  We construct mollification operators in strongly Lipschitz domains that do
not invoke non-trivial extensions, are $L^p$ stable for any real number
$p\in[1,\infty]$, and commute with the differential operators $\nabla$,
$\nabla{\times}$, and $\nabla{\cdot}$. We also construct mollification
operators satisfying boundary conditions and use them to characterize the
kernel of traces related to the tangential and normal trace of vector fields.
We use the mollification operators to build projection operators onto general
$H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming
finite element spaces, with and without homogeneous boundary conditions. These
operators commute with the differential operators $\nabla$, $\nabla{\times}$,
and $\nabla{\cdot}$, are $L^p$-stable, and have optimal approximation
properties on smooth functions.
