Characteristic of UMD spaces with Hilbert transformation of vector-valued functions on the field of $p$-adic numbers

We consider Hilbert transformation of vector-valued functions on the group of $p$-adic integers $Z_p$ taking values in Banach space $X$ , and square-integrable in Bochner sense. If Hilbert transformation $H \colon L_2(Z_p, X) \to L_2(Z_p, X)$ with $p \ne 2$ is a bounded operator, then Banach space $X$ is an UMD space.