Abstract
The classical Rellich inequalities imply that the L 2 L^2 -norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not Lipschitz. For two-dimensional domains, we obtain a sharp L p L^p -estimate for 1 > p ≤ 2 1>p\leq 2 by using a Riemann mapping and interpolation argument.
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