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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *F* be a differentiation basis in , i.e., a family of measurable sets *S* contracting to 0 such that , where is the Hardy-Littlewood maximal operator. For , we let be the complement of the Lebesgue set of *f* relative to *F*, and we show that has -capacity 0, where is a capacity associated with in much the same way as the Bessel capacity is associated with .

**[1]**A.-P. Calderón and A. Zygmund,*Local properties of solutions of elliptic partial differential equations*, Studia Math.**20**(1961), 171–225. MR**0136849****[2]**Daniel J. Deignan and William P. Ziemer,*Strong differentiability properties of Bessel potentials*, Trans. Amer. Math. Soc.**225**(1977), 113–122. MR**0422645**, 10.1090/S0002-9947-1977-0422645-7**[3]**M. deGuzmán,*Differentiation of integrals in*, Lecture Notes in Math., no. 481, Springer-Verlag, Berlin and New York, 1975.**[4]**Norman G. Meyers,*A theory of capacities for potentials of functions in Lebesgue classes.*, Math. Scand.**26**(1970), 255–292 (1971). MR**0277741****[5]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[6]**Mitchell H. Taibleson,*On the theory of Lipschitz spaces of distributions on Euclidean 𝑛-space. I. Principal properties*, J. Math. Mech.**13**(1964), 407–479. MR**0163159****[7]**S. Saks,*Remark on the differentiability of the Lebesgue indefinite integral*, Fund. Math.**22**(1934), 257-261.**[8]**N. Aronszajn and K. T. Smith,*Theory of Bessel potentials. I*, Ann. Inst. Fourier (Grenoble)**11**(1961), 385–475 (English, with French summary). MR**0143935**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
26A16,
46E35

Retrieve articles in all journals with MSC: 26A16, 46E35

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1978-0489599-X

Keywords:
Lipschitz spaces,
Lipschitz capacity,
differentiation

Article copyright:
© Copyright 1978
American Mathematical Society