Sharp trace and Korn inequalities for differential operators
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 $p=1$, a case difficult to be handled by harmonic analysis techniques, we
then classify boundary trace- and Korn-type inequalities. For $p=1$ and so
despite the failure of the Calder\'{o}n-Zygmund theory, we prove that sharp
trace estimates can be systematically reduced to full $k$-th order gradient
estimates. Moreover, for $1<p<\infty$, where sharp trace- 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 Calder\'{o}n-Zygmund estimates by extension operators might not be
possible.