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.{{cite journal | first=Tosio | last=Kato | title=Integration of the equation of evolution in a Banach space | journal= J. Math. Soc. Jpn. | volume=5 | year=1953 | issue=2 | pages=208–234 | mr=0058861 | doi=10.2969/jmsj/00520208| doi-access=free }}

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 $L\; =-\backslash mathrm\{div\}\; (A\backslash nabla)$ with bounded measurable coefficients in Rⁿ is the Sobolev space H¹(Rⁿ) in any dimension with the estimate $||\backslash sqrt\{L\}f||\_\{2\}\; \backslash sim\; ||\backslash nabla\; f||\_\{2\}$".

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.{{cite journal | first1=Pascal | last1=Auscher | first2=Steve | last2=Hofmann | first3=Michael | last3=Lacey | first4=Alan | last4=McIntosh | first5=Philippe | last5=Tchamitchian | title=The solution of the Kato square root problem for second order elliptic operators on Rⁿ | journal=Annals of Mathematics | volume=156 | issue=2 | pages=633–654 | year=2002 | doi=10.2307/3597201 | jstor=3597201 | mr=1933726}}