Fractional integration and optimal estimates for elliptic systems

Abstract

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.