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Register a new userCarleman type inequalities for fractional relativistic operators
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In this paper, we derive Carleman estimates for the fractional relativistic operator. Firstly, we consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals to deduce Carleman estimates with linear exponential weight. Our approach is based on spectral methods and functional calculus. Secondly, we use pseudo-differential calculus in order to prove Carleman estimates with quadratic exponential weight, both in parabolic and elliptic contexts. The latter also holds in the case of the fractional Laplacian.
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In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ R^m_t times R^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with time-dependent lower order terms. The key improvements are: (1) we obtain smaller observation regions compared to standard Carleman estimate results, and (2) we also prove observability when the observation point lies inside the domain. Finally, as a corollary of the observability result, we obtain improved interior controllability for the wave equation.
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