Page 13 - Textos de Matemática Vol. 34

REMARKS ON SOBOLEV EMBEDDINGS 3
An important rˆole in what follows will be played by the Hardy operator Hc (a ≤ c ≤ b) defined by
(2.2) (Hcf)(x) = Note that by H¨older’s inequality,
x c
f(t)dt, x ∈ I.
b x p
p p/p′
b
f(t)dt dx≤∥f |Lp(I)∥ |x−c|
aca
p +1 p
dx
≤(b−a)p′ ∥f|Lp(I)∥ .
It follows that, viewed as a map from Lp(I) to itself, Hc is bounded and has norm
bounded from above by (b − a). Also observe that
∥Ha | Lp(I) → Lp(I)∥ = ∥Hb | Lp(I) → Lp(I)∥
= (b − a) ∥H0 | Lp((0, 1))) → Lp((0, 1))∥
(2.3) = (b − a)γp.
The constant γp can be evaluated. In fact, the even more general quantity (2.4) γp,q := ∥H0 | Lp((0, 1)) → Lq((0, 1))∥ (p, q ∈ [1, ∞])
is known explicitly: it was determined by Erhardt Schmidt in 1940 in an apparently little-known paper [20]; the case p = q had been treated even earlier by Levin [15] in 1938. Unaware of this earlier work, much later and independently, γp was calculated in [9] and γp,q in [2] and [5]. It turns out that
(2.5) γ = r 1 −1(p′)1/pq1/q′ /B(1/p′, 1/q), p,q r
where
1=1+1−1,1 =1−1 r qps′ s
and B is the beta function given in terms of the gamma function Γ by B(s, t) = Γ(s)Γ(t)/Γ(s + t).
In particular,
(2.6) γp = γp,p = π−1p1/p′ (p′)1/p sin(π/p).