Pisier type inequalities for $K$-convex spaces

We generalize several theorems of Hytönen-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 $\{{\mathrm{\Delta }}^{-1}{D}_i{\}}_{i=1}^n$ in $L^p({\mathrm{\Omega }}_n,X)$.