In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing result by M. Bellassoued and J. Le Rousseau (2018) arguments Di Cristo, E. Francini, C.-L. Lin, S. Vessella, J.-N. Wang (2017), present an elementary method to under optimal regularity assumption on coefficients.

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A generalization of Schauder basis associated with the concept of generalized analytic functions is introduced. Corresponding concepts of density, completeness, biorthogonality and basicity are defined. Also, corresponding concept of the space of coefficients is introduced. Under certain conditions for the  corresponding operators, some properties of the space of coefficients and basicity crite...