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
|
| Online Access: | https://hdl.handle.net/1721.1/154302 |
| _version_ | 1824458200378769408 |
|---|---|
| author | Hernandez, Felipe Spector, Daniel |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Hernandez, Felipe Spector, Daniel |
| author_sort | Hernandez, Felipe |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:00:37Z |
| format | Article |
| id | mit-1721.1/154302 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2025-02-19T04:22:07Z |
| publishDate | 2024 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1543022025-01-04T04:58:09Z Fractional integration and optimal estimates for elliptic systems Hernandez, Felipe Spector, Daniel Massachusetts Institute of Technology. Department of Mathematics 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. 2024-04-29T15:05:54Z 2024-04-29T15:05:54Z 2024-04-26 2024-04-28T03:15:40Z Article http://purl.org/eprint/type/JournalArticle 0944-2669 1432-0835 https://hdl.handle.net/1721.1/154302 Calculus of Variations and Partial Differential Equations. 2024 Apr 26;63(5):117 PUBLISHER_CC en 10.1007/s00526-024-02722-8 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Science and Business Media LLC Springer Berlin Heidelberg |
| spellingShingle | Hernandez, Felipe Spector, Daniel Fractional integration and optimal estimates for elliptic systems |
| title | Fractional integration and optimal estimates for elliptic systems |
| title_full | Fractional integration and optimal estimates for elliptic systems |
| title_fullStr | Fractional integration and optimal estimates for elliptic systems |
| title_full_unstemmed | Fractional integration and optimal estimates for elliptic systems |
| title_short | Fractional integration and optimal estimates for elliptic systems |
| title_sort | fractional integration and optimal estimates for elliptic systems |
| url | https://hdl.handle.net/1721.1/154302 |
| work_keys_str_mv | AT hernandezfelipe fractionalintegrationandoptimalestimatesforellipticsystems AT spectordaniel fractionalintegrationandoptimalestimatesforellipticsystems |