This work aims to study Carleman estimates, a weighted-type of inequalities first introduced by Carleman in 1939. Such estimates are very important for proving unique continuation properties of differential and pseudo-differential operators. We first derive a Carleman estimate for the Laplacian operator as an illustrative example following the work of Jérôme Le Rousseau and Gilles Lebeau in [7] which is a summary of a much large study. We try to extend the methodology to non-local operators. In particular we aim to deal with the fractional Laplacian. The results are focused on proving unique continuation properties and showing the significance of weighted estimates and the operators involved. For this we mainly use: Fourier analysis, Symbol theory and differential and pseudodifferential analysis.

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2023-2024. Director: María Ángeles García-Ferrero i Joaquim Ortega Cerdà