Real Analysis Exchange

An Lp differentiable non-differentiable function.

J. Marshall Ash

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Abstract

There is a set $E$ of positive Lebesgue measure and a function nowhere differentiable on $E$ which is differentiable in the $L^p$ sense for every positive $p$ at each point of $E$. For every $p\in(0,\infty]$ and every positive integer $k$ there is a set $E=E(k,p)$ of positive measure and a function which for every $q< p$ has $k$ $L^q$ Peano derivatives at every point of $E$ despite not having an $L^p$ $k$th derivative at any point of $E$.

Article information

Source
Real Anal. Exchange, Volume 30, Number 2 (2004), 747 - 754 .

Dates
First available in Project Euclid: 15 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.rae/1129416459

Mathematical Reviews number (MathSciNet)
MR2177431

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

Keywords
$L^{p}$ derivative Peano derivative $L^{p}$ Peano derivative super density

Citation

Ash, J. Marshall. An L p differentiable non-differentiable function. Real Anal. Exchange 30 (2004), no. 2, 747 -- 754. https://projecteuclid.org/euclid.rae/1129416459


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References

  • A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math., 9 (1927), 212–279.
  • A.-P. Calderón, and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math., 20 (1961), 171–225.
  • J. Luke\us, J. Malý, and L. Zajíček, Fine topology methods in real analysis and potential theory, Lecture Notes in Mathematics, 1189, Springer-Verlag, Berlin, 1986.