The Kato-Ponce Inequality with Polynomial Weights

TL;DR

This paper proves that weighted Kato-Ponce inequalities with polynomial weights hold across all bilinear Lebesgue exponents and for the full range of the order s, without requiring classical weight conditions.

Contribution

It establishes the validity of weighted Kato-Ponce inequalities with polynomial weights for all exponents and orders, extending previous results and removing classical weight restrictions.

Findings

Weighted inequalities hold for all bilinear Lebesgue exponents.

Classical Muckenhoupt weight conditions are not necessary.

Results include strong-type inequalities at L^1 and L^∞ endpoints.

Abstract

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights $\langle x\rangle^a = (1+|x|^2)^{a/2}$ for $a\ge 0$. We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential $J^s = (1- \De)^{{s}/{2}}$ holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree $s$. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponce inequality. We also consider a few other variants such as commutator and mixed norm estimates, and analogous conclusions are derived. Our results contain strong-type inequalities for both $L^1$ and $L^\infty$ endpoints, which extend several existing results.