Sharp Trace and Korn Inequalities for Differential Operators

Abstract We establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to p=1 p = 1, a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For p=1 p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k -th order gradient estimates. Moreover, for 1 1 p ∞, where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.