Derivative bounds for Hardy heat kernels

Derive suitable operator or kernel estimates for $|D|^\beta e^{-tL_\kappa}$, where $L_\kappa=(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$, in order to replace the back-and-forth commutation argument used in the propagation estimates and remove the associated technical restrictions on the exponent $s$.

Background

The proof of the short-range scattering result for small $\alpha$ does not directly estimate the action of fractional momentum powers on the Hardy heat flow. Instead, it commutes $|D|^\beta$ to the left, compares powers of $L_0$ and $L_\kappa$, and then commutes $L_\kappa$ back next to the heat semigroup.

The authors identify this as the source of the technical condition $s\leq 3\alpha/2$ and suggest that direct derivative estimates for the Hardy heat kernel could simplify the argument and potentially eliminate that restriction.