In the 1960's, T. Kato posed a conjecture about finding the domain and some crucial estimates for the square root of elliptic partial differential operators in divergence form. The question attracted lots of interest because of the applications that it would have, and it turned out to be fairly tough to prove: only after around 40 years and joint efforts from different areas in Analysis (mainly PDE, Functional Analysis and Harmonic Analysis), it was finally solved by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002.

In this talk, we will illustrate the main changes and new difficulties that arise if we want to solve Kato's problem for operators in non-divergence form instead. We will present a partial solution of the problem which already faces, at least to some extent, some of the difficulties inherent in the non-divergence setting, like the lack of symmetry of the operator and the need to use weights. This is a joint work with L. Escauriaza and S. Hofmann.