33.5. HELMHOLTZ DECOMPOSITIONS 1151

Letting

Φεi j ≡

{0 if |y|< ε,Φ,i j (y) if |y| ≥ ε,

it followswi, j (x) =Cnδ i j f (x)+Φ

εi j ∗ f̃ (x)+ e(ε) .

By the theory of singular integrals, there exists a continuous linear map,

Ki j ∈L (Lp (Rn) ,Lp (Rn))

such thatKi j f ≡ lim

ε→0Φ

εi j ∗ f .

Therefore, letting ε → 0,wi, j =Cnδ i j f +Ki j f̃

whenever f ∈C∞c (U).

Now let f ∈ Lp (U), let|| fk− f ||Lp(U)→ 0,

where fk ∈C∞c (U), and let

wki (x) =

∫U

Φ,i (x−y) fk (y)dy.

Then it follows as before that wki → wi in Lp (U) and

wki, j =Cnδ i j fk +Ki j f̃k.

Now let φ ∈C∞c (U).

wi, j (φ) ≡ −∫

Uwiφ , jdx =− lim

k→∞

∫U

wki φ , jdx

= limk→∞

∫U

wki, jφdx = lim

k→∞

∫U

(Cnδ i j f̃k +Ki j f̃k

)φdx

=∫

U

(Cnδ i j f̃ +Ki j f̃

)φdx.

It followswi, j =Cnδ i j f̃ +Ki j f̃

and this proves the lemma.

Corollary 33.5.6 In the situation of Theorem 33.5.4, all weak derivatives of u of order 2are in Lp (U) and also f → u,i j is a continuous map.