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

Studia Mathematica (1996)

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

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topStokolos, 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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- [3] M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Math. Stud. 46, Amsterdam, 1981. Zbl0449.42001
- [4] V. I. Kolyada, Rearrangements of functions and embedding theorems, Uspekhi Mat. Nauk 49 (5) (1989), 61-95 (in Russian).
- [5] V. G. Maz'ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions, Izdat. Leningrad. Univ., Leningrad, 1986 (in Russian).
- [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] S. M. Nikol'skiĭ, Approximation of Functions of Several Variables and Embedding Theorems, Izdat. Nauka, Moscow, 1969 (in Russian).
- [8] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Zbl0207.13501
- [9] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1968. Zbl0157.38204

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