@article{oai:oist.repo.nii.ac.jp:00001561,
 author = {Spector, Daniel},
 issue = {2},
 journal = {Portugaliae Mathematica},
 month = {Feb},
 note = {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 that
 [Formula: see text]
for all u∈Lq(Rd) for some 1≤q<d/(1−α) such that Dαu:=∇I1−αu∈L1(Rd;Rd). 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.  [Formula: see text] cannot control the integral of u with respect to the Hausdorff content  [Formula: see text]. 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  [Formula: see text], 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).},
 pages = {153--168},
 title = {A noninequality for the fractional gradient},
 volume = {76},
 year = {2020}
}