1356 CHAPTER 41. ELLIPTIC REGULARITY

=∫

U(Dα f )zdy. (41.1.27)

Let Û1 be as indicated in the following picture

R

Rn−1U

V

U1Û1

Now apply the induction hypothesis to Û1 in order to write∣∣∣∣∣∣∣∣∂ (Dτ w)∂yr

∣∣∣∣∣∣∣∣H1(Û1)

≤ ||Dτ w||H2(Û1)

≤C(|| f ||Hk−1(U)+∑

s||hs||Hk(U)+ ||w||Hk(U)

).

Since αrs ∈ Ck−1,1(U), it follows that each term from the sum in 41.1.27 satisfies an

inequality of the form ∣∣∣∣∣∣∣∣C (τ)Dα−τ (αrs)∂ (Dτ w)

∂yr

∣∣∣∣∣∣∣∣H1(Û1)

≤

C(|| f ||Hk−1(U)+∑

s||hs||Hk(U)+ ||w||Hk(U)

)and consequently, ∣∣∣∣∣

∣∣∣∣∣∑τ<α

C (τ)Dα−τ (αrs)∂ (Dτ w)

∂yr +Dα (hs)

∣∣∣∣∣∣∣∣∣∣H1(Û1)

≤

C(|| f ||Hk−1(U)+∑

s||hs||Hk(U)+ ||w||Hk(U)

). (41.1.28)

Now consider 41.1.27. The equation remains true if you replace U with Û1 and requirethat spt(z) ⊆ Û1. Therefore, by Lemma 41.1.1 on Page 1349 there exists a constant, Cindependent of w such that

||Dα w||H2(U1)≤C

(||Dα f ||L2(Û1) +

||Dα w||H1(Û1)+