1344 CHAPTER 40. KORN’S INEQUALITY

Therefore, ∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Rn

≡ sup{∣∣∣∣∫Rn

f∂φ

∂xidx∣∣∣∣ : ||φ ||1,2,Rn ≤ 1

}

= sup{∣∣∣∣∫U−

f[

∂φ

∂xi−3

∂ψ1∂xi

+2∂ψ2∂xi

]dx∣∣∣∣ : ||φ ||1,2,Rn ≤ 1

}≤Cg

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,U−

where Cg is a constant which depends on g. This inequality along with 40.1.2 yields

n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Rn

≤Cg

(n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,U−

).

The inequality,|| f ||−1,2,Rn ≤Cg || f ||−1,2,U−

follows from 40.1.4 and the equation,∫Rn

f φdx =∫

U−f φdx−3

∫U−

f (x,yn)ψ1 (x,yn)dxdyn

+2∫

U−f (x,yn)ψ2 (x,yn)dxdyn

which results in the same way as before by changing variables using the definition of f offU−. This proves the lemma.

The next lemma is a simple application of Fourier transforms.

Lemma 40.1.3 If f ∈ L2 (Rn) , then the following formula holds.

Cn || f ||0,2,Rn =n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Rn

+ || f ||−1,2,Rn

Proof: For φ ∈C∞c (Rn)

||φ ||1,2,Rn ≡(∫

Rn

(1+ |t|2

)|Fφ |2 dt

)1/2

is an equivalent norm to the usual Sobolev space norm for H10 (Rn) and is used in the

following argument which depends on Plancherel’s theorem and the fact that F(

∂φ

∂xi

)=

tiF (φ) . ∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Rn

≡ sup{∣∣∣∣∫Rn

∂φ

∂xif dx∣∣∣∣ : ||φ ||1,2 ≤ 1

}

=Cn sup{∣∣∣∣∫Rn

ti (Fφ)(F f )dt∣∣∣∣ : ||φ ||1,2 ≤ 1

}