Hardy and Rellich inequalities for submanifolds in Hadamard spaces

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Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities on regions in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael–Simon [13] and Hoffman–Spruck [12]. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. Years later, Carron [6] obtained a Hardy inequality for submanifolds in Hadamard spaces. In this paper, we prove the general Hardy and Rellich Inequalities for submanifolds in Hadamard spaces. Some applications are given and we also analyse the equality cases.