On the full range of Zippin and inclusion indices of rearrangement-invariant spaces

Abstract

AbstractLet X be a rearrangement-invariant space on [0, 1]. It is known that its Zippin indices $$\underline{\beta }{}_X,\overline{\beta }{}_X$$ β ̲ X , β ¯ X and its inclusion indices $$\gamma _X,\delta _X$$ γ X , δ X are related as follows: $$0\le \underline{\beta }{}_X\le 1/\gamma _X \le 1/\delta _X\le \overline{\beta }{}_X\le 1$$ 0 ≤ β ̲ X ≤ 1 / γ X ≤ 1 / δ X ≤ β ¯ X ≤ 1 . We show that given $$\underline{\beta },\overline{\beta }\in [0,1]$$ β ̲ , β ¯ ∈ [ 0 , 1 ] and $$\gamma ,\delta \in [1,\infty ]$$ γ , δ ∈ [ 1 , ∞ ] satisfying $$\underline{\beta }\le 1/\gamma \le 1/\delta \le \overline{\beta }$$ β ̲ ≤ 1 / γ ≤ 1 / δ ≤ β ¯ , there exists a rearrangement-invariant space X such that $$\underline{\beta }{}_X=\underline{\beta }$$ β ̲ X = β ̲ , $$\overline{\beta }{}_X=\overline{\beta }$$ β ¯ X = β ¯ and $$\gamma _X=\gamma $$ γ X = γ , $$\delta _X=\delta $$ δ X = δ .