Extremizability of Fourier restriction to the paraboloid

Abstract

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in ${\mathbb{R}}^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case $q=2$. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.

Some typos were corrected and arguments expanded in several places