On sets of singular rotations for translation invariant differentiation bases formed by intervals
-
Giorgi Oniani
and
Kakha Chubinidze
Abstract
We study the dependence of differential properties of an indefinite
integral on a rotation of the coordinate system.
Namely, the following problem is studied:
For a summable function f, what kind of a set may be the set of rotations θ for which
Funding source: Shota Rustaveli National Science Foundation
Award Identifier / Grant number: 217282
Funding statement: The first author was supported by the Shota Rustaveli National Science Foundation (Project no. 217282).
References
[1] K. Chubinidze, On sets of singular rotations for translation invariant bases, Trans. A. Razmadze Math. Inst. 170 (2016), no. 1, 1–6. 10.1016/j.trmi.2015.12.005Search in Google Scholar
[2] K. Chubinidze and G. Oniani, Rotation of coordinate axes and differentiation of integrals with respect to translation invariant bases, Proc. A. Razmadze Math. Inst. 167 (2015), 107–112. Search in Google Scholar
[3]
M. de Guzmán,
Differentiation of Integrals in
[4]
O. S. Dragoshanskiĭ,
On the convergence of double Fourier series and Fourier integrals of functions on
[5] M. I. D’yachenko, The rotation of a coordinate system and two-dimensional classes of functions of generalized bounded variation (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2008), no. 3, 26–29, 71; translation in Moscow Univ. Math. Bull. 63 (2008), no. 3, 107–110. 10.3103/S0027132208030054Search in Google Scholar
[6] G. A. Karagulyan, A complete characterization of R-sets in the theory of differentiation of integrals, Studia Math. 181 (2007), no. 1, 17–32. 10.4064/sm181-1-2Search in Google Scholar
[7] G. Lepsveridze, On strong differentiability of integrals along different directions, Georgian Math. J. 2 (1995), no. 6, 613–630. 10.1007/BF02262858Search in Google Scholar
[8] B. López-Melero, A negative result in differentiation theory, Studia Math. 72 (1982), no. 2, 173–182. 10.4064/sm-72-2-173-182Search in Google Scholar
[9] J. M. Marstrand, A counter-example in the theory of strong differentiation, Bull. Lond. Math. Soc. 9 (1977), no. 2, 209–211. 10.1112/blms/9.2.209Search in Google Scholar
[10] G. G. Oniani, Differentiation of Lebesgue Integrals (in Russian), Tbilisi University, Tbilisi, 1998. Search in Google Scholar
[11] G. Oniani, On the integrability of strong maximal functions corresponding to different frames, Georgian Math. J. 6 (1999), no. 2, 149–168. 10.1515/GMJ.1999.149Search in Google Scholar
[12] G. Oniani, On the strong differentiation of multiple integrals along different frames, Georgian Math. J. 12 (2005), no. 2, 349–368. 10.1515/GMJ.2005.349Search in Google Scholar
[13] G. Oniani, A resonance theorem for a family of translation invariant differentiation bases, Proc. A. Razmadze Math. Inst. 168 (2015), 99–116. Search in Google Scholar
[14]
G. G. Oniani,
Erratum to: “On the differentiability of integrals with respect to the bases
[15] G. Oniani and K. Chubinidze, Characterization of sets of singular rotations for a class of differentiation bases, Trans. A. Razmadze Math. Inst., 173 (2019), 91–102. Search in Google Scholar
[16] G. G. Oniani and K. A. Chubinidze, Rotation of coordinate system and differentiation of integrals with respect to translation invariant bases (in Russian), Mat. Sb. 208 (2017), no. 4, 51–72; translation in Sb. Math. 208 (2017), no. 4, 510–530. 10.1070/SM8643Search in Google Scholar
[17] A. Stokolos, Zygmund’s program: Some partial solutions, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1439–1453. 10.5802/aif.2129Search in Google Scholar
[18] A. M. Stokolos, An inequality for equimeasurable rearrangements and its application in the theory of differentiation of integrals, Anal. Math. 9 (1983), no. 2, 133–146. 10.1007/BF01982009Search in Google Scholar
[19]
A. M. Stokolos,
On the differentiation of integrals of functions from
[20] A. M. Stokolos, On a problem of A. Zygmund (in Russian), Mat. Zametki 64 (1998), no. 5, 749–762; translation in Math. Notes 64 (1998), no. 5-6, 646–657. 10.1007/BF02316290Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
- On generalized α-ψ-Geraghty contractions on b-metric spaces
- New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
- A Tauberian theorem for the generalized Nörlund summability method
- A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
- The Robin function and conformal welding – A new proof of the existence
- Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
- The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
- Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
- On an approximate solution of a class of surface singular integral equations of the first kind
- On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
- On the geometrical properties of hypercomplex four-dimensional Lie groups
- On sets of singular rotations for translation invariant differentiation bases formed by intervals
- Certain commutativity criteria for rings with involution involving generalized derivations
- The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
- Ripplet transform and its extension to Boehmians
- Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
Articles in the same Issue
- Frontmatter
- An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
- On generalized α-ψ-Geraghty contractions on b-metric spaces
- New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
- A Tauberian theorem for the generalized Nörlund summability method
- A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
- The Robin function and conformal welding – A new proof of the existence
- Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
- The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
- Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
- On an approximate solution of a class of surface singular integral equations of the first kind
- On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
- On the geometrical properties of hypercomplex four-dimensional Lie groups
- On sets of singular rotations for translation invariant differentiation bases formed by intervals
- Certain commutativity criteria for rings with involution involving generalized derivations
- The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
- Ripplet transform and its extension to Boehmians
- Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
