### Abstract

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}(∂Ω) → H
^{1}(∂Ω) (where ∂Ω 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}(∂Ω) , of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit L
^{2}(∂Ω) → H
^{1}(∂Ω) 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, Müller, and Spence in Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem, 2017. arXiv:1608.01035).

Original language | English |
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Article number | 6 |

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Integral Equations and Operator Theory |

Volume | 91 |

Issue number | 1 |

Early online date | 31 Jan 2019 |

DOIs | |

Publication status | Published - 1 Feb 2019 |

### Keywords

- math.AP
- 31B10, 31B25, 35J05, 35J25, 65R20
- Boundary integral equation
- Helmholtz equation
- Semiclassical
- Layer-potential operators
- High frequency

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory