Resolvent estimates for the Lam\'e operator and failure of Carleman estimates

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 $-\Delta^\ast$.