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]

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References

  1. Kato, Tosio (1953). "Integration of the equation of evolution in a Banach space". J. Math. Soc. Jpn. 5 (2): 208–234. doi: 10.2969/jmsj/00520208 . MR   0058861.
  2. 1 2 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. JSTOR   3597201. MR   1933726.