sala A1-33, godz. 10:30–12:00

Loukas Grafakos (University of Missouri, Columbia, USA)
The Leibniz fractional rule of differentiation

Abstract. We present some recent results with S. Oh on the fractional Leibniz rule of differentiation combined with Hölder’s inequality. Let \(\Delta\) be the Laplacian on \(R^n\) and consider the classical Bessel potential \(J^s=(1−\Delta)^{s/2}\) and Riesz potential \(D^s=(−\Delta)^{s/2}\). We revisit the inequalities of Kato and Ponce concerning the \(L^r\) norm of the Bessel potential \(J^s\) (or the Riesz potential \(D^s\)) of the product of two functions in terms of the product of the \(L^p\) norm of one function and the \(L^q\) norm of the the Bessel potential \(J^s\) (resp. Riesz potential \(D^s\)) of the other function, i.e., $$\|J^s (fg)\|_{L^r}\leq C\big[\|f\|_{L^p} \|J^sg\|_{L^g}+\|J^sf\|_{L^p} \|g\|_{L^q}\big] $$and an analogous inequality with \(D^s\) in place of \(J^s\). Here the indices \(p,q\), and \(r\) are related as in Hölder’s inequality \(1/p+1/q=1/r\) and they satisfy \(1\leq p, q \leq\infty\) and \(1/2\leq r <\infty[/latex] and [latex]s>n/r −n\). Also the estimate is of weak-type when either \(p\) or \(q\) is equal to 1. In the case \(\)r < 1[/latex] we indicate via an example that when [latex]s\leq n/r − n[/latex] the inequality fails. We explain how these problems can be addressed via analysis of multilinear multiplier operators. We also discuss extensions of these results to the multi-parameter case.
Abstract in the pdf file
Kawa przed wykładem o 10:00 w klubie profesorskim. Seminarium wspólne z Zakładem Analizy Funkcjonalnej oraz Zakładem Teorii Funkcji Rzeczywistych.