An Estimate for the Free Helmholtz equation

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. Lp 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 ∗Dedicated to David Colton and Rainer Kress. It is a pleasure to express my gratitude for your many interesting and practical insights into inverse scattering, which have played a big role in directing my research, and that of many others. More importantly, I applaud the warm and welcoming mathematical community that you have shaped around you. It is privilege to have become part of that community. †This research was supported in part by NSF grant DMS-0355455, ONR grant N0001405-1-0716