Weighted bi-parameter fractional Leibniz rules

Fractional Leibniz rules are estimates in norm for fractional differential operators applied to the product of functions, resembling the product rule from early calculus. We obtain fractional Leibniz rules associated to partial fractional differential operators and bi-parameter Coifman–Meyer multiplier operators in the setting of weighted Lebesgue spaces, improving the range of the fractional orders of differentiation allowed in existing estimates. Our methods of proof rely on appropriate paraproduct decompositions of bilinear operators and new Nikol'skiĭ representations for weighted bi-parameter Triebel–Lizorkin spaces. As a bi-product, we also obtain bi-parameter fractional Leibniz rule in the context of Triebel–Lizorkin and Besov spaces.