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.     

On Sharp Sobolev Embedding and The Logarithmic Sobolev Inequality

The purpose of this note is to give a short proof of the Grosslogarithmic Sobolev inequality using the asymptotics of thesharp L² Sobolev constant and the product structure of Euclideanspace. Let FL^r(Rⁿ) for some positive r with ||F||_r=1. 1991 MathematicsSubject Classification 58G35.