  We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$,
$u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a
bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is
of class $C^{2,1}$ then the corresponding pressure satisfies $p\in
C^{2\gamma}_*(\Omega)$ in the case $\gamma\in (0,\frac{1}{2}]$, where
$C^{2\gamma}_*$ is the H\"older-Zygmund space, which coincides with the usual
H\"older space for $\gamma<\frac12$. This result, together with our previous
one in [11] covering the case $\gamma\in(\frac12,1)$, yields the full double
regularity of the pressure on bounded and sufficiently regular domains. The
interior regularity comes from the corresponding $C^{2\gamma}_*$ estimate for
the pressure on the whole space $\mathbb{R}^d$, which in particular extends and
improves the known double regularity results (in the absence of a boundary) in
the borderline case $\gamma=\frac{1}{2}$. The boundary regularity features the
use of local normal geodesic coordinates, pseudodifferential calculus and a
fine Littlewood-Paley analysis of the modified equation in the new coordinate
system. We also discuss the relation between different notions of weak
solutions, a step which plays a major role in our approach.
