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 Mat. Zametki, 2004, Volume 76, Issue 5, Pages 762–775 (Mi mz146)

On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases

G. G. Oniani

Abstract: It is proved that if a convex density-like differential basis $B$ is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to $B$ 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.

DOI: https://doi.org/10.4213/mzm146

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English version:
Mathematical Notes, 2004, 76:5, 711–722

Bibliographic databases:

UDC: 517.51

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

Citation in format AMSBIB
\Bibitem{Oni04} \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 \mathnet{http://mi.mathnet.ru/mz146} \crossref{https://doi.org/10.4213/mzm146} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2129342} \zmath{https://zbmath.org/?q=an:1106.26013} \transl \jour Math. Notes \yr 2004 \vol 76 \issue 5 \pages 711--722 \crossref{https://doi.org/10.1023/B:MATN.0000049670.36842.71} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000226356700012} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-10344260223} 

• http://mi.mathnet.ru/eng/mz146
• https://doi.org/10.4213/mzm146
• http://mi.mathnet.ru/eng/mz/v76/i5/p762

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This publication is cited in the following articles:
1. G. G. Oniani, “Note on Besicovitch's Theorem on the Possible Values of Upper and Lower Derivatives”, Math. Notes, 93:2 (2013), 282–287
2. Oniani G., “On the Differentiation of Integrals With Respect to Translation Invariant Convex Density Bases”, Fundam. Math., 246:2 (2019), 205–216
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