Borel reducibility and Holder($\alpha$) embeddability between Banach spaces

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\mathbb{N}}\mathrm{/}{\mathrm{\ell }}_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called H"older$(\alpha )$ embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in [1,+\mathrm{\infty })$. We also answer a problem presented by Kanovei in the affirmative by showing that $C({\mathbb{R}}^{+})\mathrm{/}{C}_0({\mathbb{R}}^{+})$ is Borel bireducible to ${\Bbb R}^{\Bbb N}/c_0$.