A SIMPLE PROOF OF SCHIPP’S THEOREM
YANBO REN AND JUNYAN REN
Department of Mathematics and Physics
Henan University of Science and Technology
Luoyang 471003, China.
EMail: ryb7945@sina.com.cn renjy03@lzu.edu.cn
Received:
13 September, 2007
Accepted:
06 November, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
60G42.
Key words:
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
Martingale inequality, Property ∆.
J
I
Abstract:
In this paper we give a simple proof of Schipp’s theorem by using a basic martingale inequality.
Page 1 of 7
Acknowledgements:
The authors thank referees for their valuable comments.
Go Back
Full Screen
Close
Contents
1
Introduction
3
2
Proof of Theorem 1.1
6
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
J
I
Page 2 of 7
Go Back
Full Screen
Close
1.
Introduction
The property ∆ of operators was introduced by F. Schipp in [1] and he proved that,
if (Tn , n ∈ P) are a series
P of operators with property ∆ and some boundedness,
then the operator T = ∞
n=1 Tn is of type (p, p) (p ≥ 2). We resume this result
as Theorem 1.1. F. Schipp applied Theorem 1.1 to prove the significant result that
the Fourier-Vilenkin expansions of the function f ∈ Lp converge to f in Lp -norms
(1 < p < ∞).
Throughout this paper P and N denote the set of positive integers and the set of
nonnegative integers, respectively. We always use C, C1 and C2 to denote constants
which may be different in different contexts.
Let (Ω, F, µ) be a complete probability space
S and {Fn , n ∈ N} an increasing
sequence of sub-σ-algebras of F with F = σ( n Fn ). Denote by E and En expectation operator and conditional expectation operators relative to Fn for n ∈ N,
respectively. We briefly write Lp instead of the complex Lp (Ω, F, µ) while the
1
norm (or quasinorm) of this space is defined by kf kp = (E[|f |p ]) p . A martingale
f = (fn , n ∈ N) is an adapted, integrable sequence with En fm = fn for all n ≤ m.
For a martingale f = (fn )n≥0 we say that f = (fn )n≥0 is Lp (1 ≤ p < ∞)-bounded
if kf kp = supn kfn kp < ∞. If 1 < p < ∞ and f ∈ Lp then f˜ = (En f )n≥0 is a Lp bounded martingale, and kf kp = f˜ (see [2]). We denote the maximal function
p
and the martingale differences of a martingale f = (fn , n ∈ N) by f ∗ = supn∈N |fn |
and dfn = fn −fn−1 (n ∈ P), df0 = f0 , respectively. We recall that for a Lp -bounded
martingale f = (fn )n≥0 (p > 1):
(1.1)
kf kp ≤ kf ∗ kp ≤ C kf kp .
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
J
I
Page 3 of 7
Go Back
Full Screen
Close
We will use the following martingale inequality (see Weisz [2]):
! 12 ∞
X
2
∗
(1.2) kf kp ≤ C1 En−1 |dfn |
n=0
p
≤ C2 kf ∗ k (2 ≤ p < ∞).
sup
|df
|
+ C1 n
p
n∈N
p
Now let ∆0 = E, ∆n = En − En−1 (n ∈ P). It is easy to see that
(1.3)
En ◦ Em = Emin(n,m) ,
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
∆n ◦ ∆m = δmn ∆n (n, m ∈ P),
where δmn is the Kronecker symbol and ◦ denotes the composition of functions. Let
{Tn , n ∈ P}, Tn : Lp → Lq (1 ≤ p, q < ∞), be a sequence of operators. We say
that the operators {Tn , n ∈ P} are uniformly of type (Fn−1 , p, q) if there exists a
constant C > 0 such that for all f ∈ Lp
1
q
1
p
(En−1 [|Tn f |q ]) ≤ C(En−1 [|f |p ]) .
A sequence of operators {Tn , n ∈ P} is said to satisfy the ∆-condition, if
Title Page
Contents
JJ
II
J
I
Page 4 of 7
Tn ◦ ∆n = ∆n ◦ Tn = Tn (n ∈ P).
Go Back
From the equations in (1.3) it is easy to see that the ∆−condition is equivalent to the
following conditions:
Full Screen
(1.4)
Tn ◦ En = En ◦ Tn = Tn ,
Tn ◦ En−1 = En−1 ◦ Tn = 0(n ∈ P).
P
Pm
∞
∗
For f ∈ Lp , set T fP=
It is obvious
n=1 Tn f and T f = sup |
n=1 Tn f |. S
p
that the operator series ∞
T
f
is
convergent
at
each
point
of
L
=
n=1 n
n L (Fn ) if
{Tn , n ∈ P} satisfy the ∆-condition, since for f ∈ Lp (FN ), Tn f = Tn ◦∆n ◦EN f =
0. We resume Schipp’s theorem as follows:
(1.5)
Close
Theorem 1.1 ([1]). Let (Tn , n ∈ P) be a sequence of operators with the property ∆,
and let p ≥ 2. If for r = 2, p and n ∈ P, the operators (Tn , n ∈ P) are uniformly
of type (Fn−1 , r, r), then the operator T is of type (r, r), i.e., there exists a constant
C > 0 such that for all f ∈ Lr :
kT f kr ≤ C kf kr .
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
J
I
Page 5 of 7
Go Back
Full Screen
Close
2.
Proof of Theorem 1.1
Proof. Let P
f ∈ Lr (r ≥ 2). Then by (1.5), it is easy to see that the stochastic
sequence ( nk=1 Tk f, Fn ) is a martingale. By (1.1) we only need to prove that
kT ∗ f kr ≤ C kf ∗ kr .
Since the operators Tn are uniformly of type (Fn−1 , 2, 2) and (Fn−1 , r, r), it follows from (1.2) and (1.4) that
! 12 X
∞
2
∗
+ C sup |Tn f |
kT f kr ≤ C E
|T
f
|
n−1
n
n∈N
n=0
r
r
1
! ∞
X
2
2
+ C sup |Tn ◦ ∆n f |
=C
E
|T
◦
∆
f
|
n−1
n
n
n∈N
n=0
r
r
1
!
∞
X
2
2
+ C sup |∆n f |
≤C
E
|∆
f
|
n−1
n
n∈N
n=0
r
r
≤ C kf ∗ kr .
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
J
I
Page 6 of 7
Go Back
Full Screen
Remark 1. The theorem is proved for r = 2 and r > 2 in a unified way, which differs
from the original proof.
Close
References
[1] F. SCHIPP, On Lp -norm convergence of series with respect to product systems,
Analysis Math., 2(1976), 49–64.
[2] F. WEISZ, Martingale Hardy spaces and their applications in Fourier analysis,
Lecture Notes in Math., Vol. 1568, Berlin: Springer, 1994.
Proof of Schipp’s Theorem
Yanbo Ren and Junyan Ren
vol. 8, iss. 4, art. 117, 2007
Title Page
Contents
JJ
II
J
I
Page 7 of 7
Go Back
Full Screen
Close