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We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtzsingle- 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 estimateson the restriction to the boundary of quasimodes of the Laplacian, building onrecent work by the first author and collaborators.  Our main motivation for considering these operators is that they appear inthe standard second-kind boundary-integral formulations, posed in $L^2(\partial\Omega)$, of the exterior Dirichlet problem for the Helmholtz equation. Our newwavenumber-explicit $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$bounds can then be used in a wavenumber-explicit version of the classiccompact-perturbation analysis of Galerkin discretisations of these second-kindequations; this is done in the companion paper [Galkowski, M\"uller, Spence,arXiv 1608.01035].

Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators