Abstract
Using a new approach, we show that, for any ideal space X with nonempty regular part, the maximal function operator M B constructed from an arbitrary quasidensity differential basis B is not compact if considered in a pair of weighted spaces (X w , X v ) generated by X. For special differential bases that include convex quasidensity bases, we prove that M B is not compact in a pair of weighted spaces (X w , X v ) generated by an arbitrary ideal space X. An example is given of a quasidensity differential basis such that the maximal function operator constructed from this basis is compact in (L∞, L∞).
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Original Russian Text Copyright © 2015 Berezhnoĭ E.I.
The author was partially supported by the Russian Foundation for Basic Research (Grant 14–01–00417).
Yaroslavl’. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 752–761, July–August, 2015; DOI: 10.17377/smzh.2015.56.403.
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Berezhnoĭ, E.I. On compactness of maximal operators. Sib Math J 56, 593–600 (2015). https://doi.org/10.1134/S0037446615040035
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DOI: https://doi.org/10.1134/S0037446615040035