We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = δ_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results were obtained in the 2000s, using intricate Fourier analysis. In 2022, a very simple deconvolution theorem was proved, but its applicability was limited. We extend the 2022 theorem to remove its limitations while maintaining its simplicity -- our main tools are Hölder's inequality, weak derivatives, and basic Fourier theory in $L^p$ space. Our motivation comes from critical phenomena in equilibrium statistical mechanics, where the convolution equation is provided by the lace expansion and $G$ is a critical two-point function. Our results significantly simplify existing proofs of critical $|x|^{-(d-2)}$ decay in high dimensions for self-avoiding walk, Ising and $\varphi^4$ models, percolation, and lattice trees and lattice animals. We also improve previous error estimates.