1152 CHAPTER 33. FOURIER ANALYSIS IN Rn

Proof:u,i (x) =

∫U

Φ,i (x−y) f (y)dy

and so u,i j ∈ Lp (U) and f → u,i j is continuous by Lemma 33.5.5.With this preparation, it is possible to consider the Helmholtz decomposition. Let F ∈

Lp (U ;Rn) and define

φ (x)≡∫

U∇Φ(x−y) ·F(y)dy. (33.5.72)

Then by Lemma 33.5.5,φ , j =CnF̃j +∑

iKi jF̃i ∈ Lp (Rn)

and the mapping F→∇φ is continuous from Lp (U ;Rn) to Lp (U ;Rn).Now suppose F ∈C∞

c (U ;Rn). Then

φ (x) =∫

U

n

∑i=1− ∂

∂yi (Φ(x−y)Fi (y))+Φ(x−y)∇ ·F(y)dy

=∫

UΦ(x−y)∇ ·F(y)dy

and so by Lemma 33.5.3,∇ ·∇φ = ∆φ =−∇ ·F.

This continues to hold in the sense of weak derivatives if F is only in Lp (U ;Rn) becauseby Minkowski’s inequality and 33.5.72 the map F→φ is continuous. Also note that for F∈C∞

c (U ;Rn),

φ (x) =∫

BΦ(y)∇ ·F(x−y)dy.

Next define π : Lp (U ;Rn)→ Lp (U ;Rn) by

πF =−∇φ , φ (x) =∫

U∇Φ(x−y) ·F(y)dy.

It was already shown that π is continuous, linear, and ∇ ·πF =∇ ·F. It is also true that π isa projection. To see this, let F ∈C∞

c (U ;Rn). Then for B large enough,

π2F(x) = −∇

∫B

Φ(z)∇ ·πF(x− z)dz

= −∇

∫B

Φ(z)∇ ·∇∫

BΦ(w)∇ ·F(x− z−w)dwdz

= −∇

∫B

Φ(z)∇ ·F(x− z)dz = πF(x).

Since π is continuous and C∞c (U ;Rn) is dense in Lp (U ;Rn), it follows that π2F =πF for all

F ∈ Lp (U ;Rn). This proves the following theorem which is the Helmholtz decomposition.