Paraproduct

Summary

A paraproduct is the part of a product $fg$ in which the high frequencies of $f$ are multiplied by the low frequencies of $g$. Using the Littlewood–Paley decomposition, every product splits as $fg = \Pi_g f + \Pi_f g + R(f, g)$, two paraproducts and a remainder in which both factors have comparable frequencies. Paraproducts are bilinear operators that behave much better than the full product in many function spaces. They appear in Bony's paradifferential calculus for nonlinear PDE and in the T(1) theorem, where they are the model non-convolution Calderón–Zygmund operators.