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Supplement to the paper 'the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)'
- Source :
- Journal of Soviet Mathematics. 42:1636-1640
- Publication Year :
- 1988
- Publisher :
- Springer Science and Business Media LLC, 1988.
-
Abstract
- Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let t∈L1(0,1), let Nt be the smallest rearrangement-invariant order ideal vector lattice in L1(0,1), containing t. In the paper one investigates the properties of the image E(Nt¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, g∈L1(0,1), such that E(Nt¦A)⊂Ng. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(t∣A)≺QE(t*∣A*), where ≺ is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1⩽Q⩽25. One formulates the problem of the smallest value of Q.
Details
- ISSN :
- 15738795 and 00904104
- Volume :
- 42
- Database :
- OpenAIRE
- Journal :
- Journal of Soviet Mathematics
- Accession number :
- edsair.doi...........5d33d52e5b12c4f47fc055c64712fdec