Fractional integration and optimal estimates for elliptic systems
In 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 $$F \in L^1(\mathbb {R}^3;\mathbb {R}^3)$$...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Science and Business Media LLC
2024
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| Online Access: | https://hdl.handle.net/1721.1/154302 |
| Summary: | In 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. |
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