Multilinear Embedding -- convolution estimates on smooth submanifolds

TL;DR

This paper develops multilinear embedding estimates for the fractional Laplacian involving convolution estimates on smooth submanifolds, extending classical Riesz potential frameworks and inspired by space-time estimates in quantum hierarchy proofs.

Contribution

It introduces new convolution estimates that include the critical endpoint index and generalizes fractional integral inequalities to smooth submanifolds.

Findings

Extended convolution algebra to include critical endpoint index

Provided new fractional integral inequalities on submanifolds

Modeled estimates on space-time techniques used in quantum hierarchy proofs

Abstract

Multilinear embedding estimates for the fractional Laplacian are obtained in terms of functionals defined over a hyperbolic surface. Convolution estimates used in the proof enlarge the classical framework of the convolution algebra for Riesz potentials to include the critical endpoint index, and provide new realizations for fractional integral inequalities that incorporate restriction to smooth submanifolds. Results developed here are modeled on the space-time estimate used by Klainerman and Machedon in their proof of uniqueness for the Gross-Pitaevskii hierarchy.