Real Analysis Exchange

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

Artur Nicolau

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The main purpose of the paper is to show that differentiability properties of a measurable function defined in 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 function \(f\) is differentiable in terms of a certain square function \(g(f)\). 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 where \(f\) is 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 of \(g(f)\) 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.

Article information

Real Anal. Exchange, Volume 43, Number 1 (2018), 155-186.

First available in Project Euclid: 2 May 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 60G46: Martingales and classical analysis

Differentiability Dyadic Martingales Quadratic Variation Sobolev Spaces


Nicolau, Artur. Divided Differences, Square Functions, and a Law of the Iterated Logarithm. Real Anal. Exchange 43 (2018), no. 1, 155--186. doi:10.14321/realanalexch.43.1.0155.

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