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$%D. E. Edmunds and M. Krbec, Decomposition and Moser’s Lemma, Rev. Mat. Complut. 15 (2002), no. 1, 5774.%$

Decomposition and Moser’s Lemma
David E. EDMUNDS and Miroslav KRBEC
Centre for Mathematical Analysis and Its Applications
University of Sussex
Falmer BN1 9QH — Great Britain
Institute of Mathematics
Acad. Sci. of the Czech Republic
Žitná 25, 115 67 Prague 1 — Czech Republic

Received: January 22, 2001
Revised: April 17, 2001


Using the idea of the optimal decomposition developed in recent papers (2000) by the same authors and in Cruz-Uribe et al. (to appear), we study the boundedness of the operator         integral 1
Tg(x)=  x g(u)du/u  , x (-  (0,1)  , and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser’s lemma, O’Neil’s convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs of uniform boundedness of exponential and double exponential integrals in the spirit of the celebrated lemma due to Moser (1971).

2000 Mathematics Subject Classification: 46E30, 26A12, 35A15, 35B10.