Variable exponents and grand Lebesgue spaces: Some optimal results

Consider p : Omega -> [1, vertical bar infinity vertical bar, a measurable bounded function on a bounded set Omega with decreasing rearrangement p(*) : [0, vertical bar Omega vertical bar] -> [1,+infinity[. We construct a rearrangement invariant space with variable exponent p(*) denoted by L-**(p*(.))(Omega). 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-Holder continuity at zero, then it is contained in the grand Lebesgue space L-p*(0))(Omega). This inclusion fails to be true if we impose a slower growth as vertical bar p(*)(t) - p(*)(0)vertical bar >= A/Ln vertical bar Ln t vertical bar at zero. Some other results are discussed.