A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has $$ \mathbb{E}\bigg(d_X \big(f(Z_n),f(Z_0) \big)^p \bigg) \leq K^p n \mathbb{E} \bigg( d_X \big(f(Z_1),f(Z_0)\big)^p \bigg) $$ This notion was introduced by K.Ball and he showed that Hilbert spaces have Markov type 2. Using an isometric embedding of "snowflaked" $L_p$ (for $p<2$) into $L_2$ he then concludes that $L_p$ has Markov type $p$.

$\mathbf{Question:}$ Assume the spheres in two Banach $X$ and $X'$ spaces are uniformly homeomorphic and that $X$ has Markov type $p$, which conditions (on the uniform homeomorphism) imply that $X'$ has Markov type $q$?

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has $$ \mathbb{E}\bigg(d_X \big(f(Z_n),f(Z_0) \big)^p \bigg) \leq K^p n \mathbb{E} \bigg( d_X \big(f(Z_1),f(Z_0)\big)^p \bigg) $$ This notion was introduced by K.Ball and he showed that Hilbert spaces have Markov type 2. Using an isometric embedding of "snowflaked" $L_p$ (for $p<2$) into $L_2$, he then concludes that $L_p$ has Markov type $p$.

$\mathbf{Question:}$ Assume the unit spheres in two Banach $X$ and $X'$ spaces are uniformly homeomorphic and that $X$ has Markov type $p$. Which conditions (on the uniform homeomorphism) imply that $X'$ has Markov type $q$? e.g. Hölder continuity?