Divided differences, square functions, and a law of the iterated logarithm

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Abstract

© 2017, Michigan State University Press. The main purpose of the paper is to show that differentiability prop- erties of a measurable function defined in the euclidean space can be described using square functions which involve its second symmetric di- vided 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 dif- ferentiable. In the nineties, Anderson and Pitt proved that the growth of the divided differences of a function in the Zygmund class obeys a ver- sion 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.
Original languageEnglish
Pages (from-to)155-186
JournalReal Analysis Exchange
Volume43
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Differentiability
  • Dyadic Martingales
  • Quadratic variation
  • Sobolev spaces

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