The problem of creating a directivity pattern within desired upper and lower bounds using a nonsuperdirective microwave antenna of prescribed aperture is studied, and a strong necessary condition for the existence of solutions is derived. The condition is tested in cases of interest in designing satellite antennas that provide regional coverages on the Earth. For these cases, aperture fields are found which closely approach directivity limits implied by the necessary condition. Realistic designs are also obtained for shaped reflectors with simple feeds, and the results confirm the strength of the necessary condition for practical problems. ¿ American Geophysical Union 1996