Norm Estimates for the Fractional Derivative of Multiple Factors

We extend the Kato-Ponce inequality to a product of m functions, proving an estimate currently missing from the literature. This study is motivated by the fact that the 3-factor Kato-Ponce does not follow directly from the 2-factor version in the full range of permissible indices. Our methodology is based upon that in [12] but our extension entails a novel decomposition that elegantly and effectively handles the technical difficulties that arise from the combinatorial complexity of the possibly large number of factors.