Kato Problem

The Kato problem, or Kato square root problem, asked whether the square root of a uniformly elliptic divergence-form operator

on has the same domain as the first-order gradient. The space is the L²-space of equivalence classes of square integrable functions with respect to Lebesgue measure, meaning . When , the notation is . Here is a bounded matrix-valued measurable function satisfying uniform ellipticity.

The answer is affirmative: the domain of is the Sobolev space , and

for . The problem originated in Kato's (1961) question whether a regularly accretive operator and its adjoint have square roots with the same domain. Auscher et al. (2002) proved the divergence-form result above in all dimensions.