A Noninequality for the Fractional Gradient
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 weaktype 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 weaktype estimate for the Riesz transforms extends to $\beta \in (0,1)$.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.05541
 Bibcode:
 2019arXiv190605541S
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis
 EPrint:
 12 pages