Kato's conjecture

Kato's conjecture is a mathematical problem named after mathematician Tosio Kato, of the University of California, Berkeley. Kato initially posed the problem in 1953.^[1]

Kato asked whether the square roots of certain elliptic operators, defined via functional calculus, are analytic. The full statement of the conjecture as given by Auscher et al. is: "the domain of the square root of a uniformly complex elliptic operator ${\displaystyle L=-\mathrm {div} (A\nabla )}$ with bounded measurable coefficients in Rⁿ is the Sobolev space H¹(Rⁿ) in any dimension with the estimate ${\displaystyle ||{\sqrt {L}}f||_{2}\sim ||\nabla f||_{2}}$".^[2]

The problem remained unresolved for nearly a half-century, until in 2001 it was jointly solved in the affirmative by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe Tchamitchian.^[2]