on interpolation of vector-valued banach lattices and calderón–lozanovskii construction
TRANSCRIPT
Math. Nachr. 278, No. 6, 735 – 737 (2005) / DOI 10.1002/mana.200510268
Erratum
On interpolation of vector-valued Banach lattices and Calderon–Lozanovskii construction
[Math. Nachr. 227, 63–80 (2001)]
Ming Fan∗1
1 Institute of Mathematics, Natural Sciences and Engineering, Dalarna University, 781 88 Borlange, Sweden
c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction and preliminaries
Inequality (1.4) on p. 65 should read as
‖T ‖F(X ),F(Y ) ≤ c ϕ(‖T ‖0, ‖T ‖1
).
2 On complex interpolation of Schechter
There is a mistake in the proof of Lemma 2.1 (on line 11 of p. 68). In fact, we need one more variant of theCθ(n)-methods, and some corresponding changes should be made in Section 2.
(a) Line 25 from the words “The equivalence” — line 26 on p. 66 should be changed as bellow:
Now we need another variant of the Cθ(±n)-methods. Let H1(Dθ, X
)be the Banach space of all functions
f : D −→ ΣX , where f are analytic in the interior of D for which f(eit
) ∈ Xj for t ∈ Iθj (j = 0, 1) and
‖f‖H1 =1
2π(1 − θ)
∫Iθ0
∥∥f(eit
)∥∥0dt ∨ 1
2πθ
∫Iθ1
∥∥f(eit
)∥∥1dt < ∞ .
We can define the interpolation space CDθ(n,1)
(X
)similar to CDθ(n)
(X
)by replacing A
(Dθ,X
)with
H1(Dθ, X
). The equivalence
Cθ(n)
(X
)= CDθ(n,1)
(X
)= CDθ(n)
(X
)for n ∈ Z . (2.1)
(b) On the last line of p. 66, “ ± 1” should be deleted, and the following sentence should be inserted:
For n = ±1, we have the norm estimate
‖x‖CDθ(±1,1) = ‖x‖CDθ(±1) ≤ ‖x‖Cθ(±1) ≤ 3√
34
‖x‖CDθ(±1,1)
by [FK, Th. 5.2].
(c) Lines 16–21 of p. 67 should be changed as below:
∗ e-mail: [email protected], Phone: +46 (0) 23778853, +46 (0) 23778054
c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
736 Fan: Erratum
Lemma 2.1 If f ∈ H1(Dθ,X
), then there exists h ∈ H1
(Dθ, X
)with h(k) = f(k) for k = 0, 1, . . . , n
such that
‖h‖H1 ≤ cn ϕθ,n
(1
2π(1 − θ)
∫Iθ0
∥∥f(eit
)∥∥0dt,
12πθ
∫Iθ1
∥∥f(eit
)∥∥1dt
), (2.4)
where c1 = 1 and cn < 3n for n ≥ 2.
P r o o f. For f ∈ H1(Dθ, X
), let
M0 =1
2π(1 − θ)
∫Iθ0
∥∥f(eit
)∥∥0dt and M1 =
12πθ
∫Iθ1
∥∥f(eit
)∥∥1dt .
(d) Line 11 of p. 68 should read
‖h‖H1 ≤ ‖φ∞‖∞‖f exp(−ψ)‖H1 = ϕθ,1
(M0,M1
).
(e) On line 14 of p. 68: “h ∈ A(Dθ,X
)” should be replaced by “h ∈ H1
(Dθ, X
)”.
(f) On line 21 of p. 68: “‖f · exp(−ψ)‖∞” should be replaced by “‖f · exp(−ψ)‖H1”.
(g) In Theorem 2.3 on p. 69: The sentences “These interpolation spaces are exact when n = ±1” in part (i)and “These inclusions are contractive” in part (ii) should be deleted. In addition, part (iii) should be inserted asbelow:
(iii) In particular, Φθ,1
(CDθ(±1,1)
(X
))0are exact interpolation spaces for the couple
(Φ0
(X0
),Φ1
(X1
)),
and the inclusion
CDθ(±1,1)
(Φ0
(X0
),Φ1
(X1
)) ⊆ Φθ,1
(CDθ(±1,1)
(X
))0
is norm decreasing.
(h) The first two paragraphs on p. 70 should be changed as below:
P r o o f. By the equivalence (2.1), we may use different variants of the Cθ(n)-methods for the different partsin the proof.
For the inclusion
X = CDθ(n,1)
(Φ0
(X0
),Φ1
(X1
)) ⊆ Φθ,n
(CDθ(n,1)
(X
))0
in parts (ii) & (iii), we can assume that x =(xν
)ν∈ Φ0
(X0
) ∩ Φ1
(X1
)with ‖x‖X < 1 by the density of
Φ0
(X0
) ∩ Φ1
(X1
)in X . Thus there exists f =
(fν
)ν∈ H1
(Dθ,
(Φ0
(X0
),Φ1
(X1
)))with xν = fν(n) and
‖f‖H1 =1
2π(1 − θ)
∫Iθ0
∥∥f(eit
)∥∥Φ0
(X0
) ∨ 12πθ
∫Iθ1
∥∥f(eit
)∥∥Φ1
(X1
) < 1
for j = 0, 1. Then xν ∈ ∆X ⊆ CDθ(n,1)
(X
)and fν ∈ H1
(Dθ,X
)for every ν. This implies by Lemma 2.1
and the definition of Φθ,n that
‖x‖Φθ,n(CDθ(n,1)(X))
=∥∥∥(
‖xν‖CDθ(n,1)(X))ν
∥∥∥Φθ,n
≤ cn
∥∥∥∥(ϕθ,n
(1
2π(1 − θ)
∫Iθ0
∥∥fν
(eit
)∥∥X0dt,
12πθ
∫Iθ1
∥∥fν
(eit
)∥∥X1dt
))ν
∥∥∥∥Φθ,n
≤ cn
∥∥∥∥(
12π(1 − θ)
∫Iθ0
∥∥fν
(eit
)∥∥X0dt
)ν
∥∥∥∥Φ0
∨∥∥∥∥(
12πθ
∫Iθ1
∥∥fν
(eit
)∥∥X1dt
)ν
∥∥∥∥Φ1
≤ cn‖f‖H1 < cn .
(i) On the first line of p. 71: “ψ =(ψν
)ν” should be inserted between the words “function” and “on”.
c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Math. Nachr. 278, No. 6 (2005) / www.mn-journal.com 737
4 Some remarks on uniform convexity, UMD property and related topics
In this section we have following corrections:
(j) On line 3 of p. 76: “n ≥ −1” should be replaced by “n ≤ 1”.
(k) On line 4 of p. 76: The word “also” should be replaced by “equivalent to”.
(l) On line 4 of p. 76: “n > 1” should be replaced by “n < −1”.
c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim