Individual convergence of paraproduct limits

Determine whether the limits of the paraproduct terms \(\Pi_f^{(t')}g\) and \(\Pi_g^{(t')}f\) exist individually in the stated generality as \(t'\to0^+\), particularly for the terms with index \(h=0\).

Background

The paper derives a paraproduct decomposition of the product $fg$ using heat-semigroup operators and proves convergence of the combined expression in $S'(G)$. The decomposition contains the terms $\Pi_f^{(t')}g$ and $\Pi_g^{(t')}f$, whose individual limits are needed if the components are to be interpreted separately rather than only through their combined contribution.

The authors explicitly identify the unresolved issue as arising from the contributions with $h=0$. Thus, the open problem concerns convergence of these two limits separately, not convergence of the full paraproduct expansion, which is established in the surrounding result.