Decoupling for two quadratic forms in three variables: a complete characterization

Abstract

We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in R 5 given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case, this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables.

Decoupling inequalities for quadratic forms

“…for which sharp p L p decoupling inequalities were previously proved in [GZ20b] and [GORYZ19], respectively. The reasons of picking these two examples will soon become clear.…”

“…Further sharp decoupling inequalities for (non-degenerate) quadratic d-surfaces of co-dimension 2 were proven, for 2 ď d ď 4, in [DGS19] and [GZ20a]. More recently, in [GORYZ19], the classification of sharp decoupling inequalities for quadratic 3-surfaces in R 5 was completed, and sharp decoupling inequalities were proved in the "degenerate" cases, which were not covered by previously mentioned works. The approach to the "degenerate" cases in [GORYZ19] stands out from the previously mentioned works, in that it relies on small cap decouplings for the parabola and the 2-surface pξ 1 , ξ 2 , ξ 2 1 , ξ 1 ξ 2 q (we refer also to [DGW20] for further discussion of small cap decouplings).…”

“…Let us turn to the tuple Q 2 . As mentioned above, sharp decoupling inequalities for Q 2 were proven in [GORYZ19]. The tuple Q 2 is an interesting example as it is perhaps the simplest nontrivial example that is degenerate, in the sense that the scale-invariant Brascamp-Lieb inequality cannot be used for any p P r2, 8q.…”

“…More recently, in [GORYZ19], the classification of sharp decoupling inequalities for quadratic 3-surfaces in R 5 was completed, and sharp decoupling inequalities were proved in the "degenerate" cases, which were not covered by previously mentioned works. The approach to the "degenerate" cases in [GORYZ19] stands out from the previously mentioned works, in that it relies on small cap decouplings for the parabola and the 2-surface pξ 1 , ξ 2 , ξ 2 1 , ξ 1 ξ 2 q (we refer also to [DGW20] for further discussion of small cap decouplings). For manifolds of co-dimension n ą 2, the only result for quadratic forms that are not monomials prior to the current article was due to Oh [Oh18], who proved sharp decoupling inequalities for non-degenerate 3-surfaces in R 6 .…”

“…The tuple Q 2 is an interesting example as it is perhaps the simplest nontrivial example that is degenerate, in the sense that the scale-invariant Brascamp-Lieb inequality cannot be used for any p P r2, 8q. To handle this degeneracy, several specialized tools were introduced in [GORYZ19], including a partial small ball decoupling inequality for the tuple pξ 1 ξ 2 , ξ 2 2 q. Moreover, the authors there had to first prove sharp decoupling inequalities at the critical points B 2 and B 3 , and then apply interpolations with trivial estimates at B 1 and B 4 to cover the whole range p ě 2.…”