Kato's conjecture
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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 with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate
".[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]