Boundary behavior of the Szegö kernel

We give a Hörmander-type localization principle for the diagonal Szegö kernel S Ω ( z ) $S_\Omega (z)$ . We also show that for each boundary point z 0 $z_0$ , S Ω ( z ) ≳ | z − z 0 | − 1 3 $S_\Omega (z)\gtrsim |z-z_0|^{-\frac{1}{3}}$ holds non-tangentially for any bounded pseudoconvex domain with smooth boundary in C 2 ${\mathbb {C}}^2$ .