  The present work has as a first goal to extend the previous results in
\cite{CFL20} to weighted uncertainty principles with nontrivial radially
symmetric weights applied to curl-free vector fields. Part of these new
inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN)
inequalities studied by Catrina and Costa in \cite{CC} from scalar fields to
curl-free vector fields. We will apply a new representation of curl-free vector
fields developed by Hamamoto in \cite{HT21}. The newly obtained results are
also sharp and minimizers are completely described.
  Secondly, we prove new sharp second order interpolation functional
inequalities for scalar fields with radial weights generalizing the previous
results in \cite{CFL20}. We apply new factorization methods being inspired by
our recent work \cite{CFL21}. The main novelty in this case is that we are able
to find a new independent family of minimizers based on the solutions of
Kummer's differential equations.
  We point out that the two types of weighted inequalities under consideration
(first order inequalities for curl-free vector fields vs. second order
inequalities for scalar fields) represent independent families of inequalities
unless the weights are trivial.

Analysis of PDEs