Grand / small lebesgue spaces : the setting, different approaches, important properties

Let Ω ⊂ Rn be of finite Lebesgue measure and 1 < p < ∞. The grand Lebesgue space Lp)(Ω) (cf. [IS92]) and the small Lebesgue space L(p(Ω) (cf. [Fio00]) are rearrangement invariant Banach function spaces. The classical Lebesgue space Lp(Ω) is embedded in Lp)(Ω), which again is embedded in every Lebesgue space Lp−ε(Ω), 0 < ε < p−1. Similarly, Lp+ε(Ω), ε > 0 is embedded in the space L(p(Ω), which again is embedded in Lp(Ω). We present a way to find their norms which are based on the decreasing rearrangement. To get there, we define specific extrapolation and interpolation constructions and use them, alone and in combination, in order to characterise the spaces. Finally, we compare them to Lorentz-Zygmund spaces.