$L^p$-Boundedness properties with exponential weights and a Parseval-type relation for the index ${ }_2F_1$-transform

We will present a unique continuation result for solutions of second order differential equations of real principal type P(x,D)u+V(x)u=0 with critical potential V in L n/2 (where n is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove L p Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its L p -L p ' boundedness properties.