In this paper we give a streamlined proof of an inequality recently obtained by the author: For every α ∈ (0, 1) there exists a constant C = C(α, d) > 0 such thatWe also give a counterexample which shows that in contrast to the case α = 1, the fractional gradient does not admit an L 1 trace inequality, i.e. D α u L 1 (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 spaceIt is an open question whether this failure of a weaktype estimate for the Riesz transforms extends to β ∈ (0, 1).