A short proof of commutator estimates

TL;DR

This paper presents a concise proof of homogeneous commutator estimates for fractional derivatives, extending known results and providing new estimates in one-dimensional cases for certain index ranges.

Contribution

It offers a short, classical proof for commutator estimates, including new results in one dimension for specific fractional derivative ranges.

Findings

New estimates for 1/3<r≤1/2 in 1D

Unified proof for L^p and weighted L^p estimates

Simplified proof approach for fractional derivative commutator estimates

Abstract

The goal of this note is to give, at least for a restricted range of indices, a short proof of homogeneous commutator estimates for fractional derivatives of a product, using classical tools. Both $L^{p}$ and weighted $L^{p}$ estimates can be proved by the same argument. When the space dimension is 1, we obtain some new estimates in the unexplored range $1/3<r\le1/2$.