Rearrangement Inequality and Chebyshev＇s Sum Inequality on Positive Tensor Products of Orlicz Sequence Space with Banach Lattice

Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Fremlin projective tensor product of lφ with a Banach lattice X, and in lφ iX, the Wittstock injective tensor product of lφ with a Banach lattice X.

A Rearrangement Inequality on Positive Tensor Products of Banach Lattices

In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a random permutation. We now consider a sequence in lp with 1 〈 p 〈 ∞, and a sequence in a Banach lattice X. Instead of normal multiplication, we consider the tensor product of lp and X. We show that in Wittstock injective tensor product, lp iX, and Fremlin projective tensor product, lp FX, the rearrangement inequality still exists.