# A noninequality for the fractional gradient

### Daniel Spector

National Chiao Tung University, Hsinchu, Taiwan; Okinawa Institute of Science and Technology Graduate University, Japan

## Abstract

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

for all $u∈L_{q}(R_{d})$ for some $1≤q<d/(1−α)$ such that $D_{α}u:=∇I_{1−α}u∈L_{1}(R_{d};R_{d})$. We also give a counterexample which shows that in contrast to the case $α=1$, the fractional gradient does not admit an $L_{1}$ 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 $H_{∞}$. 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 $L_{1}(H_{∞})$, $β∈[1,d)$. It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to $β∈(0,1)$.

## Cite this article

Daniel Spector, A noninequality for the fractional gradient. Port. Math. 76 (2019), no. 2, pp. 153–168

DOI 10.4171/PM/2031