Functionals for multilinear fractional embedding

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.