Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE theory (see the works of Taylor and Tataru).
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(CW post just to get this off the unanswered list) As Carlo Beenakker remarked, the only connection is that they are named after the same Carleman. The Carleman estimates are weighted energy estimates for partial differential equations. They are so named because of their use by Carleman in his 1939 paper on the uniqueness (unique continuation) of solutions for a certain system of first order PDEs. 

