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, structure, space, models, and change. History On ...

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 i ...

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 theo ...

, 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 In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential o ...

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, Alan McIntosh, and Philippe Tchamitchian.

References

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