Kato-Ponce inequalities on weighted and variable Lebesgue spaces. (Q331337)

The goal of this paper is to prove Kato-Ponce inequalities in the setting of weighted Lebesgue spaces and also on variable Lebesgue spaces.NEWLINENEWLINENamely, the weighted version of the result stats that, for \(1<p,q<\infty\) and \(1/2<r<\infty\) such that \(\frac{1}{r}=\frac{1}{p}+\frac{1}{q}\), if \(v\) and \(w\) are Muckenhoupt weights in \(A_p\) and \(A_q\), respectively, and \(s>\max\{0,n(\frac{1}{r}-1)\}\) or \(s\) is a non-negative even integer, being \(f,g\) in the Schwartz class \({\mathcal S}(\mathbb{R}^n)\), the following inequalities holds NEWLINE\[NEWLINE\|T^s(fg)\|_{L^r(v^{\frac{r}{p}}w^{\frac{r}{q}})} \lesssim \|T^s f\|_{L^p(v)} \|g\|_{L^q(w)}+\|f\|_{L^p(v)}\|T^s g\|_{L^q(w)}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|T^s(fg)-f T^s g\|_{L^r(v^{\frac{r}{p}}w^{\frac{r}{q}})} \lesssim \|T^s f\|_{L^p(v)} \|g\|_{L^q(w)}+\|\nabla f\|_{L^p(v)}\|T^{s-1} g\|_{L^q(w)},NEWLINE\]NEWLINE where \(T^s\) represents either the inhomogeneous \(s\)-th differentiation operator \(J^s\) or the homogeneous one \(D^s\), which are defined via the Fourier transform as NEWLINE\[NEWLINE\widehat{J^s(f)}(\xi)=(1+|\xi|^2)^{s/2} \hat{f}(\xi), \;\;\widehat{D^s(f)}(\xi)=|\xi|^{s} \hat{f}(\xi).NEWLINE\]NEWLINEAn analogous theorem in the context of Lebesgue spaces with variable exponents is obtained also for both kind of differentiation operators. The main tools to obtain the results are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem.