  A novel representation is developed as a measure for multilinear fractional
embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu
theorem and Pitt's inequality. New results are obtained for diagonal trace
restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev
inequality. Smoothing estimates are used to provide new structural
understanding for density functional theory, the Coulomb interaction energy and
quantum mechanics of phase space. Intriguing connections are drawn that
illustrate interplay among classical inequalities in Fourier analysis.
