On Quantitative Sphere Equivalence and Extrapolation Phenomenon

Let X,Y be Banach spaces.We prove that if their unit spheres S(X)and S(Y)are(α,β)-Hlder homeomorphic for someα,β∈(0,1].Then the unit spheres of Lp[Ω,μ,X)and Lq(Ω,μ,Y)are(min{p/q,α},min{q/p,β})-Hlder homeomorphic for every measure space(Ω,μ)and for every 1≤p,q<∞.In particular,the result yields an interesting application involving extrapolation phenomenon.