1271

Therefore, for such x ∈U ∩Uk,37.0.11 reduces to

vtk (x) =∫Rn

u(y)ψk (y)φ l(tk) (x+ tkak−y)dy

=∫

Uu(y)ψk (y)φ l(tk) (x+ tkak−y)dy.

It follows that for |α| ≤ m, and x ∈U ∩Uk

Dα vtk (x) =∫

Uu(y)ψk (y)Dα

φ l(tk) (x+ tkak−y)dy

=∫

UDα (uψk)(y)φ l(tk) (x+ tkak−y)dy

=∫Rn

˜Dα (uψk)(y)φ l(tk) (x+ tkak−y)dy

=∫Rn

˜Dα (uψk)(x+ tkak−y)φ l(tk) (y)dy. (37.0.12)

Actually, this formula holds for all x∈U. If x∈U but x /∈Uk, then the left side of the aboveformula equals zero because, as noted above, spt

(vtk

)⊆Uk. The integrand of the right side

equals zero unless

x ∈ B(

0,1

l (tk)

)+ spt(ψk)− tkak ⊆Uk

by 37.0.9 and here x /∈Uk.Next an estimate is obtained for

∣∣∣∣Dα vtk −Dα (uψk)∣∣∣∣

Lp(U). By 37.0.12,∣∣∣∣Dα vtk −Dα (uψk)

∣∣∣∣Lp(U)

≤(∫U

(∫Rn

∣∣∣ ˜Dα (uψk)(x+ tkak−y)− ˜Dα (uψk)(x)∣∣∣φ l(tk) (y)dy

)p

dx)1/p

≤∫Rn

φ l(tk) (y)(∫

U

∣∣∣ ˜Dα (uψk)(x+ tkak−y)− ˜Dα (uψk)(x)∣∣∣p dx

)1/p

dy

≤ ε

2k

whenever tk is taken small enough. Pick tk this small and let wk ≡ vtk . Thus

||Dα wk−Dα (uψk)||Lp(U) ≤ε

2k

and wk ∈C∞c (Rn) . Now let

J (x)≡∞

∑k=1

wk.

Since the Uk are locally finite and spt(wk)⊆Uk for each k, it follows

Dα J =∞

∑k=0

Dα wk