Smoothing Properties of the Friedrichs Operator on $L^p$ spaces

We show that the Friedrichs operator exhibits smoothing properties in the $L^p$ scale. In particular we prove that on any smoothly bounded pseudoconvex domain the Friedrichs operator maps $A^2(\mathrm{\Omega })$ to $A^p(\mathrm{\Omega })$ for some $p>2$.