37.1. EMBEDDING THEOREMS FOR W m,p (Rn) 1275

Definition 37.1.4 Let α ∈ (0,1] and K a compact subset of Rn

Cα (K)≡ { f ∈C (K) : ρα ( f )+ || f || ≡ || f ||α< ∞}

where|| f || ≡ || f ||

∞≡ sup{| f (x)| : x ∈ K}

and

ρα ( f )≡ sup{| f (x)− f (y)||x−y|α

: x,y ∈ K, x ̸= y}.

Then (Cα (K) , ||·||α) is a complete normed linear space called a Holder space.

The verification that this is a complete normed linear space is routine and is left for you.More generally, one considers the following class of Holder spaces.

Definition 37.1.5 Let K be a compact subset of Rn and let λ ∈ (0,1]. Cm,λ (K) denotesthe set of functions, u which are restrictions of functions defined on Rn to Ksuch that for|α| ≤ m,

Dα u ∈C (K)

and if |α|= m,

Dα u ∈Cλ (K) .

Thus C0,λ (K) =Cλ (K) . The norm of a function in Cm,λ (K) is given by

||u||m,λ ≡ sup|α|=m

ρλ (Dα u)+ ∑

|α|≤m||Dα u||

∞.

Lemma 37.1.6 Let m be a positive integer, K a compact subset of Rn, and let 0 < β < λ ≤1. Then the identity map from Cm,λ (K) into Cm,β (K) is compact.

Proof: First note that the containment is obvious because for any function, f , if

ρλ ( f )≡ sup

{| f (x)− f (y)||x−y|λ

: x,y ∈ K, x ̸= y

}< ∞,

Then

ρβ ( f ) ≡ sup

{| f (x)− f (y)||x−y|β

: x,y ∈ K, x ̸= y

}

= sup

{| f (x)− f (y)||x−y|λ

|x−y|λ−β : x,y ∈ K, x ̸= y

}

≤ sup

{| f (x)− f (y)||x−y|λ

diam(K)λ−β : x,y ∈ K, x ̸= y

}< ∞.

Suppose the identity map, id, is not compact. Then there exists ε > 0 and a sequence,{ fk}∞

k=1 ⊆Cm,λ (K) such that || fk||m,λ < M for all k but || fk− fl ||β ≥ ε whenever k ̸= l. By