Decomposition and Moser’s lemma

Using the idea of the optimal decomposition developed in re-cent papers [EK2] by the same authors and in [CUK] we study the boundedness of the operator Tg(x) = ∫ 1 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 bound-edness of exponential and double exponential integrals in the spirit of the celebrated lemma due to Moser [Mo].