A short course on the Kato problem (Q2776613)

The introduction starts by saying that `In these notes, we shall discuss recent progress on the so-called ``Kato problem'' or ``square root problem''. This exposition is intended to be at least somewhat accessible to graduate students, and therefore contains some rather elementary material. On the other hand, we shall sometimes gloss over certain technicalities, and resort to heuristic arguments during our discussion of some of the preliminaries, in order not to obscure the main ideas by a morass of technical complication.' This is a good description of the philosophy of this course (or survey paper). It concerns the Kato problem (also called ``square root problem''). This amounts to investigate the existence of an estimate as NEWLINE\[NEWLINE\|\sqrt Lf\|_{L^2 (\mathbb{R}^n)} \leq C\|\nabla f\|_{L^2 (\mathbb{R}^n)}NEWLINE\]NEWLINE for any \(f\in L^2 (\mathbb{R}^n)\), where \(\sqrt L\) is the formal square root of the operator \(Lu=\text{div}(A(x)\nabla u)\), where \(A(x)\) is an \(n\times n\) matrix with bounded coefficients satisfying an ellipticity condition and the constant \(C>0\) depends only on \(n\) and the ellipticity constants of \(A\). The relationship with the \(L^2\) boundedness of the Cauchy integral on a Lipschitz curve is pointed out and the important work by the author and co-authors such as Auscher, Lacey, McIntosh and Tchamitchian settling the problem in all dimensions is presented in a rather informal and readable way. Results involving the ``Gaussian property'' deserve particular emphasis.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00018].