**Authors:** José M. Conde-Alonso (ICMAT-UAM), Adrián M. González-Pérez and Javier Parcet (ICMAT-CSIC)

**Source:** *Cambridge University Press*

**Date of publication:** 20 November 2020

**Review:**

A general form of the fundamental theorem of calculus in one dimension is the following: given an integrable function *f*, Lebesgue’s differentiation theorem states that the integral averages of *f* over balls (or cubes) of shrinking radii around a given point *x* converge to the value *f*(*x*) at almost every point *x*. This *almost everywhere convergence* to original data is also known to hold in other scenarios, when replacing Euclidean balls by other averaging processes. The standard strategy to establish this almost everywhere convergence leads to prove a maximal inequality. This is nothing but a quantitative estimate for the maximal operator *M* associated with the corresponding averaging process. Indeed, the underlying maximal inequality behind Lebesgue’s differentiation is the celebrated Hardy-Littlewood maximal theorem, which is almost one hundred years old. It states that *M* maps the space of integrable functions into the space of weakly integrable functions, and it yields the best possible convergence result.

The same problem of convergence can be studied when the integral averages are taken with respect to rectangles -in two dimensions- with sides parallel to the axes instead of cubes. In this case, a new, significant difficulty appears, that comes from the fact that the rectangles may not have bounded eccentricity, and therefore the geometric arguments that apply in the case of cubes can no longer be used. The first convergence result in this context is due to Jessen-Marcinkiewicz-Zygmund. They established the convergence of integral averages for functions in *L*log*L*, a proper subclass of the integrable functions. This can be shown to be optimal. In this case, the strategy using maximal inequalities leads to studying the strong maximal operator. And the relevant maximal inequality was independently established by De Guzmán and Córdoba-Fefferman in the seventies. In higher dimensions –when replacing rectangles by parallellepipeds–, analogous results are also true, with appropriate modifications of the relevant classes of functions involved.

In this paper, José M. Conde-Alonso, Adrián M. González-Pérez and Javier Parcet considered generalizations of the two lines of results concerning the geometry of rectangles. The setting is now the theory of von Neumann algebras and noncommutative measure/Lp theory. The geometric notion of rectangle can naturally be replaced by a product structure on the von Neumann algebra under consideration.

The primary goal in the paper is to investigate noncommutative strong maximals, almost uniform convergence in several directions to initial data and applications in noncommutative harmonic analysis. The main results of the paper are a noncommutative form of Jessen-Marcinkiewicz-Zygmund theorem about almost everywhere convergence of integral averages, and a noncommutative form of De Guzmán/Córdoba-Fefferman maximal inequality.

The products to which the main results can apply may be tensor products of quantum martingale filtrations, ergodic means or subordinated Markov semigroups, but also free products and other noncommuting compositions.