  We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz
single- and double-layer boundary-integral operators as mappings from
$L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$
is the boundary of the obstacle). The new bounds are obtained using estimates
on the restriction to the boundary of quasimodes of the Laplacian, building on
recent work by the first author and collaborators.
  Our main motivation for considering these operators is that they appear in
the standard second-kind boundary-integral formulations, posed in $L^2(\partial
\Omega)$, of the exterior Dirichlet problem for the Helmholtz equation. Our new
wavenumber-explicit $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$
bounds can then be used in a wavenumber-explicit version of the classic
compact-perturbation analysis of Galerkin discretisations of these second-kind
equations; this is done in the companion paper [Galkowski, M\"uller, Spence,
arXiv 1608.01035].
