The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

TL;DR

This paper investigates a higher-dimensional Benjamin-Ono equation, establishing local well-posedness in weighted Sobolev spaces, deriving sharp unique continuation properties, and determining the optimal decay rate of solutions.

Contribution

It introduces a new commutator estimate involving Riesz transforms and analyzes the well-posedness and decay properties in weighted Sobolev spaces for the higher-dimensional model.

Findings

Local well-posedness in weighted Sobolev spaces

Sharp unique continuation properties

Optimal decay rate of solutions

Abstract

We study the initial value problem associated to a higher dimensional version of the Benjamin-Ono equation. Our purpose is to establish local well-posedness results in weighted Sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow. In consequence, optimal decay rate for this model is determined. A key ingredient is the deduction of a new commutator estimate involving Riesz transforms.