1278 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

≤(∫

Rn|wn+1 (x)|n dmn

)1/n

·(n

∏j=1

(∫Rn−1

(∫R

∣∣w j (x)∣∣n dxn+1

)dmn−1

)1/(n−1))(n−1)/n

=

(∫Rn|wn+1 (x)|n dmn

)1/n n

∏j=1

(∫Rn

∣∣w j (x)∣∣n dmn

)1/n

=n+1

∏j=1

(∫Rn

∣∣w j (x)∣∣n dmn

)1/n

This proves the lemma.

Lemma 37.1.9 If φ ∈C∞c (Rn) and n≥ 1, then

||φ ||n/(n−1) ≤1n√

n

n

∑j=1

∣∣∣∣∣∣∣∣ ∂φ

∂x j

∣∣∣∣∣∣∣∣1.

Proof: The case where n = 1 is obvious if n/(n−1) is interpreted as ∞. Assume thenthat n > 1 and note that for ai ≥ 0,

nn

∏i=1

ai ≤

(n

∑j=1

ai

)n

In fact, the term on the left is one of many terms of the expression on the right. Therefore,taking nth roots

n

∏i=1

a1/ni ≤ 1

n√

n

n

∑j=1

ai.

Then observe that for each j = 1,2, · · · ,n,

|φ (x)| ≤∫

∞

−∞

∣∣∣φ , j (x)∣∣∣dx j

so

||φ ||n/(n−1)n/(n−1) ≡

∫Rn|φ (x)|n/(n−1) dmn

≤∫Rn

n

∏j=1

(∫∞

−∞

∣∣∣φ , j (x)∣∣∣dx j

)1/(n−1)

dmn

and from Lemma 37.1.8 this is dominated by

≤n

∏j=1

(∫Rn

∣∣∣φ , j (x)∣∣∣dmn

)1/(n−1)

.