  We consider $L^p$-$L^q$ estimates for the spherical harmonic projection
operators and obtain sharp bounds on a certain range of $p$, $q$. As an
application, we provide a proof of off-diagonal Carleman estimates for the
Laplacian, which extends the earlier results due to Jerison and Kenig
\cite{JK}, and Stein \cite{St-append}.
