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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Tosio Kato was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis. Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption o ...

, of the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. 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 im ...

s, defined via

functional calculus In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral t ...

, 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 Pascal Auscher is a French mathematician working at University of Paris-Sud. Specializing in harmonic analysis and operator theory, he is mostly known for, together with Steve Hofmann, Michael Lacey Michael Lacey may refer to: * Michael Lacey ( ...

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 Alan McIntosh (born 29 July 1939) was a Welsh amateur football outside forward who played in the Football League for Cardiff City Cardiff City Football Club ( cy, Clwb Pêl-droed Dinas Caerdydd) is a professional association football cl ...

, 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) (born 2000), Brazilian footballer * Prince Philippe, Count ...

References

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