Divided Differences, Square Functions, and a Law of the Iterated Logarithm

• Published in: Real analysis exchange Vol. 43; no. 1; pp. 155 - 186
• Main Author: Nicolau, Artur
• Format: Journal Article
• Language: English
• Published: East Lansing Michigan State University Press 01.01.2018
• Subjects: Euclidean space; Graphs; Harmonic functions; Integers; Lebesgue measures; Logarithms; Martingales; Mathematical functions; Mathematical theorems; Natural logarithms; Sobolev spaces; Symmetry
• ISSN: 0147-1937; 1930-1219
• DOI: 10.14321/realanalexch.43.1.0155

The main purpose of the paper is to show that differentiability properties of a measurable function defined in the euclidean space can be described using square functions which involve its second symmetric divided differences. Classical results of Marcinkiewicz, Stein and Zygmund describe, up to sets of Lebesgue measure zero, the set of points where a functionfis differentiable in terms of a certain square functiong(f). It is natural to ask for the behavior of the divided differences at the complement of this set, that is, on the set of points wherefis not differentiable. In the nineties, Anderson and Pitt proved that the growth of the divided differences of a function in the Zygmund class obeys a version of the classical Kolmogorov's Law of the Iterated Logarithm (LIL). A square function, which is the conical analogue ofg(f) will be used to state and prove a general version of the LIL of Anderson and Pitt as well as to prove analogues of the classical results of Marcinkiewicz, Stein and Zygmund. Sobolev spaces can also be described using this new square function. Mathematical Reviews subject classification: Primary: 26A24, 60G46 Key words: Differentiability, Dyadic Martingales, Quadratic Variation, Sobolev Spaces