  Let $P(D)$ be the Laplacian $\Delta,$ or the wave operator $\square$. The
following type of Carleman estimate is known to be true on a certain range of
$p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot
x}P(D)u\|_{L^p(\mathbb{R}^d)} \] with $C$ independent of $v\in \mathbb{R}^d$.
The estimates are consequences of the uniform Sobolev type estimates for second
order differential operators due to Kenig-Ruiz-Sogge \cite{KRS} and
Jeong-Kwon-Lee \cite{JKL}. The range of $p,q$ for which the uniform Sobolev
type estimates hold was completely characterized for the second order
differential operators with nondegenerate principal part. But the optimal range
of $p,q$ for which the Carleman estimate holds has not been clarified before.
When $P(D)=\Delta$, $\square$, or the heat operator, we obtain a complete
characterization of the admissible $p,q$ for the aforementioned type of
Carleman estimate. For this purpose we investigate $L^p$-$L^q$ boundedness of
related multiplier operators. As applications, we also obtain some unique
continuation results.
