Author:
Hernandez Felipe,Spector Daniel
Abstract
AbstractIn this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div–Curl system: If $$F \in L^1(\mathbb {R}^3;\mathbb {R}^3)$$
F
∈
L
1
(
R
3
;
R
3
)
satisfies $$\text {div}F=0$$
div
F
=
0
in the sense of distributions, then the function $$Z=\text {curl} (-\Delta )^{-1} F$$
Z
=
curl
(
-
Δ
)
-
1
F
satisfies $$\begin{aligned} \text {curl } Z&= F \\ \text {div } Z&= 0 \end{aligned}$$
curl
Z
=
F
div
Z
=
0
and there exists a constant $$C>0$$
C
>
0
such that $$\begin{aligned} \Vert Z\Vert _{L^{3/2,1}(\mathbb {R}^3;\mathbb {R}^3)} \le C\Vert F\Vert _{L^{1}(\mathbb {R}^3;\mathbb {R}^3)}. \end{aligned}$$
‖
Z
‖
L
3
/
2
,
1
(
R
3
;
R
3
)
≤
C
‖
F
‖
L
1
(
R
3
;
R
3
)
.
Our proof relies on a new endpoint Hardy–Littlewood–Sobolev inequality for divergence free measures which we obtain via a result of independent interest, an atomic decomposition of such objects.
Funder
Ministry of Education
Ministry of Science and Technology, Taiwan
Hertz Foundation
Publisher
Springer Science and Business Media LLC
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