Fractional Leibniz rules in quasi-Banach function spaces and weighted bi-parameter settings

Abstract

Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, describing properties of fractional differential operators applied to a product of functions. In particular, these inequalities traditionally give estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivatives. Fractional Leibniz rules have applications in the study of the existence of solutions to PDEs such as Euler and Navier-Stokes equations. Moreover, partial fractional differential operators satisfy estimates whose structure resembles the Leibniz rule for classical partial derivatives and that also have applications in understanding the well-posedness of PDEs.

Results of this type have been studied in a variety of settings. For instance, Lebesgue norms can be replaced with norms associated to other function spaces. For single parameter differential operators, settings studied have included Hardy, Triebel-Lizorkin, Besov, and mixed Lebesgue spaces, as well as their weighted counterparts and weighted Lebesgue spaces. Bi-parameter results have been demonstrated in weighted and mixed Lebesgue spaces. Fractional Leibniz rules in each of these settings can be further generalized by replacing the product of functions with a bilinear pseudodifferential operator, with varying assumptions on the smoothness of the associated multiplier.

In this manuscript, we present fractional Leibniz rules associated to Coifman-Meyer bilinear pseudodifferential operators in a broad context that unifies many existing results in addition to obtaining new ones. In particular, the use of Nikol'skiĭ representations allows for a flexible approach by first obtaining estimates at the level of Triebel-Lizorkin spaces based on the function spaces of interest. We prove such estimates for Coifman-Meyer multiplier operators in the setting of Triebel-Lizorkin and Besov spaces based on quasi-Banach function spaces, which imply fractional Leibniz rules in the setting of quasi-Banach function spaces. As corollaries, we obtain results in weighted mixed Lebesgue, weighted Morrey, and variable Lebesgue spaces. Another example is the class of rearrangement invariant quasi-Banach function spaces, which includes weighted Lebesgue, Lorentz, and Orlicz spaces. We further demonstrate the flexibility of this method by using it to prove bi-parameter fractional Leibniz rules in the setting of weighted Lebesgue spaces.