A noninequality for the fractional gradient

In this paper we give a streamlined proof of an inequality recently obtained by the author: For every a a α∈(0,1) there exists a constant C = C(α,d)>0 such that for all u∈ L^q(R^d) for some 1 ≤ q < d/(1-α) such that D^αu :ΔI_1-αu∈L¹(R^d; R^d). We also give a counterexample which shows that in contrast to the case α=1, the fractional gradient does not admit an L¹ trace inequality, i.e. ||D^αu||_{L¹(R^d; R^d)} cannot control the integral of u with respect to the Hausdorff content H_∞ ^d-α. The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space L¹(H_∞ ^d-α),β∈[1,d).It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to β∈(0,1).