  A short intrinsic proof is given for the Bourgain-Brezis-Mironescu theorem
with an extension for higher-order gradient forms. This argument illustrates
the role of functional geometry and Fourier analysis for obtaining embedding
estimates. New Hausdorff-Young inequalities are obtained for fractional
embedding as an extension of the classical Aronszajn-Smith formula. These
results include bilinear fractional embedding as suggested by the Landau
collision operator in plasma dynamics.

Analysis of PDEs