The Christoffel-Darboux kernel has a sharp dichotomy. Inside the support of a measure, the kernel grows linearly with polynomial degree. Outside, it grows exponentially. This makes the kernel excellent for detecting support boundaries — the transition from linear to exponential growth marks the edge precisely.
But the sharpness that makes boundary detection easy makes density recovery hard. Recovering the density inside the support requires dividing by the kernel, and the sharp transition at the boundary creates numerical instabilities. The inverse problem is ill-posed precisely because the forward problem is too well-posed — the kernel knows the boundary too precisely.
The paper mollifies the kernel — replaces the sharp Christoffel-Darboux kernel with a smoothed version that trades boundary precision for interior regularity. Inside the support, the mollified kernel provides uniform bounds instead of sharp growth estimates. The exponential detection of the exterior is sacrificed, but what's gained is a well-conditioned density recovery problem.
The through-claim: sharpness in one direction creates ill-posedness in the reverse direction. A perfect boundary detector cannot be a good density estimator, because the information that distinguishes “inside” from “outside” with exponential confidence overwhelms the gentler information about how much is inside. Smoothing the boundary trades detection for estimation.
This is a general pattern in inverse problems. The features that make a forward map informative — strong contrast, sharp transitions, high sensitivity — are exactly the features that make inversion unstable. The well-conditioned inverse requires a forward map that doesn't amplify differences, which means accepting a blurrier view of the boundaries in exchange for a clearer view of the interior. You can see where the thing is, or what the thing contains, but not both at maximum resolution.