Page 15 - Textos de Matemática Vol. 34

REMARKS ON SOBOLEV EMBEDDINGS 5
It turns out that the extremals, when 1 < p < ∞, are precisely the constant multiples of
f(x) = cosp(πpx/2).
Whenp=1or∞,γp =1.Therearenoextremalswhenp=1;allnon-zero
constants are extremals when p = ∞. 0
Now let E0 = E0(I) : W1p(I) → Lp(I) be the natural embedding. Theorem 2.1. Let 1<p≤∞ and put c=(a+b)/2. Then
∥E0∥= 1γp(b−a). 2
Proof. Given ε > 0, there exists φ ∈ Lp ((c, b)) such that
∥Hbφ | Lp((c, b))∥ ≥ (∥Hb | Lp((c, b)) → Lp((c, b))∥ − ε) ∥φ | Lp((c, b))∥
1
= 2γp(b−a)−ε ∥φ|Lp((c,b))∥,
the equality following from (2.3). Define a function ψ on I by φ(2c−t), a<t≤c,
and put
ψ(t) = φ(t),
(Haψ)(t),
c < t < b,
a<t≤c, c < t < b.
Ψ(t) = −(Hbψ)(t), Then ψ and Ψ are symmetric about c,
∥ψ | Lp(I)∥ = 21/p ∥φ | Lp((c, b))∥ ∥Ψ | Lp(I)∥ = 21/p ∥Ψ | Lp((c, b))∥ = 21/p ∥Hbφ | Lp((c, b))∥
1/p 1
≥2 2γp(b−a)−ε ∥φ|Lp((c,b))∥
1 1 ′
= 2γp(b−a)−ε ∥ψ|Lp(I)∥= 2γp(b−a)−ε ∥Ψ |Lp(I)∥.
Since Ψ ∈ W0 1(I) we see that ∥E ∥ ≥ γp (b − a). p02
and