# Endpoint $L^1$ estimates for Hodge systems

@inproceedings{Hernndez2021EndpointE, title={Endpoint \$L^1\$ estimates for Hodge systems}, author={Felipe Hern{\'a}ndez and Bogdan Raiță and Daniel Spector}, year={2021} }

In this paper we give a simple proof of the endpoint Besov-Lorentz estimate ‖IαF‖Ḃ0,1 d/(d−α),1 (Rd;Rk) ≤ C‖F‖L1(Rd;Rk) for all F ∈ L1(Rd;Rk) which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with L1 data via fractional integration for exterior derivatives.

#### References

SHOWING 1-10 OF 30 REFERENCES

Estimates for L-1 vector fields under higher-order differential conditions

- Mathematics
- 2008

We prove that an L-1 vector field whose components satisfy some condition on k-th order derivatives induce linear functionals on the Sobolev space W-1,W-n(R-n). Two proofs are provided, relying on… Expand

A note on div curl inequalities

- Mathematics
- 2005

(Note: We state this result, and others below, for smooth functions or forms of compact support. More general formulations then follow by standard limiting arguments). The result above is remarkable… Expand

New estimates for the Laplacian, the div-curl, and related Hodge systems

- Mathematics
- 2004

Abstract We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To… Expand

New Directions in Harmonic Analysis on $L^1$

- Mathematics
- 2019

The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there… Expand

New estimates for elliptic equations and Hodge type systems

- Mathematics
- 2007

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension n, with data in L1. We also present related results concerning differential… Expand

Limiting Fractional and Lorentz Space Estimates of Differential Forms

- Mathematics
- 2010

We obtain estimates in Besov, Triebel-Lizorkin and Lorentz spaces of differential forms on R-n in terms of their L-1 norm.

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

Fractional Integration and Optimal Estimates for Elliptic Systems.

- Physics, Mathematics
- 2020

In this paper we prove the following optimal Lorentz embedding for the Riesz potentials: Let $\alpha \in (0,d)$. There exists a constant $C=C(\alpha,d)>0$ such that \[ \|I_\alpha F… Expand

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

Function Spaces and Potential Theory

- Mathematics
- 1995

The subject of this book is the interplay between function space theory and potential theory. A crucial step in classical potential theory is the identification of the potential energy of a charge… Expand