Artur Nicolau

The main goal of the lectures is to describe differentiability properties of measurable functions defined in the euclidean space using square functions which involve their  second symmetric divided differences.  Classical results of Marcinkiewicz, Stein and Zygmund will be discussed as well as their discrete versions which can be stated in terms of dyadic martingales. Results of Burkholder, Gundy and Stout relating the assymptotic behavior of a dyadic martingale with the size of its quadratic variation will also be discussed. An averaging argument will be used to transfer the Fatou type results and the Law of the Iterated Logarithm which hold in the discrete setting to the continuous one.