1280 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

Also, n−1n −

p−1p = n−p

np and so, dividing both sides by the last term yields

(∫|φ |

npn−p dmn

) n−pnp

≤ rn√

n

n

∑i=1

(∫ ∣∣φ ,i

∣∣p)1/p

≤ rn√

n||φ ||1,p,Rn .

Letting q = npn−p , it follows 1

q = n−pnp = 1

p −1n and

||φ ||q ≤r

n√

n||φ ||1,p,Rn .

Now let f ∈ W m,p (Rn) and let ||φ k− f ||1,p,Rn → 0 as k → ∞. Taking another sub-sequence, if necessary, you can also assume φ k (x)→ f (x) a.e. Therefore, by Fatou’slemma,

|| f ||q ≤ lim infk→∞

(∫Rn|φ k (x)|

q dmn

)1/q

≤ lim infk→∞

rn√

n||φ k||1,p,Rn = || f ||1,p,Rn .

This proves the theorem.

Corollary 37.1.11 Suppose mp < n. Then W m,p (Rn) ⊆ Lq (Rn) where q = npn−mp and the

identity map, id : W m,p (Rn)→ Lq (Rn) is continuous.

Proof: This is true if m = 1 according to Theorem 37.1.10. Suppose it is true for m−1where m > 1. If u ∈W m,p (Rn) and |α| ≤ 1, then Dα u ∈W m−1,p (Rn) so by induction, forall such α,

Dα u ∈ Lnp

n−(m−1)p (Rn) .

Thus u ∈W 1,q1 (Rn) whereq1 =

npn− (m−1) p

By Theorem 37.1.10, it follows that u ∈ Lq (Rn) where

1q=

n− (m−1) pnp

− 1n=

n−mpnp

.

This proves the corollary.There is another similar corollary of the same sort which is interesting and useful.

Corollary 37.1.12 Suppose m≥ 1 and j is a nonnegative integer satisfying jp < n. Then

W m+ j,p (Rn)⊆W m,q (Rn)

forq≡ np

n− jp(37.1.16)

and the identity map is continuous.