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Portugaliae Mathematica

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Volume 76, Issue 2, 2019, pp. 153–168
DOI: 10.4171/PM/2031

Published online: 2020-02-13

A noninequality for the fractional gradient

Daniel Spector[1]

(1) National Chiao Tung University, Hsinchu, Taiwan and Okinawa Institute of Science and Technology Graduate University, Jap

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 $$\|u\|_{L^{d/(d-\alpha),1}(\mathbb{R}^d)} \leq C \| D^\alpha u\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)}$$ 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 $\beta \in (0,1)$.

Keywords: $L^1$-Sobolev inequality, Lorentz spaces, trace inequality

Spector Daniel: A noninequality for the fractional gradient. Port. Math. 76 (2019), 153-168. doi: 10.4171/PM/2031