On the basic sequence structure of variable exponent Lebesgue spaces

Abstract We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces L P $L_{\boldsymbol {P}}$ upper L Subscript bold italic upper P built from index functions P : Ω → ( 0 , ∞ ] $\boldsymbol {P}\colon \Omega \to (0,\infty ]$ bold italic upper P colon normal upper Omega right arrow left parenthesis 0 comma infinity right bracket on σ $\sigma $ sigma -finite measure spaces ( Ω , Σ , μ ) $(\Omega ,\Sigma ,\mu )$ left parenthesis normal upper Omega comma normal upper Sigma comma mu right parenthesis . Specifically, we prove that if P $\boldsymbol {P}$ bold italic upper P is bounded away from infinity, then any complemented subsymmetric basic sequence of L P $L_{\boldsymbol {P}}$ upper L Subscript bold italic upper P is equivalent to the canonical basis of ℓ r $\ell _r$ script l Subscript r for some r ≥ 1 $r\ge 1$