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On the continuity of functions in $ W\sp{1}\sb{p}$ which are monotonic in one direction


Author: Casper Goffman
Journal: Proc. Amer. Math. Soc. 42 (1974), 581-582
MSC: Primary 26A54; Secondary 46E35
MathSciNet review: 0327998
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Abstract: It was previously shown that, for $ n = 2$, if f is such that its distribution derivatives are measures, and f is monotonically nondecreasing in one variable for almost all values of the other variable then f is equivalent to a continuous function. This is now shown to be false for $ n > 2$. It is true for $ f \in W_p^1,p > n - 1$ and may be false for $ f \in W_{n - 1}^1$.


References [Enhancements On Off] (What's this?)

  • [1] Casper Goffman, Decomposition of functions whose partial derivatives are measures, Mathematika 15 (1968), 149–152. MR 0241583
  • [2] Casper Goffman and William P. Ziemer, Higher dimensional mappings for which the area formula holds, Ann. of Math. (2) 92 (1970), 482–488. MR 0271283
  • [3] Casper Goffman and Fon-che Liu, On the localization property of square partial sums for multiple Fourier series, Studia Math. 44 (1972), 61–69. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. MR 0312147

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DOI: https://doi.org/10.1090/S0002-9939-1974-0327998-9
Article copyright: © Copyright 1974 American Mathematical Society