# A noninequality for the fractional gradient

@article{Spector2019ANF, title={A noninequality for the fractional gradient}, author={Daniel Spector}, journal={arXiv: Classical Analysis and ODEs}, year={2019} }

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… Expand

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#### References

SHOWING 1-10 OF 25 REFERENCES

Some remarks on L1 embeddings in the subelliptic setting

- Mathematics
- 2019

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ be the homogeneous dimension of $G$ and… Expand

A boxing Inequality for the fractional perimeter

- Mathematics
- 2017

We prove the Boxing inequality: $$\mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{\alpha+d}},$$ for every… Expand

An $L^1$-type estimate for Riesz potentials

- Mathematics
- 2014

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $\alpha \in (0, N)$: \[
\|I_\alpha u\|_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C… Expand

On limiting trace inequalities for vectorial differential operators

- Mathematics, Physics
- Indiana University Mathematics Journal
- 2021

We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},d\mu)} \leq c… Expand

Fractional vector analysis based on invariance requirements (critique of coordinate approaches)

- Physics
- Continuum Mechanics and Thermodynamics
- 2019

The paper discusses the fractional operators $$\begin{aligned} \nabla ^\alpha , \quad {\mathrm{div}}^\alpha , \quad (-\,\Delta )^{\alpha /2}, \end{aligned}$$ ∇ α , div α , ( - Δ ) α / 2 , where… Expand

An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces

- Mathematics
- 2002

Abstract. An elementary proof of the Brezis and Mironescu theorem on the boundedness and continuity of the composition operator:
$ W^{s,p}({\bf R}^n)\cap W^{1,sp} ({\bf R}^n)\to W^{s,p}({\bf R}^n) $… Expand

Regularity for a fractional p-Laplace equation

- Mathematics
- 2016

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural… Expand

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

- Mathematics
- Journal of Functional Analysis
- 2019

Abstract We introduce the new space B V α ( R n ) of functions with bounded fractional variation in R n of order α ∈ ( 0 , 1 ) via a new distributional approach exploiting suitable notions of… Expand

$L^p$-theory for fractional gradient PDE with VMO coefficients

- Mathematics
- 2015

In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends… Expand

Non-linear ground state representations and sharp Hardy inequalities

- Mathematics, Physics
- 2008

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a… Expand