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 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 it was jointly solved in 2001 by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and .[2]
References[]
- ^ Kato, Tosio (1953). "Integration of the equation of evolution in a Banach space". J. Math. Soc. Jpn. 5: 208–234. doi:10.2969/jmsj/00520208. MR 0058861.
- ^ a b Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Philippe (2002). "The solution of the Kato square root problem for second order elliptic operators on Rn". Annals of Mathematics. 156 (2): 633–654. doi:10.2307/3597201. MR 1933726.
Categories:
- Differential operators
- Operator theory
- Conjectures that have been proved
- Mathematical analysis stubs