  In this paper, we consider the Lam\'e operator $-\Delta^\ast$ and study
resolvent estimate, uniform Sobolev estimate, and Carleman estimate for
$-\Delta^\ast$. First, we obtain sharp $L^p$--$L^q$ resolvent estimates for
$-\Delta^\ast$ for admissible $p,q$. This extends the particular case $q=\frac
p{p-1}$ due to Barcel\'o et al. \cite{BFPRV} and Cossetti \cite{Co19}.
Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for
$-\Delta^\ast$. For the purpose we directly analyze the Fourier multiplier of
the resolvent. This allows us to prove not only the upper bound but also the
lower bound on the resolvent, so we get the sharp $L^p$--$L^q$ bounds for the
resolvent of $-\Delta^\ast$. Strikingly, the relevant uniform Sobolev and
Carleman estimates turn out to be false for the Lam\'e operator $-\Delta^\ast$
even though the uniform resolvent estimates for $-\Delta^\ast$ are valid for
certain range of $p, q$. This contrasts with the classical result regarding the
Laplacian $\Delta$ due to Kenig, Ruiz, and Sogge \cite{KRS87} in which the
uniform resolvent estimate plays crucial role in proving the uniform Sobolev
and Carleman estimates for $\Delta$. We also describe locations of the
$L^q$-eigenvalues of $-\Delta^\ast+V$ with complex potential $V$ by making use
of the sharp $L^p$--$L^q$ resolvent estimates for $-\Delta^\ast$.
