37.1. EMBEDDING THEOREMS FOR W m,p (Rn) 1279

Hence ∏ni=1 a1/n

i ≤ 1n√n ∑

nj=1 ai

||φ ||n/(n−1) ≤n

∏j=1

(∫Rn

∣∣∣φ , j (x)∣∣∣dmn

)1/n

≤ 1n√

n

n

∑j=1

∫Rn

∣∣∣φ , j (x)∣∣∣dmn

=1n√

n

n

∑j=1

∣∣∣∣φ ,i

∣∣∣∣1

and this proves the lemma.The above lemma is due to Gagliardo and Nirenberg.With this lemma, it is possible to prove a major embedding theorem which follows.

Theorem 37.1.10 Let 1≤ p < n and 1q = 1

p −1n . Then if f ∈W 1,p (Rn) ,

|| f ||q ≤1n√

n(n−1) p

n− p|| f ||1,p,Rn .

Proof: From the definition of W 1,p (Rn) , C1c (Rn) is dense in W 1,p. Here C1

c (Rn) is thespace of continuous functions having continuous derivatives which have compact support.The desired inequality will be established for such φ and then the density of this set inW 1,p (Rn) will be exploited to obtain the inequality for all f ∈W 1,p (Rn). First note thatthe case where p= 1 follows immediately from the above lemma and so it is only necessaryto consider the case where p > 1.

Let φ ∈C1c (Rn) and consider |φ |r where r > 1. Then a short computation shows |φ |r ∈

C1c (Rn) and ∣∣∣|φ |r,i∣∣∣= r |φ |r−1 ∣∣φ ,i

∣∣ .Therefore, from Lemma 37.1.9,(∫

|φ |rn

n−1 dmn

)(n−1)/n

≤ rn√

n

n

∑i=1

∫|φ |r−1 ∣∣φ ,i

∣∣dmn

≤ rn√

n

n

∑i=1

(∫ ∣∣φ ,i

∣∣p)1/p(∫ (|φ |r−1

)p/(p−1)dmn

)(p−1)/p

.

Now choose r such that(r−1) p

p−1=

rnn−1

.

That is, let r = p(n−1)n−p > 1 and so rn

n−1 = npn−p . Then this reduces to(∫

|φ |np

n−p dmn

)(n−1)/n

≤ rn√

n

n

∑i=1

(∫ ∣∣φ ,i

∣∣p)1/p(∫|φ |

npn−p dmn

)(p−1)/p

.