# Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

Authors: Ushangi Goginava; Károly Nagy
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The main aim of this paper is to prove that the logarithmic
means of quadratical partial sums of the double Walsh-Kaczmarz series
does not improve the convergence in measure. In other words, we prove
that for any Orlicz space, which is not a subspace ofL log+ L(I2), the
set of the functions the logarithmic means of quadratical partial sums
of the double Walsh-Kaczmarz series of which converge in measure is
of first Baire category.


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## Body

## 1. Definitions and Notations

We denote byL0=L0(I2) the Lebesgue space of functions that are measurable and finite almost everywhere on I2=[0,1)×[0,1). mes(A) is the Lebesgue measure of the set A⊂I2. The constants appearing in this paper are denoted by c.LetLΦ=LΦ(I2) be the Orlicz space [1] generated by Young’s function Φ; that is, Φ is convex, continuous, even function such that Φ(0)=0 and(1.1)lim⁡u→+∞Φ(u)u=+∞,lim⁡u→0Φ(u)u=0.This space is endowed with the norm(1.2)‖f‖LΦ(I2)=inf⁡{k>0:∫I2Φ(|f(x,y)|k)dxdy≤1}.
In particular case, if Φ(u)=uln(1+u), u>0, then the corresponding space is denoted by Llog+L(I2).We denote the set of nonnegative integers byN. By a dyadic interval in I:=[0,1) we mean one of the form [p/2n,(p+1)/2n) for some p∈N, 0≤p<2n. Given n∈N and x∈[0,1), let In(x) denote the dyadic interval of length 2-n which contains the point x.Every pointx∈I can be written in the following form:(1.3)x=∑k=0∞xk2k+1=:(x0,x1,…,xn,…),xk∈{0,1}.
In that case when there are two different forms we choose the one for which limk→∞xk=0. We use the notation(1.4)ej:=12j+1=(0,…,0,xj=1,0,…).It is well-known that [2, 3](1.5)In(x0,…,xn-1):=In(x)=[p2n,p+12n),
where p=∑j=0n-1xj2n-1-j.Letr0(x) be a function defined by(1.6)r0(x)={1,ifx∈[0,12),-1,ifx∈[12,1),r0(x+1)=r0(x).
The Rademacher functions are defined by(1.7)rn(x)=r0(2nx),n≥0,x∈[0,1).Letw0,w1,… represent the Walsh functions, that is, w0(x)=1, and if n=∑i=0∞ni2i is a positive integer with ni∈{0,1}, then(1.8)wn(x):=∏i=0∞(ri(x))ni.
The order of the n is denoted by |n|:=max{j∈N:nj≠0}.The Walsh-Kaczmarz functions are given byκ0:=1, and for n≥1(1.9)κn(x):=r|n|(x)∏k=0|n|-1(r|n|-1-k(x))nk.
For A∈N and x∈I, Skvorcov [4] defined a transformation τA:I→I by(1.10)τA(x):=∑k=0A-1xA-k-12-(k+1)+∑j=A∞xj2-(j+1).
By the definition of τA, we have (see [4])(1.11)κn(x)=r|n|(x)wn-2|n|(τ|n|(x))(n∈N,x∈I).The Dirichlet kernels are defined by(1.12)Dnα(x):=∑k=0n-1αk(x),
where αk=wk for all k∈N or κk for all k∈N.It is well-known that [2, 4](1.13)Dnκ(x)=D2|n|(x)+w2|n|(x)Dn-2|n|w(τ|n|(x)),(1.14)D2n(x):=D2nw(x)=D2nκ(x)={2n,ifx∈[0,12n),0,ifx∈[12n,1).Thenth Fejér means of the Walsh-(Kaczmarz-)Fourier series of function f are given by(1.15)σnα(f,x):=1n∑j=1nSjα(f,x),
where(1.16)Snα(f,x)=∑k=0n-1f̂α(k)αk(x)
is the nth partial sum of the Fourier series and f̂α(n):=∫Ifαn(n∈N) is said to be the nth Walsh-(Kaczmarz-)Fourier coefficient of the function f.The Nörlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows:(1.17)tnα(f,x)=1ln∑k=1n-1Skα(f,x)n-k,Fnα(t)=1ln∑k=1n-1Dkα(t)n-k,
where ln=∑k=1n-11/k.The Kronecker product (αm,n:n,m∈N) of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,(1.18)αm,n(x,y)=αm(x)αn(y).Iff∈L(I2), then the number f̂α(m,n):=∫I2fαm,n(n,m∈N) is said to be the (m,n)th Walsh-(Kaczmarz-)Fourier coefficient of f. The rectangular partial sums of double Fourier series with respect to both system are defined by(1.19)Sm,nα(f,x,y)=∑i=0m-1∑j=0n-1f̂α(i,j)αi(x)αj(y).The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by(1.20)tnα(f,x,y)=1ln∑k=1n-1Sk,kα(f,x,y)n-k,Fnα(x,y)=1ln∑k=1n-1Dkα(x)Dkα(y)n-k.It is evident that(1.21)tnα(f,x,y)-f(x,y)=∫I2[f(x⊕t,y⊕s)-f(x,y)]Fnα(t,s)dtds,
where ⊕ denotes the dyadic addition [3]. The nth Marcinkiewicz kernels are given by(1.22)Knα(x,y):=1n∑k=1nDkα(x)Dkα(y).In 1948 Šneĭder [5] showed that the inequality(1.23)limsup⁡n→∞Dnκ(x)log⁡n≥C>0
holds a.e. for the Walsh-Kaczmarz Dirichlet kernel. This inequality shows that the behavior of the Walsh-Kaczmarz system is worse than the behavior of the Walsh system in the Paley enumeration. This “spreadness” property of the kernel makes it easier to construct examples of divergent Fourier series [6]. On the other hand, Schipp [7] and Young [8] in 1974 proved that the Walsh-Kaczmarz system is a convergence system. Skvorcov in 1981 [4] showed that the Fejér means with respect to the Walsh-Kaczmarz system converge uniformly to f for any continuous functions f. For any integrable functions, Gát [9] proved that the Fejér means with respect to the Walsh-Kaczmarz system converge a.e. to the function. Recently, Gát’s result was generalized by Simon [10, 11]. The a.e. convergence of the Walsh-Kaczmarz-Marcinkiewicz means of integrable functions was discussed by the second author [12].The partial sumsSnw(f) of the Walsh-Fourier series of a function f∈L(I), I=[0,1) converge in measure on I [2]. The condition f∈Llog+L(I2) provides convergence in measure on I2 of the rectangular partial sums Sn,mw(f) of double Walsh-Fourier series [13]. The first example of a function from classes wider than Llog+L(I2) with Sn,nw(f) divergent in measure on I2 was obtained in [14, 15]. Moreover, in [16] Tkebuchava proved that in each Orlicz space wider than Llog+L(I2) the set of functions where quadratic Walsh-Fourier sums converge in measure on I2 is of first Baire category (see Goginava [17] for Walsh-Kaczmarz-Fourier series). In the paper [18] it was showed that the Nörlund logarithmic means tnκf of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that tnκf converge in measure on I for all functions f∈L(I) and tn,mκf converge in measure on I2 for all functions f∈Llog+L(I2).At last, we note that the Walsh-Nörlund logarithmic means are closer to the partial sums than to the classical logarithmic means or the Fejér means. Namely, it was proved that there exists a function in a certain class of functions and a set with positive measure, such that the Walsh-Nörlund logarithmic means of the function diverge on the set [19].For results with respect to logarithmic means of cubical and rectangular partial sums of two-dimensional Walsh-Fourier series, see [17, 19–22].In the present paper we investigate convergence in measure of logarithmic means of quadratical partial sums(1.24)1ln∑i=1n-1Si,i(f,x,y)n-i
of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of Llog+L(I2), the set of the functions where logarithmic means converges in measure is of first Baire category. From this result it follows that (Corollary 2.2) in classes wider than Llog+L(I2) there exists functions f for which logarithmic means tnκ(f) of quadratical partial sums of double Walsh-Kaczmarz series diverge in measure.Thus, in question of convergence in measure logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series differ from the Marcinkiewicz means and are like the usual quadratical partial sums of double Walsh-Fourier series.


## 2. Main Results

The main results of this paper are presented in the following proposition.Theorem 2.1.
LetLΦ(I2) be an Orlicz space, such that
(2.1)LΦ(I2)⊈Llog⁡+L(I2).
Then the set of the functions in the Orlicz spaceLΦ(I2) with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on I2 is of first Baire category in LΦ(I2).Corollary 2.2.
Letφ:[0,∞[→[0,∞[ be a nondecreasing function satisfying for x→+∞ the condition
(2.2)φ(x)=o(xlog⁡x).
Then there exists a functionf∈L(I2) such that
(a)
(2.3)∫I2φ(|f(x,y)|)dxdy<∞,(b)
logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series off diverge in measure on I2.


## 3. Auxiliary Results

It is well-known [2, 3] for the Dirichlet kernel function that(3.1)|Dnw(x)|<1x
for any 0<x<1. Then for these x’s we also get(3.2)|Fnw(x)|<1x,
where n∈N is a nonnegative integer. It is also well-known for the Walsh-Paley-Dirichlet kernel functions the following lower pointwise estimates holds. Let pA=22A+⋯+22+20 (A∈N). Then for any 2-2A-1≤x<1 and A∈N, we have(3.3)|DpAw(x)|≥14x.Since this inequality plays a prominent role in the proofs of some divergence results concerning the partial sums of the Fourier series, then it seems that it would be useful to get a similar inequality also for the logarithmic kernels. In [20] the first author, Gát, and Tkebuchava proved the inequality(3.4)|FpAw(x)|≥clog⁡(1/x)xlog⁡pA
for all 1≤A∈N, and x∈(2-2A-1,1)∖J. We have the exceptional set J, such that it is “rare around zero.” For t=t0,t0+1,…,2A,t0=inf{t:⌊(lp[t/2]-1/16)-215⌋>1} set t̃:=⌊(lp[t/2]-1/16)-215⌋ (where ⌊u⌋ denotes the lower integral part of u), and we take a “small part’’ of the interval It∖It+1=[2-t-1,2-t). This way the intervals are defined as follows:(3.5)Δt:=[12t+1,12t+1+12t+t̃).
The exceptional set J is given by(3.6)J:=⋃t=t0∞Δt.For the logarithmic kernelsFpAκ of quadratical partial sums of double Walsh-Kaczmarz-Fourier series, we prove an analogue result. To do this we need the following lemma of the first author, Gát and Tkebuchava [22].Lemma 3.1 (see [22]).
Lets,t=0,1,…,2A, s≤t, and (x,y)∈(Js×Jt)∖(Qt+s̃(s,t)∪(Δs×Δt)). Then one has
(3.7)|FpAw(x,y)|≥c2t+sslog⁡pA,
where Jt:=It∖It+1 and
(3.8)QB(s,t):=⋃xs+1+⋯+xt-1≤1⋃xt=01⋯⋃xB-1=01IB(0,…,0,1,xs+1,…,xB-1)×IB(0,…,0,1,yt+1=xt+1,…,yB-1=xB-1).Now, for the Walsh-Kaczmarz logarithmic kernelsFpAκ, we prove the following.Lemma 3.2.
Letx∈I2A(1,x1,…,xt-2,xt-1=1,xt,…,xs-1,xs=1,xs+1=1,0,…,0)=:I2At,s,  y∈I2A(1,y1,…,yt-2,yt-1=0,yt=1,yt+1=1,0,…,0)=:Ĩ2At for s,t=2,3,…,A and t≤s. Then
(3.9)|FpAκ(x,y)|≥c24A-t-s.Proof.
Set(x,y)∈I2At,s×Ĩ2At. Let
(3.10)GpAα(x,y):=lpAFpAα(x,y),GpAα(x):=lpAFpAα(x)
for α=w or κ. Now, we write that
(3.11)GpAκ(x,y)=∑j=122ADjκ(x)Djκ(y)pA-j+∑j=22A+1pA-1Djκ(x)Djκ(y)pA-j:=I+II.
First, we discussII. We use equality  (1.13)(3.12)II=∑j=1pA-1-1Dj+22Aκ(x)Dj+22Aκ(y)pA-1-j=lpA-1D22A(x)D22A(y)+D22A(x)r2A(y)GpA-1w(τ2A(y))+D22A(y)r2A(x)GpA-1w(τ2A(x))+r2A(x)r2A(y)GpA-1w(τ2A(x),τ2A(y)).
If(x,y)∈I2At,s×Ĩ2At, then (see (1.14))(3.13)D22A(x)=D22A(y)=0,τ2A(x)=(0,…,0,xs+1=1,xs=1,xs-1,…,xt,xt-1=1,xt-2,…,x1,x0=1,x2A,…),τ2A(y)=(0,…,0,yt+1=1,yt=1,yt-1=0,yt-2,…,y1,y0=1,y2A,…).
Moreover, by Lemma 3.1, we have
(3.14)|II|=|GpA-1w(τ2A(x),τ2A(y))|≥c(2A-s)24A-t-s.
Now, we discussI by the help of  (1.13)(3.15)I=∑l=02A-1∑j=2l2l+1-1Djκ(x)Djκ(y)pA-j+D22A(x)D22A(y)pA-1=∑l=02A-1∑j=02l-1Dj+2lκ(x)Dj+2lκ(y)pA-j-2l=∑l=02A-1∑j=02l-1D2l(x)D2l(y)pA-j-2l+∑l=02A-1D2l(x)∑j=02l-1rl(y)Djw(τl(y))pA-j-2l+∑l=02A-1D2l(y)∑j=02l-1rl(x)Djw(τl(x))pA-j-2l+∑l=02A-1∑j=02l-1rl(x)rl(y)Djw(τl(x))Djw(τl(y))pA-j-2l.
Since x0=y0=1, D2l(x)=D2l(y)=0 for all l≥1, thus,
(3.16)I=1pA-1+∑l=12A-1rl(x)rl(y)∑j=12l-1Djw(τl(x))Djw(τl(y))pA-j-2l=:1pA-1+∑l=12A-1Il.
We use Abel’s transformation forIl (l≥1)(3.17)Il=rl(x)rl(y)∑j=12l-2(1pA-j-2l-1pA-j-2l-1)jKjw(τl(x),τl(y))+rl(x)rl(y)(2l-1)K2l-1w(τl(x),τl(y))pA-2l+1+1=:Il1+Il2.
We write
(3.18)|Il1|≤c24A∑j=12l-2j|Kjw(τl(x),τl(y))|,|Il2|≤c22A(2l-1)|K2l-1w(τl(x),τl(y))|.
Since |Di|≤i, we have
(3.19)n|Knw|≤|∑i=0nDiwDiw|≤cn3.
For l≤s+2, we obtain that
(3.20)|Il1|≤c24l24A,|Il2|≤c23l22A.
Moreover,
(3.21)∑l=1s+2|Il1|≤c24A∑l=1s+224l≤c24s24A<c,∑l=1s+2|Il2|≤c22A∑l=1s+223l≤c23s22A≤c2A.
Let s+2<l<2A. Then we have
(3.22)τl(x)=(0,…,0,1,1,xs-1,…,xt-1=1,…,x1,1,0,…,0,x2A,…),τl(y)=(0,…,0,1,1,yt-1=0,yt-2,…,y1,1,0,…,0,y2A,…),
that is, τl(x)∈Il-s-3∖Il-s-2, τl(y)∈Il-t-3∖Il-t-2.
Now, we introduce the following notation:(3.23)Ka,bw(x,y):=∑j=aa+b-1Djw(x)Djw(y),
and n(s):=∑i=s∞ni2i (n,s∈N). A simple consideration gives
(3.24)nKnw(x,y)=∑s=0|n|nsKn(s+1),2s(x,y)+Dnw(x)Dnw(y).
It is known [23] that if t≤k and (x,y)∈(It∖It+1)×(Ik∖Ik+1), then
(3.25)|Kn(s+1),2sw(x,y)|≤c2s+t+k.
Using this inequality, we have that
(3.26)|Il1|≤c24A∑k=0l-1∑j=2k2k+1-1j|Kjw(τl(x),τl(y))|≤c24A∑k=0l-1∑j=2k2k+1-1(∑ν=0k|Kj(ν+1),2νw(τl(x),τl(y))|+|Dj(τl(x))Dj(τl(y))|)≤c24A∑k=0l-1∑j=2k2k+1-1(∑ν=0k2ν+2l-t-s+j2)≤c24l-t-s24A+c23l24A.
Moreover,
(3.27)∑l=s+32A-1|Il1|≤c28A-t-s24A+c26A24A≤c24A-t-s.
Analogously,
(3.28)|Il2|≤c22A(∑ν=0l-12ν+2l-t-s+22l)≤c23l-t-s22A+c22l22A,(3.29)∑l=s+32A-1|Il2|≤c24A-t-s.
Taking into account (3.11), we get
(3.30)|FpAκ(x,y)|≥|II|-|I|lpA≥c1|II|-|I|2A.
Therefore by (3.14)–(3.29), it follows that
(3.31)|FpAκ(x,y)|≥c⋅c1(1-s2A)24A-t-s-c̃2A24A-t-s
with suitable constants c,c1,c̃>0. It is clear that for A large enough we have
(3.32)|FpAκ(x,y)|≥c*24A-t-s,
where c*>0 is an absolute constant.We apply the reasoning of [24] formulated as the following proposition in particular case.Theorem A.
LetH:L1(I2)→L0(I2) be a linear continuous operator, which commutes with family of translations ℰ, that is, for all E∈ℰ for all f∈L1(I2)HEf=EHf. Let ∥f∥L1(I2)=1 and λ>1. Then for any 1≤r∈ℕ under condition, mes{(x,y)∈I2:|Hf|>λ}≥1/r, there exist E1,…,Er,E1′,…,Er′∈ℰ and εi=±1, i=1,…,r, such that
(3.33)mes{(x,y)∈I2:|H(∑i=1rεif(Eix,Ei′y))|>λ}≥18.Theorem B.
Let{Hm}m=1∞ be a sequence of linear continuous operators, acting from Orlicz space LΦ(I2) into the space L0(I2). Suppose that there exist the sequence of functions {ξk}k=1∞ from unit bull SΦ(0,1) of space LΦ(I2), and sequences of integers {mk}k=1∞ and {νk}k=1∞ increasing to infinity such that
(3.34)ε0=inf⁡kmes{(x,y)∈I2:|Hmkξk(x,y)|>νk}>0.
ThenB—the set of functions f from space LΦ(I2), for which the sequence {Hmf} converges in measure to an a.e. finite function—is of first Baire category in space LΦ(I2).Theorem GGT.
LetLΦ be an Orlicz space, and let φ:[0,∞)→[0,∞) be measurable function with condition φ(x)=o(Φ(x)) as x→∞. Then there exists Orlicz space Lω, such that ω(x)=o(Φ(x))as x→∞, and ω(x)≥φ(x) for x≥c≥0.The proof of Theorems B and GGT can be found in [20].


## 4. Proof of the Theorem

Proof of Theorem2.1.
By Theorem B the proof of Theorem2.1 will be complete if we show that there exist sequences of integers {Ak:k≥1} and {νk:k≥1} increasing to infinity, and a sequence of functions {ξk:k≥1} from the unit bull SΦ(0,1) of Orlicz space LΦ(I2), such that for all k(4.1)mes{(x,y)∈I2:|tpAkκ(ξk,x,y)|>νk}≥18.
First, we prove that there existsc>0 such that(4.2)mes{(x,y)∈I2:|tpAkκ(D22A+1⊗D22A+1;x,y)|>c23A}>cA23A.
It is easy to show that(4.3)tpAkκ(D22A+1⊗D22A+1;x,y)=S22A+1,22A+1(FpAκ;x,y)=FpAκ(x,y).
Hence, from Lemma 3.2, we can write
(4.4)|tpAkκ(D22A+1⊗D22A+1;x,y)|=|FpAκ(x,y)|≥c24A-t-s,
for (x,y)∈I2At,s×Ĩ2At.
Since(4.5)c24A-t-s>c23A
for t+s<A and 0≤t≤s<A, from (4.4) we conclude that
(4.6)mes{|tpAκ(D22A+1⊗D22A+1;x,y)|(x,y)∈I2:|tpAκ(D22A+1⊗D22A+1;x,y)|>c23A}≥c∑s=[A/2]A∑t=0A-smes(I2At,s×Ĩ2At)≥c∑s=[A/2]A∑t=0A-s2s22A2t22A≥c∑s=[A/2]A2s22A2A-s22A≥cA23A.
Hence, (4.2) is proved.
From the condition of the theorem, we write [1](4.7)liminf⁡u→∞Φ(u)ulog⁡u=0.
Consequently, there exists a sequence of integers {Ak}k=1∞ increasing to infinity, such that
(4.8)lim⁡k→∞Φ(24Ak)2-4AkAk-1=0,Φ(24Ak)24Ak≥4,∀k.
From (4.2) we write(4.9)mes{(x,y)∈I2:|tpAkκ(D22Ak+1⊗D22Ak+1;x,y)|>c23Ak}>cAk23Ak.
Then by the virtue of Theorem A, there exists e1,…,er,e1′,…,er′∈[0,1] and ε1,…,εr=±1 such that
(4.10)mes{(x,y)∈I2:|∑i=1rεitpAkκ(D22Ak+1⊗D22Ak+1,ei⊕x,ei′⊕y)|>23Ak}>18,
where r=[23Ak/(cAk)]+1.
Set(4.11)ξk(x,y)=24Ak-1Φ(24Ak)Mk(x,y),
where
(4.12)Mk(x,y)=1r∑i=1rεiD22Ak+1(ei⊕x)D22Ak+1(ei′⊕y).
Denote
(4.13)νk=27Ak-1rΦ(24Ak).
It is easy to show that
(4.14)mes{(x,y)∈I2:|tpAkκ(ξk,x,y)|>νk}=mes{(x,y)∈I2:|∑i=1rεitpAkκ(D22Ak+1⊗D22Ak+1,ei⊕x,ei′⊕y)|>23Ak}>18.
Hence, (4.1) is proved.
Since(4.15)‖Mk‖L∞(I2)≤24mk+2,‖Mk‖L1(I2)≤1.
Moreover,
(4.16)|∫I2fg|≤12[∫I2Φ(2|f|)+1],Φ(u)u<Φ(u′)u′(0<u<u′).
From (4.8) we can write
(4.17)‖ξk‖LΦ(I2)≤12[1+∫I2Φ(24Ak|Mk(x,y)|Φ(24Ak))dxdy]≤12[1+∫I2Φ(24Ak24Ak+2/(Φ(24Ak)))24Ak(24Ak+2/(Φ(24Ak)))24Ak|Mk(x,y)|Φ(24Ak)dxdy]≤12[1+∫I2Φ(24Ak)24Ak24Ak|Mk(x,y)|Φ(24Ak)dxdy]≤1.
Hence,ξk∈SΦ(0,1), and Theorem 2.1 is proved.The validity of Corollary2.2 follows immediately from Theorems 2.1 and GGT.



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*Source: 582726-2012-01-04.xml*