Page 17 - Textos de Matemática Vol. 34

REMARKS ON SOBOLEV EMBEDDINGS 7
Proof. Let k ∈ N and put lk = (b − a)/k. We decompose the interval I = (a, b) in the form I = ∪kj=0Ik, where
I0 =(a,a1), Ij =[aj,aj+1)forj=1,...,k−1, andIk =[ak,b),
a1 =a+lk/2, aj =a1+(j−1)lk forj=1,...,k−1, andak =b−lk/2.
with
Now let f ∈ C01(I) and write
Pf(t) = k−1 f(cj)χj(t), j=1
wherecj =1(aj+1+aj)andχj isthecharacteristicfunctionofIj.Definec0 =a 2
and ck = b. Then by (2.3),
p k ′ p γp ′ p
∥E0f−Pf|Lp(I)∥ = j=0Hcjf |Lp(Ij) ≤ 2k(b−a)∥f |Lp(I)∥ . 0
Since C01(I) is dense in W1p(I) it follows that
∥E0 − P ∥ ≤ γp (b − a),
andasrankP ≤k−1weseethat
2k ak(E0)≤ γp(b−a).
2k
To obtain the reverse inequality, decompose I into k non-overlapping intervals
Jj of equal length, so that I = ∪kj=1Jj and each Jj has length (b − a)/k. Let 0 < ε < γp (b − a), let φ1, ..., φk be functions which are translations of each other
2k
and are such that φj ∈ W1p(Jj) and
0
∥E0φj |Lp(Jj)∥
for j = 1,...,k; let M be the subspace of W1p(I) spanned by the φj. It is clear that
dim M = k and for any φ ∈ M, φ ̸= 0, we have ∥φ′|Lp(I)∥ ≤ 1 .
φ′j|Lp(Jj) ≤ 1
0
γp(b−a)−ε 2k
Hence
We claim that
∥E0φ | Lp(I)∥ γp (b − a) − ε 2k
μ γp(b−a),I≥k. 0 2k
μ0(ε, I) ≤ ν0(ε, I).