Kato's conjecture is a

mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...

named after mathematician Tosio Kato, of the University of California, Berkeley. Kato initially posed the problem in 1953. Kato asked whether the square roots of certain

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which i ...

s, 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 (A\nabla)

with bounded measurable coefficients in Rⁿ is the Sobolev space ''H''¹(Rⁿ) in any dimension with the estimate

, , \sqrtf, , _ \sim , , \nabla f, , _

". The problem remained unresolved for nearly a half-century, until in 2001 it was jointly solved in the affirmative by Pascal Auscher,

Steve Hofmann Steve Hofmann is a mathematician who helped solve the famous Kato's conjecture. Said Hofmann, “It's a problem that has interested me since I was a graduate student... It was one of the biggest open problems in my field and everybody thought it wa ...

Michael Lacey Michael Lacey may refer to: * Michael Lacey (mathematician) (born 1959), an American mathematician * Michael Lacey (editor), American newspaper editor * Michael Pearse Lacey (1916–2014), Canadian bishop * Mick Lacey, Irish hurler See also * M ...

, Alan McIntosh, and

Philippe Tchamitchian Philippe is a masculine sometimes feminin given name, cognate to Philip. It may refer to: * Philippe of Belgium (born 1960), King of the Belgians (2013–present) * Philippe (footballer) Philippe Almeida Costa (born 1 March 2000) is a Brazilian ...

References

Differential operators Operator theory Conjectures that have been proved {{mathanalysis-stub