Weighted gradient inequalities and unique continuation problems

TL;DR

This paper establishes weighted gradient inequalities using Pitt inequalities for the Fourier transform, leading to Carleman-type estimates that support unique continuation for solutions of first order differential equations.

Contribution

It introduces a new weighted gradient inequality derived from Pitt inequalities, providing novel tools for unique continuation problems.

Findings

Proves a weighted gradient inequality using Pitt inequalities.

Derives Carleman-type estimates for differential equations.

Supports unique continuation for first order systems.

Abstract

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$ \|e^{-\tau\ell(\cdot)} u^{\frac 1q} f\|_q\leq c_\tau\| e^{-\tau\ell(\cdot)} v^{\frac 1p}\, \nabla f\|_p, \quad f\in C^\infty_0( R^n). $$ This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.