A SIMPLE PROOF OF SCHIPP’S THEOREM

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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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: Martingale inequality, Property∆.

Abstract: In this paper we give a simple proof of Schipp’s theorem by using a basic mar- tingale inequality.

Acknowledgements: The authors thank referees for their valuable comments.

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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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

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1. Introduction

The property∆of operators was introduced by F. Schipp in [1] and he proved that, if (T_n, n ∈ P) are a series of operators with property ∆ and some boundedness, then the operator T = P∞

n=1T_n is of type (p, p) (p ≥ 2). We resume this result as Theorem1.1. F. Schipp applied Theorem1.1 to prove the significant result that the Fourier-Vilenkin expansions of the functionf ∈ L^p converge to f inL^p-norms (1< p <∞).

Throughout this paperP andNdenote the set of positive integers and the set of nonnegative integers, respectively. We always useC,C₁ andC₂to denote constants which may be different in different contexts.

Let (Ω,F, µ) be a complete probability space and {F_n, n ∈ N} an increasing sequence of sub-σ-algebras of F with F = σ(S

nF_n). Denote by E and En ex- pectation operator and conditional expectation operators relative toF_n for n ∈ N, respectively. We briefly write L^p instead of the complex L^p(Ω,F, µ) while the norm (or quasinorm) of this space is defined by kfk_p = (E[|f|^p])^p1. A martingale f = (f_n, n∈ N)is an adapted, integrable sequence withEⁿf_m =f_n for alln ≤m.

For a martingalef = (f_n)_n≥0 we say thatf = (f_n)_n≥0 isL_p (1≤p <∞)-bounded ifkfk_p = sup_nkf_nk_p <∞. If1< p < ∞andf ∈L^p thenf˜= (Eⁿf)n≥0is aL^p- bounded martingale, and kfk_p =

f˜

p

(see [2]). We denote the maximal function and the martingale differences of a martingalef = (f_n, n∈N)byf^∗ = sup_n∈N|f_n| anddf_n=f_n−fn−1(n ∈P), df₀ =f₀, respectively. We recall that for aL_p-bounded martingalef = (f_n)n≥0 (p >1):

(1.1) kfk_p ≤ kf^∗k_p ≤Ckfk_p.

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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We will use the following martingale inequality (see Weisz [2]):

(1.2) kf^∗k_p ≤C₁

∞

X

n=0

Eⁿ⁻¹

|df_n|²

!¹₂ p

+C₁

sup

n∈N

|df_n| p

≤C₂kf^∗k_p (2≤p <∞).

Now let∆₀ =E,∆_n =En−Eⁿ⁻¹(n ∈P). It is easy to see that (1.3) Eⁿ◦E^m =E^min(n,m), ∆_n◦∆_m =δ_mn∆_n(n, m∈P),

whereδ_mnis the Kronecker symbol and◦denotes the composition of functions. Let {T_n, n ∈ P}, T_n : L^p → L^q (1 ≤ p, q < ∞), be a sequence of operators. We say that the operators{T_n, n ∈ P} are uniformly of type (Fn−1, p, q) if there exists a constantC > 0such that for allf ∈L^p

(En−1[|T_nf|^q])^1q ≤C(En−1[|f|^p])^1p.

A sequence of operators{T_n, n∈P}is said to satisfy the∆-condition, if (1.4) T_n◦∆_n= ∆_n◦T_n=T_n(n ∈P).

From the equations in (1.3) it is easy to see that the∆−condition is equivalent to the following conditions:

(1.5) T_n◦En=En◦T_n=T_n, T_n◦En−1 =En−1◦T_n= 0(n∈P).

For f ∈ L^p, set T f = P∞

n=1T_nf and T^∗f = sup|Pm

n=1T_nf|. It is obvious that the operator seriesP∞

n=1T_nf is convergent at each point ofL =S

nL^p(F_n)if {T_n, n∈P}satisfy the∆-condition, since forf ∈L^p(F_N),T_nf =T_n◦∆_n◦ENf = 0. We resume Schipp’s theorem as follows:

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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Theorem 1.1 ([1]). Let(T_n, n∈P)be a sequence of operators with the property∆, and letp ≥ 2. If forr = 2, pandn ∈ P,the operators (T_n, n ∈ P)are uniformly of type(Fn−1, r, r), then the operatorT is of type(r, r), i.e., there exists a constant C >0such that for allf ∈L^r:

kT fk_r ≤Ckfk_r.

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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2. Proof of Theorem 1.1

Proof. Let f ∈ L^r (r ≥ 2). Then by (1.5), it is easy to see that the stochastic sequence(Pn

k=1T_kf,F_n)is a martingale. By (1.1) we only need to prove that kT^∗fk_r ≤Ckf^∗k_r.

Since the operatorsTnare uniformly of type(Fn−1,2,2)and(Fn−1, r, r), it fol- lows from (1.2) and (1.4) that

kT^∗fk_r ≤C

∞

X

n=0

Eⁿ⁻¹

|T_nf|²

!¹₂ r

+C

sup

n∈N

|T_nf| r

=C

∞

X

n=0

Eⁿ⁻¹

|T_n◦∆_nf|²

!¹₂ r

+C

sup

n∈N

|T_n◦∆_nf| r

≤C

∞

X

n=0

Eⁿ⁻¹

|∆_nf|²

!¹₂ r

+C

sup

n∈N

|∆_nf| r

≤Ckf^∗k_r.

Remark 1. The theorem is proved forr= 2andr >2in a unified way, which differs from the original proof.

Proof of Schipp’s Theorem Yanbo Ren and Junyan Ren vol. 8, iss. 4, art. 117, 2007

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References

[1] F. SCHIPP, OnL^p-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.