On the continuity of functions in $W^{1}_{p}$ which are monotonic in one direction
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- by Casper Goffman
- Proc. Amer. Math. Soc. 42 (1974), 581-582
- DOI: https://doi.org/10.1090/S0002-9939-1974-0327998-9
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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
- Casper Goffman, Decomposition of functions whose partial derivatives are measures, Mathematika 15 (1968), 149โ152. MR 241583, DOI 10.1112/S0025579300002497
- Casper Goffman and William P. Ziemer, Higher dimensional mappings for which the area formula holds, Ann. of Math. (2) 92 (1970), 482โ488. MR 271283, DOI 10.2307/1970629
- Casper Goffman and Fon-che Liu, On the localization property of square partial sums for multiple Fourier series, Studia Math. 44 (1972), 61โ69. MR 312147, DOI 10.4064/sm-44-1-61-69
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- 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