Bergman-Szeg\H{o} kernel asymptotics in weakly pseudoconvex finite type cases

We construct a pointwise Boutet de Monvel-Sj\"ostrand parametrix for the Szeg\H{o} kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert