Lq-structure of variable exponent spaces

It is shown that a separable variable exponent (or Nakano) function space L p ( ⋅ ) ( Ω ) has a lattice-isomorphic copy of l q if and only if q ∈ R p ( ⋅ ) , the essential range set of the exponent function p ( ⋅ ) . Consequently R p ( ⋅ ) is a lattice-isomorphic invariant set. The values of q such that l q embeds isomorphically in L p ( ⋅ ) ( Ω ) is determined. It is also proved the existence of a bounded orthogonal l q -projection in the space L p ( ⋅ ) ( Ω ) , for every q ∈ R p ( ⋅ ) .