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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
DOI: https://doi.org/10.1090/S0002-9939-1974-0327998-9
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] C. Goffman, Decomposition of functions whose partial derivatives are measures, Mathematika 15 (1968), 149-152. MR 39 #2923. MR 0241583 (39:2923)
  • [2] C. Goffman and W. P. Ziemer, Higher dimensional mappings for which the area formula holds, Ann. of Math. (2) 92 (1970), 482-488. MR 42 #6166. MR 0271283 (42:6166)
  • [3] C. Goffman and F. C. Liu, On the localization of square partial sums for multiple Fourier series, Studia Math. 44 (1972), 61-69. MR 0312147 (47:709)

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

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