Journal of Inequalities in Pure and
Applied Mathematics
APPLICATION LEINDLER SPACES TO THE REAL
INTERPOLATION METHOD
volume 5, issue 3, article 68,
2004.
VADIM KUKLIN
Department of Mathematics
Voronezh State University
Voronezh, 394693, Russia.
EMail: craft_kiser@mail.ru
Received 30 March, 2004;
accepted 21 April, 2004.
Communicated by: H. Bor
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
071-04
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Abstract
The paper is devoted to the important section the Fourier analysis in one variable (AMS subject classification 42A16). In this paper we introduce Leindler
space of Fourier - Haar coefficients, so we generalize [2, Theorem 7.a.12] and
application to the real method spaces.
Application Leindler Spaces to
the Real Interpolation Method
2000 Mathematics Subject Classification: 26D15, 40A05, 42A16, 40A99, 46E30,
47A30, 47A63.
Key words: Leindler sequence space of Fourier - Haar coefficients, Lorentz space,
Haar functions, real method spaces.
Vadim Kuklin
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Lemmas and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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3
6
9
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J. Ineq. Pure and Appl. Math. 5(3) Art. 68, 2004
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1.
Introduction
A Banach space E[0, 1] is said to be a rearrangement invariant space (r.i) provided f ∗ (t) ≤ g ∗ (t) for any t ∈ [0, 1] and g ∈ E implies that f ∈ E and
kf kE ≤ kgkE , where g ∗ (t) is the rearrangement of |g(t)|. Denote by ϕE
the fundamental function of (r.i) space E such that ϕE = kκe (t)k (see, [1,
p. 137]). Given τ > 0, the dilation operator στ f (t) = f ( τt ), t ∈ [0, 1] and
min(1, τ ) ≤ kστ kE→E ≤ max(1, τ ). Denote by
ln kστ kE→E
αE = lim
,
τ →+0
ln τ
ln kστ kE→E
βE = lim
τ →∞
ln τ
the Boyd indices of E. In general, 0 ≤ αE ≤ βE ≤ 1.
0
The
R 1 associated space to E is the space of all measurable functions f (t) such
that 0 f (t)g(t)dt < ∞ for every g(t) ∈ E endowed with the norm
Z
kf (t)kE 0 =
sup
kg(t)kE ≤1
1
f (t)g(t)dt.
0
For every (r.i) space E space the embedding E ⊂ E 00 is isometric. If an (r.i)
space E is separable, then (χkn ) is everywhere dense in E.
Denote by Ψ the set of increasing concave functions ψ(t) ≥ 0 on [0, 1] with
ψ(0) = 0. Then each function ψ(t) ∈ Ψ generates the Lorentz space Λ(ψ)
endowed with the norm
Z 1
kg(t)kΛ(ψ) =
g ∗ (t)dϕ(t) < ∞.
0
Application Leindler Spaces to
the Real Interpolation Method
Vadim Kuklin
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For every (r.i) space E space the embedding E ⊂ E 00 is isometric.
Let be Ω the set of (n, k) such that 1 ≤ k ≤ 2n , n ∈ N∪ {0}. Put χ00 ≡ 1. If
(n, k) ∈ Ω,

k−1
< t < 2k−1
,

2n
2n+1
 1,
χkn (t) =
−1, 2k−1
< t < 2kn ,
2n+1


0,
for any t ∈ k−1
, 2kn .
2n
The set of functions χkn is called the Haar functions, normalized
in L∞ [0, 1]
k
(see [2, p. 15-18]). If an (r.i) space E is separable, then χn everywhere dense
in E. Given f (t) ∈ L1 . The Fourier-Haar coefficients are given by
Z 1
n
cn,k (f ) = 2
f (t)χkn (t)dt.
Application Leindler Spaces to
the Real Interpolation Method
Vadim Kuklin
0
Put g(t) =
cn,k χkn
P
for any g ∈ L1 [0, 1].
(n,k)∈Ω
A Banach sequence space E is said to be a rearrangement invariant space
(r.i) provided that k(an )kE ≤ k(a∗n )kE ,where a∗n the rearrangement of sequence
(an )n∈N i.e.
(
)
a∗n = inf
sup |ai | : J ⊂ N, card(J) <n .
i∈N\J
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It is maximal if the unit ball BE is closed in the poinwise convergence topology inducted by the space A of all real sequences. This condition is equivalent
to E # = E 0 , where
(
)
∞
X
#
E = (bn )n∈N ⊂ A :
|an bn | < ∞, (an )n∈N ⊂ E
n=1
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J. Ineq. Pure and Appl. Math. 5(3) Art. 68, 2004
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is the Kother dual of E. Clearly, E # is a maximal Banach space under the norm
(∞
)
X
k(bn )kE # = sup
|an bn | < ∞ : k(an )kE ≤ 1 .
n=1
Denoting λ = (λn )∞
n=1 be a sequence of positive numbers. We shall use the
following notation (see [3, pp. 517-518]):
Λn =
∞
X
λk and
Λ(c)
n
k=n
=
∞
X
Application Leindler Spaces to
the Real Interpolation Method
λk Λ−c
k , (Λ1 < ∞);
k=n
Vadim Kuklin
furthermore, for c ≥ 0. By analogy with [3, pp. 517-518] we define Leindler
sequence space of Fourier-Haar coefficients, for p > 0, c ≥ 0, with the norm:
k(cn,k )∞
n=1 kλ(p,c)
=
∞
X
n=1
!p ! p1
n
λn Λ−c
n
2
X
−n
|cn,k | 2
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< ∞.
k=1
Why do we consider the sequence (cn,k )∞
this question
n=1 ? The answer to
−n
p
follows from [2, Theorem 7.a.3], i.e. g ∈ Λ(ψ) ⇔ sup 2 cn,1 (g) < ∞.
0<t≤1
Here, as usual, X ,→ Y stands for the continuous embedding, that is, kgkY ≤
C kgkX for some C > 0 and every g∈X. The sign ∼
= means that these spaces
coincide to with within equivalence of norms.
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2.
Problems
By [2, Theorem 7.a.12] for p = 2 we have
! 12
X
∞
2n
X
X
cn,k χkn 2−n
c2n,k
.
=
(n,k)∈Ω
n=1
k=1
L2
If for (cn,k )∞
we put p = 2, c = 0, λn = 1, then
n=1 λ(p,c)
k(cn,k )∞
n=1 kλ(2,0)
Application Leindler Spaces to
the Real Interpolation Method
X
k
≤M
cn,k χn (n,k)∈Ω
.
L2
Denote by

T
cn,k χkn  = (cn,k )(n,k)∈Ω .
(n,k)∈Ω
Hence by [1, Chapter 2, §5, Theorem 5.5] we have the operator bounded from
Λ(ψ) into λ(2, 0). In general we consider
Problem 1. Let 0 < c < 1, 1 < p < ∞. Whether there exists a operator T
bounded from Λ(ψ) into λ(p, c)?
Let (E0 , E1 ) be a compatible pair of Banach spaces. We recall
K(t, g) = K(t, g, E0 , E1 ) =
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
X
Vadim Kuklin
inf
g=g0 +g1 ,gi ∈Ei (i=0,1)
kg0 kE0 + t kg1 kE1 .
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Here g ∈ E0 + E1 , 0 < t ≤ 1. If 0 < θ < 1, 1 ≤ p ≤ ∞, then the spaces
(E0 , E1 )θ,p endowed with the norm
Z
kgk(E0 ,E1 )θ,p =
1
−θ p dt
(K(t, g)t )
0
t
p1
< ∞, iff p < ∞
and
kgk(E0 ,E1 )θ,p = sup K(t, g)t−θ < ∞, iff p = ∞
0<t<1
are called real method spaces. Let 0 ≤ α0 < α1 < 1, ψ0 (t) = tα0 , ψ1, (t) = tα1 ,
∼
0 < θ < 1, 1 ≤ p ≤ ∞, ψ(t) =
when does the equivalence
t
.
ψ(t)
In [5, §2, p. 174] the problem was solved:
∼
∼ (Λ(ψ0 ), Λ(ψ1 ))θ,p ∼
= M (ψ0 ), M (ψ1 )
Application Leindler Spaces to
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Vadim Kuklin
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θ,p
holds?
∼
∼ We consider the embedding (Λ(ψ0 ), Λ(ψ1 ))θ,p ,→ M (ψ0 ), M (ψ1 )
. Let
θ,p
0 ≤ α0 = α1 < 1, ψ(t) = tα , 0 < θ < 1, 1 < p ≤ ∞.
Problem 2. Whether there exists 0 < c < 1, 1 < p < ∞ such that
T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p ?
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In this article we consider Leindler sequence space of Fourier-Haar coefficients λ(p, c).
To prove our theorems we need the following Theorem 1 (see [4]).
J. Ineq. Pure and Appl. Math. 5(3) Art. 68, 2004
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Theorem 1. If p > 1, 0 ≤ c < 1, then
!p p X
∞
n
∞
X
X
p
−c
p−c p
λn Λn
|ak | ≤
λ1−p
n Λn an .
1
−
c
n=1
n=1
k=1
The constant is best possible.
Application Leindler Spaces to
the Real Interpolation Method
Vadim Kuklin
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3.
Lemmas and Theorems
n
Lemma 3.1. Let 1 < p < ∞, 0 ≤ c < 1 and sup 2− p cn,1 (g) < ∞. Then the
0<t≤1
operator T is bounded from Λ(ψ) into λ(p, c).
Proof. By [2, Theorem 4.a.1] for 1 < p < ∞ we have
p
p
p
Z 1 X
Z 1 X
Z 1 X
2n
∞ X
2n
k
k
p
k
cn,k χn dt ≤
cn,k χn dt ≤ 2
cn,k χn dt,
0 0 0 k=1
n=l k=1
(n,k)∈Ω
where n ≤ l ≤ ∞.
On the other hand,
p
Z 1
Z 1
2n
2n
2n
X
X
X
p
cn,k χkn dt =
2−n
|cn,k | dt = 2−n
|cn,k |p .
0 0
k=1
k=1
Lp
Therefore,
! p1
2n
2−n
X
k=1
|cn,k |p
≤ 2 kgkLp .
Application Leindler Spaces to
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From the above and [1, Chapter 2, §5, Theorem 5.5] we get
! p1
∞
X
k(cn,k )∞
λn Λ−c
kgkΛ(ψ) .
n=1 kλ(p,c) ≤ 2
n
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n=1
Hence the operator T is bounded from Λ(ψ) into λ(p, c). This proves the assertion.
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Remark 3.1. In the Lemma 3.1 the condition 0 < c < 1, 1 < p < ∞ is
necessary for the operator T.
We shall formulate the sufficient condition of boundedness of the operator T
from Λ(ψ) into λ(p, c).
n
Theorem 3.2. Let 0 ≤ c < 1, sup 2− p cn,1 (g) < ∞. For of boundedness the
0<t≤1
operator T bounded from Λ(ψ) into λ(p, c) is sufficient that 2 ≤ p < ∞.
Proof. By Theorem 1 and Hölder’s inequality we have

 p1
∞
1
c
X
X
−1 1−
p
k(cn,k )∞
λnp Λn p 
|cn,k |p 2−n  .
n=1 kλ(p,c) ≤
1 − c n=1
Application Leindler Spaces to
the Real Interpolation Method
Vadim Kuklin
(n,k)∈Ω
Now using [2, Theorem 7.a.12 (c. 2)] and [1, Chapter 2, §5, Theorem 5.5] we
obtain that
∞
1
c X
X
−1 1− p
p
p
k
(cn,k )∞ λ
Λ
c
χ
.
≤
n
n
n,k
n
n=1 λ(p,c)
1 − c n=1
(n,k)∈Ω
Λ(ψ)
This finishes the proof.
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Remark 3.2. If 1 ≤ p < 2, 0 < c < 1, then by [2, Theorem 7.a.12 (c. 1)]
T : Λ(ψ) 9 λ(p, c).
n
Theorem 3.3. Let 0 ≤ c < 1, 2 ≤ p ≤ ∞, sup 2− p cn,1 (g) < ∞. Then
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0<t≤1
T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p .
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Proof. Clearly, by Hölder’s inequality the estimate
(cn,k )∞ ≤
n=1 λ(p,c)
∞
p X p1 −1 1− pc λn Λn (cn,k )∞
n=1 `2
1 − c n=1
holds. It is known that the operator T is bounded from L2 into `2 . Then from
the above and [1, Chapter 2, §5, Theorem 5.5] we obtain
K(t, (cn,k )∞
n=1 , λ(p, c), λ(p, c)) ≤ K(t, g, Λ(ψ), Λ(ψ)).
Hence T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p . This completes the proof.
Application Leindler Spaces to
the Real Interpolation Method
Vadim Kuklin
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References
[1] S.G. KREIN, YU. I. PETUNIN AND E.M. SEMENOV, Interpolation linear
operators, Nauka, Moscow, 1978, in Russian; Math. Mono., Amer. Math.
Soc., Providence, RI, 1982, English translation.
[2] I. NOVIKOV AND E. SEMENOV, Haar Series and Linear Operators,
Mathematics and Its Applications, Kluwer Acad. Publ., 1997.
[3] L. LEINDLER, Hardy - Bennett - Type Theorems, Math. Ineq. and Appl.,
4 (1998), 517–526.
[4] L. LEINDLER, Two theorems of Hardy-Bennett - type, Acta Math. Hung.,
79(4) (1998), 341–350.
[5] E. SEMENOV, On the stability of the real interpolation method in the class
of rearrangement invariant spaces, Israel Math. Proceedings, 13 (1999),
172–182.
Application Leindler Spaces to
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Vadim Kuklin
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