Extend the Laplace-transform continuation theorem beyond finite kappa_K(0)

Establish corresponding extensions of the Laplace-transform continuation results in Section 4.2 without assuming that the support-function value \(\kappa_K(0)\) is finite.

Background

The paper develops continuation criteria involving a coefficient interpolant $\phi$, its Laplace transform $\Phi$, and a closed set $K$ in the pararadial compactification. The principal refinement in Section 4.2 assumes $\kappa_K(0)<\infty$, which controls the relevant geometry and growth near the positive real direction.

The author explicitly asks whether analogous results remain valid when this finiteness condition is removed. Such an extension would broaden the scope of the Laplace-transform characterization to continuation problems involving more general, potentially unbounded singularity configurations.