Pointwise differentiability and absolute continuity

Bagby, Thomas; Ziemer, William P.

doi:10.1090/S0002-9947-1974-0344390-6

Pointwise differentiability and absolute continuity

Authors: Thomas Bagby and William P. Ziemer
Journal: Trans. Amer. Math. Soc. 191 (1974), 129-148
MSC: Primary 26A54; Secondary 46E35
DOI: https://doi.org/10.1090/S0002-9947-1974-0344390-6
MathSciNet review: 0344390
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Abstract: This paper is concerned with the relationships between ${L_p}$ differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in ${L_p}$, and $0 \leq l \leq k$, then f has an ${L_p}$ derivative of order l everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an ${L_p}$ derivative everywhere except for a set small in capacity and if these derivatives are in ${L_p}$, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.

David R. Adams and John C. Polking, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc. 37 (1973), 529–534. MR 328109, DOI https://doi.org/10.1090/S0002-9939-1973-0328109-5
N. Aronszajn and K. T. Smith, Functional spaces and functional completion, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 125–185. MR 80878
S. Bochner, Green-Goursat theorem, Math. Z. 63 (1955), 230–242. MR 72223, DOI https://doi.org/10.1007/BF01187935
A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
A. P. Calderón, Uniqueness of distributions, Rev. Un. Mat. Argentina 25 (1970/71), 37–65. MR 343018
A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI https://doi.org/10.4064/sm-20-2-181-225
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI https://doi.org/10.1512/iumj.1972.22.22013
Bent Fuglede, On a theorem of F. Riesz, Math. Scand. 3 (1955), 283–302 (1956). MR 80133, DOI https://doi.org/10.7146/math.scand.a-10448
Casper Goffman, On functions with summable derivative, Amer. Math. Monthly 78 (1971), 874–875. MR 286949, DOI https://doi.org/10.2307/2316480
Walter Littman, Polar sets and removable singularities of partial differential equations, Ark. Mat. 7 (1967), 1–9 (1967). MR 224960, DOI https://doi.org/10.1007/BF02591673
Norman G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292 (1971). MR 277741, DOI https://doi.org/10.7146/math.scand.a-10981
Norman G. Meyers, Taylor expansion of Bessel potentials, Indiana Univ. Math. J. 23 (1973/74), 1043–1049. MR 348482, DOI https://doi.org/10.1512/iumj.1974.23.23085
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
C. J. Neugebauer, Smoothness and differentiability in $L_{p}$, Studia Math. 25 (1964/65), 81–91. MR 181715, DOI https://doi.org/10.4064/sm-25-1-81-91

S. Saks, Theory of the integral, 2nd rev. ed., Monografie Mat., vol. 7, PWN, Warsaw, 1937.

J. Serrin (to appear)

Victor L. Shapiro, The divergence theorem for discontinuous vector fields, Ann. of Math. (2) 68 (1958), 604–624. MR 100725, DOI https://doi.org/10.2307/1970158
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095