Bilinear embedding theorems for differential operators in $\mathbb{R}^2$

TL;DR

This paper establishes bilinear inequalities for differential operators in two-dimensional space, exploring their implications for embedding theorems and Strichartz estimates, with a focus on both elliptic and non-elliptic cases.

Contribution

It provides a comprehensive analysis of bilinear inequalities for differential operators in , including complete results for elliptic cases and initial insights into non-elliptic cases.

Findings

Complete analysis for elliptic differential operators in 

Initial results and discussion for non-elliptic cases related to Strichartz estimates

Implications for anisotropic embedding theorems and summation exponents

Abstract

We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However, here we study the phenomenon in itself. We consider elliptic case, where our analysis is complete, and non-elliptic, where it is not. The latter case is related to Strichartz estimates in a very easy case of two dimensions.