Variable exponents and grand Lebesgue spaces: Some optimal results

Consider p : Ω → [1,+∞[, a measurable bounded function on a bounded set Ω with decreasing rearrangement p∗ : [0, |Ω| ] → [1,+∞[. We construct a rearrangement invariant space with variable exponent p∗ denoted by Lp∗(·)∗∗ (Ω). According to the growth of p∗, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p∗(·) satisfies the log-H¨older continuity at zero, then it is contained in the grand Lebesgue space Lp∗(0))(Ω). This inclusion fails to be true if we impose a slower growth as |p∗(t) − p∗(0)| ≥ ALn |Ln t| at zero. Some other results are discussed.