An existence family for Husimi operator

We prove the well-posedness of the quantum Liouville equation in L-1(R-2n), provided that the potential of the Schrodinger equation lies in some H-s(R -n). By regularizing the Wigner function we obtain the Husimi equation whic h is ill-posed in any L-p(R-2n) spaces. We show that for the Husimi operato r the maximal and minimal extensions coincide and we construct a C-existenc e family in the sense of R. deLaubenfels which is a new tool for studying t his operator.