  We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$
dimensions, for focusing and defocusing interactions. We introduce new higher
order energy functionals and prove that they are conserved for solutions of
energy subcritical defocusing, and $L^2$ subcritical (de)focusing GP
hierarchies, in spaces also used by Erd\"os, Schlein and Yau in
\cite{esy1,esy2}. By use of this tool, we prove a priori $H^1$ bounds for
positive semidefinite solutions in those spaces. Moreover, we obtain global
well-posedness results for positive semidefinite solutions in the spaces
studied in the works of Klainerman and Machedon, \cite{klma}, and in
\cite{chpa2}. As part of our analysis, we prove generalizations of Sobolev and
Gagliardo-Nirenberg inequalities for density matrices.
