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

References

^ Kato, Tosio (1953). "Integration of the equation of evolution in a Banach space". J. Math. Soc. Japan. 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.

External links

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

[:0-2] 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. MR 1933726.

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