FRACTIONAL DIFFERENTIATION: LEIBNIZ MEETS H ¨ OLDER LOUKAS GRAFAKOS Abstract. Abstract: We discuss how to estimate the fractional derivative of the product of two functions, not in the pointwise sense, but on Lebesgue spaces whose indices satisfy H¨ older’s inequality 1. Introduction We recall Leibniz’s product rule of differentiation (1) (fg) (m) = m X k=0  m k  f (m-k) g (k) which is valid for C m functions f,g on the real line. Here g (k) denotes the kth derivative of the function g on the line. This rule can be extended to functions of n variables. For a given multiindex α =(α 1 ,...,α n ) ∈ (Z + ∪{0}) n we set ∂ α f = ∂ α 1 1 ··· ∂ αn n f = ∂ α 1 ∂x 1 α 1 ··· ∂ αn ∂x n αn f. The nth dimensional extension of the Leibniz rule (2) is (2) ∂ α (fg)= X β≤α  α β  (∂ α-β f )(∂ β g) where β ≤ α means β j ≤ α j for all j =1,...,n, and (3)  α β  =  α 1 β 1  ···  α n β n  = n Y j =1 α j ! β j !(α j - β j )! . Identity (3) can be used to control the Lebesgue norm of ∂ α (fg) in terms of Lebesgue norms of partial derivatives of f and g via H¨older’s inequality: kFGk L r (R n ) ≤kF k L p (R n ) kGk L q (R n ) where 0 < p, q, r ≤∞ and 1/r =1/p +1/q. Unlike convolution, which captures the smoothness of its smoother input, multi- plication inherits the smoothness of the rougher function. In this note we study the smoothness of the product of two functions of equal smoothness. The results we prove are quantitative and we measure smoothness in terms of Sobolev spaces. We focus Date : August 1, 2016. 1991 Mathematics Subject Classification. Primary 42B20. Secondary 35Axx. Key words and phrases. Kato-Ponce inequality, bilinear operators, Riesz and Bessel potentials. Grafakos acknowledges support from the Simons Foundation. 1