Pisier type inequalities for $K$-convex spaces

We generalize several theorems of Hyt\"onen-Naor \cite{HN} using the approach from \cite{IVHV}. In particular, we give yet another necessary and sufficient condition (see (3.2)) to be a $K$-convex space, where the sufficiency was proved by Naor--Schechtman \cite{NS}. This condition is in terms of the boundedness of the second order Riesz transforms $\{\Delta^{-1} D_i\}_{i=1}^n$ in $L^p(\Omega_n, X)$.