Carleman estimates for second-order elliptic operators with limiting weights: An elementary approach

By using some deep tools from microlocalanalysis, Kenig et al. (2007), Imanuvilov et al. (2010), Dos Santos Ferreira et al. (2009), and Guillarmou and Tzou (2011) have successfully established various Carleman estimates for elliptic operators that possess the limiting Carleman weight. In this paper, we revisit these problems and present a unified and fundamental approach for deriving these estimates. The main tool we employ is an elementary pointwise estimate for second-order elliptic operators.