Volume 8 (2007), Issue 4, Article 119, 3 pp.
A SIMPLE PROOF OF SCHIPP’S THEOREM
YANBO REN AND JUNYAN REN
D EPARTMENT OF M ATHEMATICS AND P HYSICS
H ENAN U NIVERSITY OF S CIENCE AND T ECHNOLOGY
L UOYANG 471003, C HINA .
ryb7945@sina.com.cn
renjy03@lzu.edu.cn
Received 13 September, 2007; accepted 06 November, 2007
Communicated by S.S. Dragomir
A BSTRACT. In this paper we give a simple proof of Schipp’s theorem by using a basic martingale inequality.
Key words and phrases: Martingale inequality, Property ∆.
2000 Mathematics Subject Classification. 60G42.
1. I NTRODUCTION
The property ∆ of operators was introduced by F. Schipp in [1] and he proved that, if (Tn , n ∈
P)
P∞are a series 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 and {Fn , n ∈ N} an increasing sequence
S
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 norm (or quasinorm) of this space is defined by
1
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
p
˜
f = (En f )n≥0 is a L -bounded martingale, and kf k = f˜ (see [2]). We denote the maximal
p
p
function and the martingale differences of a martingale f = (fn , n ∈ N) by f ∗ = supn∈N |fn |
The authors thank referees for their valuable comments.
298-07
2
YANBO R EN AND J UNYAN R EN
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 .
We will use the following martingale inequality (see Weisz [2]):
(1.2)
! 12 ∞
X
2
∗
kf kp ≤ C1 |dfn |
≤ C2 kf ∗ kp (2 ≤ p < ∞).
En−1 |dfn |
+C1 sup
n∈N
n=0
p
p
Now let ∆0 = E, ∆n = En − En−1 (n ∈ P). It is easy to see that
(1.3)
En ◦ Em = Emin(n,m) ,
∆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
1
(En−1 [|Tn f |q ]) q ≤ C(En−1 [|f |p ]) p .
A sequence of operators {Tn , n ∈ P} is said to satisfy the ∆-condition, if
(1.4)
Tn ◦ ∆n = ∆n ◦ Tn = Tn (n ∈ P).
From the equations in (1.3) it is easy to see that the ∆−condition is equivalent to the following
conditions:
(1.5)
Tn ◦ En = En ◦ Tn = Tn ,
Tn ◦ En−1 = En−1 ◦ Tn = 0(n ∈ P).
P
Pm
∗
For fP∈ Lp , set T f = ∞
T
f
and
T
f
=
sup
|
n
n=1
n=1
S Tnpf |. It is obvious 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
p
∆-condition, since for f ∈ L (FN ), Tn f = Tn ◦ ∆n ◦ EN f = 0. We resume Schipp’s theorem
as follows:
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 .
2. P ROOF OF T HEOREM 1.1
r
Proof.
Pn Let f ∈ L (r ≥ 2). Then by (1.5), it is easy to see that the stochastic sequence
( k=1 Tk f, Fn ) is a martingale. By (1.1) we only need to prove that
kT ∗ f kr ≤ C kf ∗ kr .
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 119, 3 pp.
http://jipam.vu.edu.au/
A S IMPLE P ROOF OF S CHIPP ’ S T HEOREM
3
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
n−1
n∈N
n=0
r
r
!1 ∞
X
2
2
=C
+
C
sup
|T
◦
∆
f
|
En−1 |Tn ◦ ∆n f |
n
n
n∈N
n=0
r
r
! 12 X
∞
2
+
C
sup
≤C
En−1 |∆n f |
|∆
f
|
n
n∈N
n=0
r
r
≤ C kf ∗ kr .
Remark 2.1. The theorem is proved for r = 2 and r > 2 in a unified way, which differs from
the original proof.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 119, 3 pp.
http://jipam.vu.edu.au/