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Transactions of the American Mathematical Society

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Strong differentiability of Lipschitz functions

Author: C. J. Neugebauer
Journal: Trans. Amer. Math. Soc. 240 (1978), 295-306
MSC: Primary 26A16; Secondary 46E35
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\Vert {{M_F}f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _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

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