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. Education and career Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of To ...

, of the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

. 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 theo ...

, are

analytic Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical ...

. 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, Michael Lacey, Alan McIntosh, and Philippe Tchamitchian.

References

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