On differentiation of integrals with respect to bases of convex sets.

A. Stokolos

Studia Mathematica (1996)

  • Volume: 119, Issue: 2, page 99-108
  • ISSN: 0039-3223

Abstract

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Differentiation of integrals of functions from the class L i p ( 1 , 1 ) ( I 2 ) with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in L i p ( 1 , 1 ) ( I N ) , N ≥ 3, and H 1 ω ( I 2 ) with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.

How to cite

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Stokolos, A.. "On differentiation of integrals with respect to bases of convex sets.." Studia Mathematica 119.2 (1996): 99-108. <http://eudml.org/doc/216295>.

@article{Stokolos1996,
abstract = {Differentiation of integrals of functions from the class $Lip(1,1)(I^2)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)(I^N)$, N ≥ 3, and $H^\{ω\}_\{1\}(I^2)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.},
author = {Stokolos, A.},
journal = {Studia Mathematica},
keywords = {differentiation of integrals; differentiation basis; Lipschitz class},
language = {eng},
number = {2},
pages = {99-108},
title = {On differentiation of integrals with respect to bases of convex sets.},
url = {http://eudml.org/doc/216295},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Stokolos, A.
TI - On differentiation of integrals with respect to bases of convex sets.
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 2
SP - 99
EP - 108
AB - Differentiation of integrals of functions from the class $Lip(1,1)(I^2)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)(I^N)$, N ≥ 3, and $H^{ω}_{1}(I^2)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.
LA - eng
KW - differentiation of integrals; differentiation basis; Lipschitz class
UR - http://eudml.org/doc/216295
ER -

References

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  1. [1] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs Math., Birkhäuser, Boston, 1984. Zbl0545.49018
  2. [2] M. de Guzmán, Differentiation of Integrals in ℝ n , Lecture Notes in Math. 481, Springer, 1975. 
  3. [3] M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Math. Stud. 46, Amsterdam, 1981. Zbl0449.42001
  4. [4] V. I. Kolyada, Rearrangements of functions and embedding theorems, Uspekhi Mat. Nauk 49 (5) (1989), 61-95 (in Russian). 
  5. [5] V. G. Maz'ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions, Izdat. Leningrad. Univ., Leningrad, 1986 (in Russian). 
  6. [6] O. Nikodym, Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fund. Math. 10 (1927), 116-168. Zbl53.0176.02
  7. [7] S. M. Nikol'skiĭ, Approximation of Functions of Several Variables and Embedding Theorems, Izdat. Nauka, Moscow, 1969 (in Russian). 
  8. [8] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Zbl0207.13501
  9. [9] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1968. Zbl0157.38204

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