Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [1] P. L. Butzer and H. Berens, Semi-groups of operators and approximations, Springer-Verlag, New York, 1967. MR 0230022 (37:5588)
  • [2] P. L. Butzer and K. Scherer, On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and theorems of M. Zamansky and S. B. Stečkin, Aequationes Math. 3 (1969), 170-185. MR 0264301 (41:8897)
  • [3] P. L. Butzer and H. J. Wagner, Walsh-Fourier series and the concept of a derivative, Applicable Anal. 3 (1973), 29-46. MR 0404978 (53:8774)
  • [4] C. W. Onneweer, Differentation on a p-adic or p-series field, in Linear Spaces and Approximation, edited by P. L. Butzer and B. Sz.-Nagy, Birkhauser Verlag, Basel, 1978, pp. 187-198. MR 0511077 (58:23356)
  • [5] -, On the definition of dyadic differentiation, Applicable Anal. (to appear). MR 553959 (80m:43011)
  • [6] J. Pál, On the concept of a derivative among functions defined on the dyadic field, SIAM J. Math. Anal. 8 (1977), 375-391. MR 0620817 (58:29799)
  • [7] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
  • [8] M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space, I. Principal properties, J. Math. Mech. 13 (1964), 407-480. MR 0163159 (29:462)
  • [9] -, Harmonic analysis on n-dimensional vector spaces over local fields, I. Basic results on fractional integration, Math. Ann. 176 (1968), 191-207. MR 0226394 (37:1984)
  • [10] -, Harmonic analysis on n-dimensional vector spaces over local fields, II. Generalized Gauss kernels and the Littlewood-Paley function, Math. Ann. 186 (1970), 1-19. MR 0264394 (41:8989)
  • [11] -, Fourier analysis on local fields, Mathematical Notes, Princeton Univ. Press, Princeton, N. J., 1975. MR 0487295 (58:6943)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A70, 26A33

Retrieve articles in all journals with MSC: 43A70, 26A33

Additional Information

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

American Mathematical Society