An estimate for the free Helmholtz equation that scales

Wavelength plays a distinguished role in classical electromagnetic and acoustic scattering. Most significant features of the far field patterns radiated by a collection of sources or scatterers are related to their sizes and relative distances, measured in wavelengths. These significant features are reflected in the invariance of the Helmholtz equation with respect to translation, and its homogeneous scaling with respect to dilations. The weighted norms that were first developed to capture the correct decay properties of waves in Rⁿ do not scale homogeneously and are not invariant with respect to translation. L^p estimates scale homogeneously and commute with translations and rotations. However, their scaling properties give estimates with a weaker dependence on wavenumber (for bounded sources and scatterers with support that extends over many wavelengths). We introduce some norms and estimates that commute with translations and scale homogeneously under dilations, while retaining the same sharp dependence on wavelength for extended sources as that of the weighted estimates.