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We obtain a complete characterization of $L^p-L^q$ Carleman estimates withweight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends theCarleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge.Consequently, we obtain new unique continuation properties of higher orderSchr\"odinger equations relaxing the integrability assumption on the solutionspaces.

Carleman inequalities and unique continuation for the polyharmonic operators