Page 16 - Textos de Matemática Vol. 34

6 DAVID E. EDMUNDS 0
On the other hand, given any f ∈ W1p(I) ,
f(t) = (Haf′)(t) = −(Hbf′)(t)
and so
∥f |Lp(I)∥p =∥Haf′ |Lp((a,c))∥p +∥Hbf′ |Lp((c,b))∥p
p γp p′ pγp p01
≤ 2(b−a) ∥f |Lp(I)∥ = 2(b−a) f|Wp(I) .
This shows that ∥E0∥ ≤ γp (b − a) and completes the proof. 2

Remark 2.1. Let W1 (I), Lp,C(I) be the quotient spaces Wp1(I)/C, Lp(I)/C p,C
respectively, endowed with the norms
[f]|W1 (I):=∥f′ |L (I)∥
Let EC : W1 p,C
p,C p
and
(2.8) ∥EC∥=∥Hc |Lp(I)→Lp(I)∥= γp(b−a), 2
where c = (a + b)/2.
We can now give precise formulae for the approximation numbers of the embed-
ding E0. It is convenient at this point to introduce the numbers
(2.9) μ0(ε, I) := max{dim S : θ(S) ≤ 1/ε} and ν0(ε, I) := max{m : am(E0) ≥ ε} for all small enough ε > 0. Here
θ(S) := sup ∥f′ | Lp(I)∥ f∈S,f̸=0 ∥f | Lp(I)∥
0
for all linear subspaces S of W1p(I) and am(E0) is the mth approximation number
of the embedding E0.
Theorem2.2. Supposethat1<p<∞.Thenforallk∈N,
(2.10) ak(E0) = γp(b − a). 2k
∥[f ] | Lp,C (I )∥ := inf ∥f − c | Lp (I )∥ . c∈C
(I) → Lp,C(I) be the natural embedding. Then arguments similar to those given in the proof above show that