The Kato-Ponce inequality with polynomial weights

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 \(J^s = (1- \Delta )^{{s}/{2}}\) 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.