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 \((F*G)(x) = \delta _{0,x}\) 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.
