A conformal density defines, in a natural way, a function on the null cone in the ambient space.
The GJMS operator is defined by taking density ƒ of the appropriate weight k − n/2 and extending it arbitrarily to a function F off the null cone so that it still retains the same homogeneity.
The function ΔkF, where Δ is the ambient Laplace–Beltrami operator, is then homogeneous of degree −k − n/2, and its restriction to the null cone does not depend on how the original function ƒ was extended to begin with, and so is independent of choices.
The GJMS operator also represents the obstruction term to a formal asymptotic solution of the Cauchy problem for extending a weight k − n/2 function off the null cone in the ambient space to a harmonic function in the full ambient space.
In even dimension n, these are the operators Ln/2 that take a true function on the manifold and produce a multiple of the volume form.