Abstract:
It is proved that if a convex density-like differential basis BB is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to BB can boundedly diverge only on a set of measure zero (this generalizes a theorem of Guzmán and Menarguez); it is established that both translation and homothety invariances are necessary.
Citation:
G. G. Oniani, “On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases”, Mat. Zametki, 76:5 (2004), 762–775; Math. Notes, 76:5 (2004), 711–722
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\by G.~G.~Oniani
\paper On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases
\jour Mat. Zametki
\yr 2004
\vol 76
\issue 5
\pages 762--775
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\jour Math. Notes
\yr 2004
\vol 76
\issue 5
\pages 711--722
\crossref{https://doi.org/10.1023/B:MATN.0000049670.36842.71}
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Linking options:
https://www.mathnet.ru/eng/mzm146
https://doi.org/10.4213/mzm146
https://www.mathnet.ru/eng/mzm/v76/i5/p762
This publication is cited in the following 5 articles:
Irakli Japaridze, Giorgi Oniani, “Approximation and Transfer of Properties Between Translation Invariant Convex Differentiation Bases”, Real Anal. Exchange, 49:1 (2024)
Irakli Japaridze, “On the comparison of translation invariant convex differentiation bases”, Georgian Mathematical Journal, 31:2 (2024), 261
Paul Hagelstein, Giorgi Oniani, “On the finiteness of strong maximal functions associated to functions whose integrals are strongly differentiable”, Journal of Mathematical Analysis and Applications, 523:1 (2023), 127083
Oniani G., “On the Differentiation of Integrals With Respect to Translation Invariant Convex Density Bases”, Fundam. Math., 246:2 (2019), 205–216
G. G. Oniani, “Note on Besicovitch's Theorem on the Possible Values of Upper and Lower Derivatives”, Math. Notes, 93:2 (2013), 282–287
Citation in format AMSBIB
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