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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Strong differentiability of Lipschitz functions


Author: C. J. Neugebauer
Journal: Trans. Amer. Math. Soc. 240 (1978), 295-306
MSC: Primary 26A16; Secondary 46E35
DOI: https://doi.org/10.1090/S0002-9947-1978-0489599-X
MathSciNet review: 489599
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Abstract: Let F be a differentiation basis in ${R^n}$, i.e., a family of measurable sets S contracting to 0 such that ${\left \| {{M_F}f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p}$, where ${M_F}$ is the Hardy-Littlewood maximal operator. For $f \in \Lambda _\alpha ^{pq}$, we let ${E_F}(f)$ be the complement of the Lebesgue set of f relative to F, and we show that ${E_F}$ has $L_\alpha ^{pq}$-capacity 0, where $L_\alpha ^{pq}$ is a capacity associated with $\Lambda _\alpha ^{pq}$ in much the same way as the Bessel capacity ${B_{\alpha p}}$ is associated with $L_\alpha ^p$.


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Keywords: Lipschitz spaces, Lipschitz capacity, differentiation
Article copyright: © Copyright 1978 American Mathematical Society