UCLA

This dissertation has two parts. In the first part, we extend the direct approachto the semiclassical Bergman kernel asymptotics, developed recently in [12] for real analytic exponential weights, to the smooth case. Similar to [12], our approach avoids the use of the Kuranishi trick and it allows us to construct the amplitude of the asymptotic Bergman projection by means of an asymptotic inversion of an explicit Fourier integral operator. In the second part, we carry out a construction of a globally defined weight function on the phase space, associated to a class of non-self-adjoint semiclassical pseudodifferential operators with double characteristics.