# A Noninequality for the Fractional Gradient

## TL;DR

This paper provides a simplified proof of a fractional gradient inequality, explores its limitations regarding trace inequalities, and presents a counterexample related to Riesz transforms in fractional Hausdorff content spaces.

## Contribution

The paper offers a streamlined proof of a fractional gradient inequality and introduces a counterexample showing the non-existence of an $L^1$ trace inequality for fractional gradients.

## Key findings

- Established a fractional gradient inequality with a specific constant.
- Showed that fractional gradients do not admit an $L^1$ trace inequality unlike the classical case.
- Provided a counterexample demonstrating the failure of weak-type Riesz transform estimates on certain fractional Hausdorff content spaces.

## Abstract

In this paper we give a streamlined proof of an inequality recently obtained by the author: For every $\alpha \in (0,1)$ there exists a constant $C=C(\alpha,d)>0$ such that \begin{align*} \|u\|_{L^{d/(d-\alpha),1}(\mathbb{R}^d)} \leq C \| D^\alpha u\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \end{align*} for all $u \in L^q(\mathbb{R}^d)$ for some $1 \leq q<d/(1-\alpha)$ such that $D^\alpha u:=\nabla I_{1-\alpha} u \in L^1(\mathbb{R}^d;\mathbb{R}^d)$. We also give a counterexample which shows that in contrast to the case $\alpha =1$, the fractional gradient does not admit an $L^1$ trace inequality, i.e. $\| D^\alpha u\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)}$ cannot control the integral of $u$ with respect to the Hausdorff content $\mathcal{H}^{d-\alpha}_\infty$. 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 $L^1(\mathcal{H}^{d-\beta}_\infty)$, $\beta \in [1,d)$. It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to $\beta \in (0,1)$.