Table Of ContentMAXIMAL OPERATORS OF VILENKIN-NÖRLUND MEANS
L. E. PERSSON, G. TEPHNADZE AND P. WALL
Abstract. Inthispaperweproveanddiscusssomenew(H ,weak L )typeinequalities
p p
−
5 ofmaximaloperatorsofVilenkin-Nörlundmeanswithmonotonecoefficients. Wealsoapply
1
these results to prove a.e. convergence of such Vilenkin-Nörlund means. It is also proved
0
that these results are the best possible in a special sense. As applications, both some well-
2
known and new results are pointed out.
n
a
J
2000 Mathematics Subject Classification. 42C10, 42B25.
6
2 Key words and phrases: Vilenkin system, Vilenkin group, Nörlund means, martingale
Hardy space, weak L spaces, maximal operator, Vilenkin-Fourier series.
] p
A −
C
1. Introduction
.
h
t
a
m The definitions and notations used in this introduction can be found in our next Section.
[ In the one-dimensional case the first result with respect to the a.e. convergence of Fejér
1 is due to Fine [13]. Later, Schipp [25] for Walsh series and Pál, Simon [24] for bounded
v Vilenkin series showed that the maximal operator of Fejér means σ∗ is of weak type (1,1),
6
from which the a. e. convergence follows by standard argument [16]. Fujji [14] and Simon
9
3 [27] verified that σ∗ is bounded from H to L . Weisz [38] generalized this result and proved
1 1
5
boundedness of σ∗ from the martingale space H to the space L , for p > 1/2. Simon [26]
0 p p
. gave a counterexample, which shows that boundedness does not hold for 0 < p < 1/2. A
3
0 counterexample for p = 1/2 was given by Goginava [9] (see also [31]). In [30] it was proved
5 even stronger result than the maximal operator σ∗ is unbounded. In fact, it was proved that
1
there exists a martingale f H , such that Fejér means of Vilenkin-Fourier series of the
: 1/2
v ∈
martingale f are not uniformly bounded in the space L . Moreover, Weisz [40] proved that
i 1/2
X the following is true:
r
a Theorem W1. The maximal operator of the Fejér means σ∗ is bounded from the Hardy
space H to the space weak-L .
1/2 1/2
Riesz‘s logarithmic means with respect to the trigonometric system was studied by several
authors. We mention, for instance, the papers by Szàsz [29] and Yabuta [36]. These means
with respect to the Walsh and Vilenkin systems were investigated by Simon [26] and Gát [5].
Blahota and Gát [3] considered norm summability of Nörlund logarithmic means andshowed
that Riesz‘s logarithmic means R have better approximation properties on some unbounded
n
Vilenkin groups, than the Fejér means. In [33] it was proved that the maximal operator of
Riesz‘s means R∗ is not bounded from the Hardy space H to the space weak L , but
1/2 1/2
−
is not bounded from the Hardy space H to the space L , when 0 < p 1/2. Since the set of
p p
≤
The research was supported by Shota Rustaveli National Science Foundation grant no.13/06 (Geometry
of function spaces, interpolation and embedding theorems.
1
2 L. E. PERSSON,G. TEPHNADZEAND P. WALL
Vilenkin polynomials is dense in L , by well-known density argument due to Marcinkiewicz
1
and Zygmund [16], we have that R f f, a.e. for all f L .
n 1
→ ∈
Móricz and Siddiqi [19] investigated the approximation properties of some special Nörlund
means of Walsh-Fourier series of L function in norm. The case when q = 1/k was excluded,
p k
since the methods of Móricz and Siddiqi are not applicable to Nörlund logarithmic means.
In [6] Gát and Goginava proved some convergence and divergence properties of the Nörlund
logarithmicmeans of functions in the class of continuous functions and in the Lebesgue space
L . Among other things, they gave a negative answer to a question of Móricz and Siddiqi
1
[19]. Gát and Goginava [7] proved that for each measurable function φ(u) = u√logu ,
◦
there exists an integrable function f such that
(cid:0) (cid:1)
φ( f (x) )dµ (x) <
| | ∞
ZGm
and there exists a set with positive measure, such that the Walsh-logarithmic means of the
function diverges on this set. In [32] it was proved that there exists a martingale f H ,
p
∈
(0 < p 1),suchthatthemaximaloperatorofNörlundlogarithmicmeansL∗ isnotbounded
≤
in the space L .
p
In [11] Goginava investigated the behaviour of Cesáro means of Walsh-Fourier series in
detail. In the two-dimensional case approximation properties of Nörlund and Cesáro means
was considered by Nagy (see [20]-[23]). The a.e. convergence of Cesáro means of f L
1
∈
was proved in [12]. The maximal operator σα,∗ (0 < α < 1) of the (C,α) means of Walsh-
Paley system was investigated by Weisz [41]. In his paper Weisz proved that σα,∗ is bounded
from the martingale space H to the space L for p > 1/(1+α). Goginava [10] gave a
p p
counterexample, which shows that boundedness does not hold for 0 < p 1/(1+α).
≤
Recently, Weisz and Simon [28] showed the following statement:
Theorem SW1. The maximal operator σα,∗ (0 < α < 1) of the (C,α) means is bounded
from the Hardy space H to the space weak L .
1/(1+α) 1/(1+α)
−
In this paper we derive some new (H ,L )-type inequalities for the maximal operators of
p p
Nörlund means, with monotone coefficients.
The paper is organized as following: In Section 3 we present and discuss the main results
and in Section 4 the proofs can be found. Moreover, in order not to disturb our discussions
in these Sections some preliminaries are given in Section 2.
2. Preliminaries
Denote by N the set of the positive integers, N := N 0 . Let m := (m m ,...) be a
+ + 0, 1
∪{ }
sequence of the positive integers not less than 2. Denote by
Z := 0,1,...,m 1
mk { k − }
the additive group of integers modulo m .
k
Define the group G as the complete direct product of the groups Z with the product
m mi
of the discrete topologies of Z ‘s.
mj
The direct product µ of the measures
µ ( j ) := 1/m (j Z )
k { } k ∈ mk
is the Haar measure on G with µ(G ) = 1.
m m
NÖRLUND MEANS 3
In this paper we discuss bounded Vilenkin groups, i.e. the case when sup m < .
n n ∞
The elements of G are represented by sequences
m
x := (x ,x ,...,x ,...), x Z .
0 1 j j ∈ mj
It is easy to give a base for the neighborhood of G(cid:0) : (cid:1)
m
I (x) := G , I (x) := y G y = x ,...,y = x ,
0 m n m 0 0 n−1 n−1
{ ∈ | }
where x G , n N.
m
∈ ∈
Denote I := I (0) for n N , and I := G I .
n n + n m n
∈ \
If we define the so-called generalized number system based on m in the following way :
M := 1, M := m M (k N),
0 k+1 k k
∈
then every n N can be uniquely expressed as n = ∞ n M , where n Z (j N )
∈ j=0 j j j ∈ mj ∈ +
and only a finite number of n ‘s differ from zero.
j
P
Next, we introduce on G an orthonormal system which is called the Vilenkin system. At
m
first, we define the complex-valued function r (x) : G C, the generalized Rademacher
k m
→
functions, by
r (x) := exp(2πix /m ), i2 = 1,x G , k N .
k k k m
− ∈ ∈
Now, define the Vilenkin system ψ := (ψ :(cid:0)n N) on G as: (cid:1)
n m
∈
∞
ψ (x) := rnk (x), (n N).
n k ∈
k=0
Y
Specifically, we call this system the Walsh-Paley system when m 2.
≡
The norm (or quasi-norm) of the space L (G ) (0 < p < ) is defined by
p m
∞
f p := f pdµ.
k kp | |
ZGm
The space weak L (G ) consists of all measurable functions f, for which
p m
−
f p := supλpµ(f > λ) < + .
k kweak−Lp ∞
λ>0
The Vilenkin system is orthonormal and complete in L (G ) (see [35]).
2 m
Now, we introduce analogues of the usual definitions in Fourier-analysis. If f L (G )
1 m
∈
we can define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet
kernels with respect to the Vilenkin system in the usual manner:
f (n) := fψ dµ (n N),
n ∈
ZGm
bn−1 n−1
S f := f (k)ψ , D := ψ , (n N )
n k n k +
∈
k=0 k=0
X X
respectively. b
4 L. E. PERSSON,G. TEPHNADZEAND P. WALL
Recall that
M , if x I ,
(1) D (x) = n ∈ n
Mn 0, if x / I .
n
(cid:26) ∈
The σ-algebra generated by the intervals I (x) : x G will be denoted by ̥ (n N).
n m n
{ ∈ } ∈
Denote by f = f(n),n N a martingale with respect to ̥ (n N). (for details see e.g.
n
∈ ∈
[37]).
(cid:0) (cid:1)
The maximal function of a martingale f is defined by
f∗ = sup f(n) .
n∈N
(cid:12) (cid:12)
(cid:12) (cid:12)
For 0 < p < the Hardy martingale spaces H (G ) consist of all martingales for which
p m
∞
f := f∗ < .
k kHp k kp ∞
If f = f(n),n N is a martingale, then the Vilenkin-Fourier coefficients must be defined
∈
in a slightly different manner:
(cid:0) (cid:1)
f (i) := lim f(k)ψ dµ.
i
k→∞
ZGm
b
Let q : k 0 be a sequence of nonnegative numbers. The n-th Nörlund means for a
k
{ ≥ }
Fourier series of f is defined by
n
1
(2) t f = q S f,
n n−k k
Q
n
k=1
X
where Q := n−1q .
n k=0 k
We always assume that q > 0 and lim Q = . In this case it is well-known that
P 0 n→∞ n
∞
the summability method generated by q : k 0 is regular if and only if
k
{ ≥ }
q
n−1
lim = 0.
n→∞ Qn
Concerning this fact and related basic results, we refer to [18].
If q 1, we get the usual Fejér means
k
≡
n
1
σ f := S f .
n k
n
k=1
X
The (C,α)-means of the Vilenkin-Fourier series are defined by
n
1
σαf = Aα−1S f,
n Aα n−k k
nk=1
X
where
(α+1)...(α+n)
Aα = 0, Aα = , α = 1, 2,...
0 n n! 6 − −
NÖRLUND MEANS 5
The n-th Riesz‘s logarithmic mean R and the Nörlund logarithmic mean L are defined
n n
by
n−1 n−1
1 S f 1 S f
k k
R f := , L f := ,
n n
l k l n k
n n
k=1 k=1 −
X X
respectively, where l := n−11/k.
n k=1
For the martingale f we consider the following maximal operators:
P
t∗f := sup t f , σ∗f := sup σ f , σα,∗f := sup σαf ,
| n | | n | | n |
n∈N n∈N n∈N
R∗f := sup R f and L∗f := sup L f .
n n
| | | |
n∈N n∈N
A bounded measurable function a is a p-atom, if there exists a interval I, such that
adµ = 0, a µ(I)−1/p, supp(a) I.
k k∞ ≤ ⊂
ZI
We also need the following auxiliary results:
Lemma 1. [39] A martingale f = f(n),n N is in H (0 < p 1) if and only if there
p
∈ ≤
exists sequence (a ,k N) of p-atoms and a sequence (µ ,k N), of real numbers, such
k k
∈ (cid:0) (cid:1) ∈
that, for every n N,
∈
∞
(3) µ S a = f(n),
k Mn k
k=0
X
∞
µ p < .
k
| | ∞
k=0
X
Moreover,
∞ 1/p
f ∽ inf µ p ,
k kHp | k|
!
k=0
X
where the infimum is taken over all decomposition of f of the form (3).
Lemma 2. [39] Suppose that an operator T is σ-linear and for some 0 < p 1 and
≤
supρpµ x I : Ta > ρ c < ,
p
∈ | | ≤ ∞
ρ>0
(cid:8) (cid:9)
for every p-atom a, where I denotes the support of the atom. If T is bounded from L to
∞
L , then
∞
Tf c f .
k kweak−Lp ≤ pk kHp
and if 0 < p < 1, then T is of weak type-(1,1):
Tf c f .
k kweak−L1 ≤ k k1
6 L. E. PERSSON,G. TEPHNADZEAND P. WALL
3. Main Results
Our first main result reads:
Theorem 1. a) Let q : k 0 be a sequence of nonnegative numbers, q > 0 and
k 0
{ ≥ }
lim Q = .
n
n→∞ ∞
The summability method (2) generated by q : k 0 is regular if and only if
k
{ ≥ }
q
n−1
(4) lim = 0.
n→∞ Qn
Next, we state our main result concerning the maximal operator of the summation method
(2), which we also show is in a sense sharp.
Theorem 2. a) The maximal operator t∗ of the summability method (2) with nondecreasing
sequence q : k 0 , is bounded from the Hardy space H to the space weak L .
k 1/2 1/2
{ ≥ } −
The statement in a) is sharp in the following sense:
b) Let 0 < p < 1/2 and q : k 0 is nondecreasing sequence, satisfying the condition
k
{ ≥ }
q c
0
(5) , (c > 0).
Q ≥ n
n
Then there exists a martingale f H , such that
p
∈
sup t f = .
n∈Nk n kweak−Lp ∞
Our next result shows that the statement in b) above hold also for nonincreasing sequences
and now without any restriction like (5).
Theorem 3. Let 0 < p < 1/2. Then, for all Nörlund means with nonincreasing sequence
q : k 0 , there exists a martingale f H , such that
k p
{ ≥ } ∈
sup t f = .
n∈Nk n kweak−Lp ∞
Up to now we have considered the case 0 < p < 1/2, but in our final main result we
consider the case when p = 1/(1+α), 0 < α 1, so that 1/2 p < 1. Also this result is
≤ ≤
sharp in two different important senses.
Theorem 4. a) Let 0 < α 1. Then the maximal operator t∗ of summability method (2)
≤
with non-increasing sequence q : k 0 , satisfying the condition
k
{ ≥ }
nαq q q
0 n n+1
(6) = O(1), | − | = O(1), as n .
Q nα−2 → ∞
n
is bounded from the Hardy space H to the space weak L .
1/(1+α) 1/(1+α)
−
The parameter 1/(1+α) in a) is sharp in the following sense:
b) Let 0 < p < 1/(1+α), where 0 < α 1 and q : k 0 be a non-increasing sequence,
k
≤ { ≥ }
satisfying the condition
q c
0
(7) , (c > 0).
Q ≥ nα
n
NÖRLUND MEANS 7
Then
sup t f = .
n∈Nk n kweak−Lp ∞
Also the condition (6) is "sharp" in the following sense:
c) Let q : k 0 be a non-increasing sequence, satisfying the condition
k
{ ≥ }
q nα
0
(8) lim = , (0 < α 1).
n→∞ Qn ∞ ≤
Then
sup t f = .
n∈Nk n kweak−L1/(1+α) ∞
A number of special cases of our results are of particular interest and give both well-known
and new information. We just give the following examples of such Corollaries:
Corollary 1. (See [32]) The maximal operator of the Fejér means σ∗ is bounded from the
Hardy space H to the space weak L but is not bounded from H to the space weak L ,
1/2 1/2 p p
− −
when 0 < p < 1/2.
Corollary 2. (See [33]) The maximal operator of the Riesz‘s means R∗ is bounded from the
Hardy space H to the space weak L but is not bounded from H to the space weak L ,
1/2 1/2 p p
− −
when 0 < p < 1/2.
Corollary 3. The maximal operator of the (C,α)-means σα,∗ is bounded from the Hardy
space H to the space weak L but is not bounded from H to the space weak L ,
1/(1+α) 1/(1+α) p p
− −
when 0 < p < 1/(1+α).
Corollary 4. (See [32]) The maximal operator of the Nörlund logarithmic means L∗ is not
bounded from the Hardy space H to the space weak L , when 0 < p < 1.
p p
−
Corollary 5. Let f L and t be the Nörlund means, with nondecreasing sequence q :
1 n k
∈ {
k 0 . Then
≥ }
t f f, a.e., as n .
n
→ → ∞
Corollary 6. Let f L and t be Nörlund means, with non-increasing sequence q : k
1 n k
∈ { ≥
0 and satisfying condition (6). Then
}
t f f, a.e., as n .
n
→ → ∞
Corollary 7. Let f L . Then
1
∈
σ f f, a.e., as n ,
n
→ → ∞
R f f, a.e. as n ,
n
→ → ∞
σαf f, a.e., as n , when 0 < α < 1.
n → → ∞
Remark 1. The statements in Corollary 7 are known (see [12], [13], [28], [16], [34]), but
this unified approach to prove them is new.
8 L. E. PERSSON,G. TEPHNADZEAND P. WALL
4. Proofs of the Theorems
Proof of Theorem 1. The proof is similar as in the case with Walsh system (see [18]),
so we omit the details.
Proof of Theorem 2. By using Abel transformation we obtain that
n−1 n n−1
(9) Q := q = q 1 = (q q )j +q n,
n j n−j n−j n−j−1 0
· −
j=0 j=1 j=1
X X X
and
n−1
1
(10) t f = (q q )jσ f +q nσ f .
n n−j n−j−1 j 0 n
Q −
n !
j=1
X
Let the sequence q : k 0 be non-decreasing. By combining (9) with (10) and using
k
{ ≥ }
Abel transformation we get that
n
1
t f q S f
n n−j j
| | ≤ (cid:12)Qn (cid:12)
(cid:12) Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12)1 n−1 (cid:12)
(cid:12) q (cid:12) q j σ f +q n σ f
n−j n−j−1 j 0 n
≤ Q | − | | | | |
n !
j=1
X
n−1
c
(q q )j +q n σ∗f cσ∗f
n−j n−j−1 0
≤ Q − ≤
n !
j=1
X
so that
(11) t∗f cσ∗f.
≤
If we apply (11) and Theorem W1 we can conclude that the maximal operators t∗ are
bounded from the Hardy space H to the space weak L . It follows that (see Theorem
1/2 1/2
−
W1) t∗ has weak type-(1,1) and t f f, a.e.
n
→
In the proof of the second part of Theorem 2 we mainly follow the method of Blahota,
Gát and Goginava (see [1], [2]).
Let 0 < p < 1/2 and α : k N be an increasing sequence of positive integers such that:
k
{ ∈ }
∞
(12) 1/αp < ,
k ∞
k=0
X
k−1 1/p 1/p
M M
(13) λ αη < αk ,
α α
η k
η=0
X
1/p 1/p−2
32λM M
(14) αk−1 < αk ,
α α
k−1 k
where λ = sup m .
n n
NÖRLUND MEANS 9
We note that such an increasing sequence α : k N which satisfies conditions (12)-(14)
k
{ ∈ }
can be constructed.
Let
(15) f(A) = λ θ ,
k k
{k;Xλk<A}
where
λ
(16) λ =
k
α
k
and
1/p−1
M
(17) θ = αk D D .
k λ Mαk+1 − Mαk
(cid:16) (cid:17)
It is easy to show that the martingale f = f(1),f(2)...f(A)... H . Indeed, since
1/2
∈
(cid:0) (cid:1)
a , if α < A,
S θ = k k
MA k 0, if α A,
k
(cid:26) ≥
supp(θ ) = I , θ dµ = 0, θ M1/p = (supp a )1/p,
k αk k k kk∞ ≤ αk k
ZIαk
if we apply Lemma 1 and (12) we can conclude that f H , (0 < p < 1/2).
p
∈
Moreover, it is easy to show that
Mα1/kp−1, if j M ,...,M 1 , k = 0,1,2...,
(18) f(j) = αk ∈ { ∞αk αk+1 − }
0, if j / M ,...,M 1 .
∈ { αk αk+1 − }
k=1
b
S
We can write
1 Mαk q
0
t f = q S f + S f := I +II.
Mαk+1 Q j j Q Mαk+1
Mαk+1 j=0 Mαk+1
X
Let M j M , where s = 0,...,k 1. Moreover,
αs ≤ ≤ αs+1 −
D D j M λM , (s N)
j − Mαs ≤ − αs ≤ αs ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
10 L. E. PERSSON,G. TEPHNADZEAND P. WALL
so that, according to (1) and (18), we have that
Mαs−1+1−1 j−1
(19) S f = f(v)ψ + f(v)ψ
j v v
| | (cid:12) (cid:12)
(cid:12)(cid:12) Xv=0 v=XMαs (cid:12)(cid:12)
(cid:12) b b (cid:12)
s−1 Mαη+(cid:12)1−1 M1/p−1 M1/p−1 (cid:12)
(cid:12) αη ψ + αs D (cid:12)D
v j M
≤ (cid:12) α (cid:12) α − αs
(cid:12)(cid:12)Xη=0 v=XMαη η (cid:12)(cid:12) s (cid:12)(cid:0) (cid:1)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
s−1 1/p−1 1/p−1
(cid:12) M (cid:12) M
= (cid:12) αη D (cid:12) D + αs D D
M M j M
(cid:12) αη αη+1 − αη (cid:12) αs − αs
(cid:12)Xη=0 (cid:16) (cid:17)(cid:12) (cid:12)(cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
(cid:12) s−1 M1/p λM1/p 2λM1/p(cid:12) λM1/p (cid:12) 4λM1/p (cid:12)
λ(cid:12) αη + αs αs−(cid:12)1 + αs αk−1.
≤ α α ≤ α α ≤ α
η s s−1 s k−1
η=0
X
Let M +1 j M , where s = 1,...,k. Analogously to (19) we can prove that
αs−1+1 ≤ ≤ αs
Mαs−1+1−1 s−1 Mαη+1−1 M1/p−1
S f = f(v)ψ = αη ψ
j v v
| | (cid:12) (cid:12) (cid:12) α (cid:12)
(cid:12)(cid:12) Xv=0 (cid:12)(cid:12) (cid:12)(cid:12)Xη=0 v=XMαη η (cid:12)(cid:12)
(cid:12) b (cid:12) (cid:12) (cid:12)
s−1 1/p−1 1/p 1/p
M (cid:12) (cid:12) (cid:12) 2λM 4λM(cid:12)
= α(cid:12)η D D(cid:12) (cid:12) αs−1 (cid:12)αk−1.
M M
(cid:12) αη αη+1 − αη (cid:12) ≤ αs−1 ≤ αk−1
(cid:12)Xη=0 (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Hence
1 Mαk 4λM1/p 1 Mαk 4λM1/p
(20) I q S f αk−1 q αk−1.
j j j
| | ≤ Q | | ≤ α Q ≤ α
Mαk+1 j=0 k−1 Mαk+1 j=0 k−1
X X
If we now apply (18) and (19) we get that
1/p−1
q M
(21) II = 0 αk ψ +S f
| | QMαk+1 (cid:12)(cid:12) αk Mαk Mαk (cid:12)(cid:12)
(cid:12) (cid:12)
1/p−1
q (cid:12)M (cid:12)
= 0 (cid:12) αk ψ +S (cid:12) f
QMαk+1 (cid:12)(cid:12) αk Mαk Mαk−1+1 (cid:12)(cid:12)
(cid:12) (cid:12)
1/p−1
q (cid:12) M (cid:12)
0 (cid:12) αk ψ S (cid:12) f
≥ QMαk+1 (cid:12)(cid:12) αk Mαk(cid:12)(cid:12)−(cid:12) Mαk−1+1 (cid:12)!
(cid:12) (cid:12) (cid:12) (cid:12)
1/p−1 1/p 1/p−1
q (cid:12)M 4λM(cid:12) (cid:12) q(cid:12) M
0 (cid:12) αk (cid:12) αk−1 0 αk .
≥ Q α − α ≥ Q 4α
Mαk+1 k k−1 ! Mαk+1 k
