Backward Uniqueness of Kolmogorov Operators

TL;DR

This paper proves the backward uniqueness of a class of Kolmogorov operators using a novel Carleman inequality derived through Littlewood-Paley decomposition, advancing understanding in partial differential equations.

Contribution

It establishes the backward uniqueness for Kolmogorov operators with a new approach employing weak Carleman inequalities and Littlewood-Paley techniques.

Findings

Backward uniqueness is proved for the specified Kolmogorov operators.

A weak Carleman inequality is developed for the analysis.

The method applies to global backward uniqueness problems.

Abstract

The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.