund operators acting on noncommutative L_p-spaces. Our first result is a criterion for the weak type (1, 1) estimate of noncommutative maximal Calderón–Zygmund operators; as an application, we obtain the weak type (1, 1) estimates of operator-valued maximal singular integrals of convolution type under proper regularity conditions. These are the first noncommutative maximal inequalities for families of truly non-positive linear operators. For homogeneous singular integrals, the strong type (p, p) (1 < p< ∞) maximal estimates are shown to be true even for rough kernels. As a byproduct of the criterion, we obtain the noncommutative weak type (1, 1) estimate for Calderón–Zygmund operators with integral regularity condition that is slightly stronger than the Hörmander condition; this provides somewhat an affirmative answer to an open question in the noncommutative Calderón–Zygmund theory.