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Transactions of the American Mathematical Society

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Fractional differentiation and Lipschitz spaces on local fields

Author: C. W. Onneweer
Journal: Trans. Amer. Math. Soc. 258 (1980), 155-165
MSC: Primary 43A70; Secondary 26A33
MathSciNet review: 554325
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Abstract: In this paper we continue our study of differentiation on a local field K. We define strong derivatives of fractional order $ \alpha \, > \,0$ for functions in $ {L_r}(\textbf{K})$, $ 1\, \leqslant \,r\, < \,\infty $. After establishing a number of basic properties for such derivatives we prove that the spaces of Bessel potentials on K are equal to the spaces of strongly $ {L_r}(\textbf{K})$-differentiable functions of order $ \alpha \, > \,0$ when $ 1\, \leqslant \,r\, \leqslant \,2$. We then focus our attention on the relationship between these spaces and the generalized Lipschitz spaces over K. Among others, we prove an inclusion theorem similar to a wellknown result of Taibleson for such spaces over $ {\textbf{R}^n}$.

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Keywords: Local fields, fractional derivatives, Bessel potentials, generalized Lipschitz spaces
Article copyright: © Copyright 1980 American Mathematical Society