An improved spectral inequality for sums of eigenfunctions

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We establish a new spectral inequality for the quantified estimation of the H s H^s -norm, s ≥ 0 s\ge 0 of a finite linear combination of eigenfunctions in a domain in terms of its H s H^s -norm in a strictly open subset of the whole domain. The corresponding upper bound depends exponentially on the square root of the frequency number associated to the linear combination.