Analysis Mathematica, 28(2002), 1–23
Ces` aro summability of the character system of the p-series field in the Kaczmarz rearrangement ´ 1 and K. NAGY2 G. GAT
A b s t r a c t . Let Gp be the p-series field. In this paper we prove the a.e. convergence σn f → f (n → ∞) for an integrable function f ∈ L1 (Gp ), where σn f is the nth (C, 1) mean of f with respect to the character system in the Kaczmarz rearrangement. We define the maximal operator σ ∗ by σ ∗ f := supn |σn f |. We prove that σ ∗ is of type (q, q) for all 1 < q ≤ ∞ and of weak type (1, 1). Moreover, we prove that kσ ∗ f k1 ≤ ckf kH , where H is the Hardy space on the Gp .
1. Introduction This paper is devoted to the problem of a.e. convergence of the (C, 1) means of integrable functions with respect to the character system of the p-series field in the Kacmarz rearrangement. The Walsh system in the Kaczmarz rearrangement (on the Walsh–Paley group) was studied by a number of authors (see [Sch1], [Sch2], [Sk1], [Sk2], [Bal], [SWS], [Y]). In [Sn] it has been pointed out that the behavior of the Dirichlet kernel of the Walsh–Kaczmarz system is worse than that of the kernel of the Walsh– Paley system considered more often. Namely, it is proved [Sn] that for the Dirichlet kernel Dn (x) of the Walsh–Kaczmarz system the inequality Dn (x) ≥C>0 n→∞ log n holds a.e. The “spreadness” of this system makes easier to construct examples of divergent Fourier series [Bal]. A number of pathological properties are due to this “spreadness” property of the kernel. For example, for Fourier lim sup
Received July 8, 1999; in revised form July 12, 2000. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grants no. F0157652 , F0203341 , and M36511/2001, by the foundation “Pro Regione” Grant no. 1538-I/13/19972 , by the Hungarian “M˝ uvel˝ od´esi ´es K¨ ozoktat´ asi Miniszt´erium ”, Grants no. PFP 5306/19972 , FKFP 0710/19972,1 and 0182/20002,1 and by the Bolyai fellowship of the Hungarian Academy of Sciences, Grant no. BO/00320/991 . 0133–3852/02/$ 15.00 c 2002 Akad´emiai Kiad´o, Budapest
2
G. G´ at and K. Nagy
series with respect to the Walsh–Kaczmarz system it is impossible to establish any local test for convergence at any point or on any interval, since the principle of localization does not hold for this system. On the other hand, the global behavior of the Fourier series with respect to this system is similar in many aspects to the case of the Walsh–Paley system. Schipp [Sch2] and Wo-Sang Young [Y] proved that the Walsh– Kaczmarz system is a convergence system. Skvorcov proved for continuous functions f that the Fej´er means converge uniformly to f . One of the authors [G´at] proved the a.e. convergence σn f → f (n → ∞) as well as that the maximal operator defined as σ ∗ f := sup |σn f | n
is of type (q, q) for all 1 < q ≤ ∞ and of weak type (1, 1). A certain kind of (H, L) typeness of σ ∗ is also proved [G´at]. Namely, kσ ∗ f k1 ≤ ck|f |kH . In the present paper we generalize these results for the p-series field. In the case of (H, L) typeness we prove even more: kσ ∗ f k1 ≤ ckf kH (see the absence of absolute value marks). Let P denote the set of positive integers, N := P ∪ {0}. Let 2 ≤ p ∈ N and denote by Zp the pth cyclic group, that is, Zp can be represented by the set {0, 1, . . . , p − 1},where the group operation is the mod p addition and every subset is open. The Haar measure on Zp is given in the way that 1 µk ({j}) := (j ∈ Zp ). j Let Gp be the complete direct product of the countable infinite copies of the compact groups Zp . The compact Abelian group Gp is called p-series field. The elements of Gp are of the form x = (x0 , x1 , . . . , xk , . . .)
with 0 ≤ xk < p (xk ∈ N, k ∈ N).
The group operation on Gp is the coordinate-wise addition, the normalised Haar measure µ is the product measure. The topology on Gm is the product topology, a base for the neighborhoods of Gp can be given in the following way: I0 (x) := Gp ,
In (x) := {y ∈ Gp : y = (x0 , . . . , xn−1 , yn , yn+1 . . .)}
(x ∈ Gp , n ∈ P). Let 0 = (0 : i ∈ N) ∈ Gp denote the null element of Gp , In := In (0) (n ∈ N). Let I := {In (x) : x ∈ Gp , n ∈ N}.
Ces` aro summability of the character system
3
The elements of I are called the intervals on Gp . Set ei = (0, . . . , 0, 1, 0, . . .) ∈ Gp the ith coordinate of which is 1, the rest are zeros. Furthermore, let Lq (Gp ) (1 ≤ q ≤ ∞) denote the usual Lebesgue spaces (k · kq the corresponding norms) on Gp , An the σ-algebra generated by the sets In (x) (x ∈ Gp ), and En the conditional expectation operator with respect to An (n ∈ N and f ∈ L1 (Gp )). Define the Hardy space H 1 = H 1 (Gp ) as follows. Let f ∗ := sup |En f | n∈N
1
be the maximal function of f ∈ L (Gp ). Set H 1 (Gp ) := {f ∈ L1 (Gp ) : f ∗ ∈ L1 (Gp )}, H 1 is a Banach space with the norm kf kH 1 := kf ∗ k1 . A good property of the space H 1 (Gp ) is the atomic structure ([SWS]). A function a ∈ L∞ (Gp ) is called an atom if a = 1 or a satisfies the following three properties: supp a ⊆ Ia ,
Z
kak∞ ≤ 1/µ(Ia ) and
Ia
a = 0, for some Ia ∈ I.
A function f belongs to the Hardy space H(Gp ) if there exist λi ∈ C and ai atoms (i ∈ N) such that ∞ X
|λi | < ∞
and
f=
i=0
∞ X
λi ai .
i=0
H = H(Gp ) is a Banach space with the norm kf kH := inf
∞ X
|λi |,
i=0
where the infimum is taken over all decompositions of f=
∞ X
λi ai .
i=0
Moreover,
H 1 (Gp ) = H(Gp ) and kf kH 1 ∼ kf kH . We will use often the following. For any f ∈ H 1 and t ∈ Gp we have that f (· + t) ∈ H 1
and
kf kH 1 ∼ kf (· + t)kH 1 .
This follows from that if a is an atom, then so is the function a(· + t). Let Γ(p) denote the character group of Gp . We arrange the elements of Γ(p) as follows. For k ∈ N and x ∈ Gp denote by rk the kth generalized Rademacher function: 2πix √ k rk (x) := exp (i := −1, x ∈ Gp , k ∈ N). p
4
G. G´ at and K. Nagy
Let n ∈ N. Then n=
∞ X
n i pi ,
where 0 ≤ ni < p (ni , i ∈ N),
i=0
n is expressed in the number system with base p. Denote by |n| := max(j ∈ N : nj 6= 0),
i.e.,
p|n| ≤ n < p|n|+1 .
Now, we define the sequence of functions ψ := (ψn : n ∈ N) by ∞ Y
ψn (x) :=
(rk (x))nk
(x ∈ Gp , n ∈ N).
k=0
We remark that Γ(p) = {ψn : n ∈ N} is a complete orthonormal system with respect to the normalized Haar measure on Gp . The character group Γ(p) can be given in the Kaczmarz rearrangement as follows: Γ(p) = {χn : n ∈ N}, where |n|−1
n
χn (x) := r|n||n| (x)
Y
(r|n|−1−k (x))nk
(x ∈ Gp , n ∈ P),
k=0
χ0 (x) = 1 (x ∈ Gp ). Let the transformation τA : Gp → Gp be defined as follows: τA (x) := (xA−1 , xA−2 , . . . , x0 , xA , xA+1 , . . .). The transformation τA is measure-preserving and τA (τA (x)) = x. By the definition of τA , we have n
χn (x) = r|n||n| (x)ψn−n|n| p|n| (τ|n| (x))
(n ∈ N, x ∈ Gp ).
For a function f in L1 (Gp ) the Fourier coefficients, the partial sums of the Fourier series, the Diriclet kernels, the Fej´er means, the Fej´er kernels and the maximal functions of the Fej´er means are defined as follows: fbα (n) := σnα f :=
Z Gp
f αn ,
n 1X S α f, n k=1 k
Snα f := Knα :=
n−1 X
fbα (k)αk ,
k=0 n X
1 Dα , n k=1 k
Dnα :=
n−1 X
αk ,
k=0
σ ∗α f := sup |σnα f |, n∈P
and D0α = K0α := 0, where α is either ψ or χ. It is well known that Snα f (x)
Z
=
Gp
f (x +
t)Dnα (t) dµ(t),
σnα f (x)
Z
=
Gp
f (x + t)Knα (t) dµ(t),
where x ∈ Gp , n ∈ N. The pn th Dirichlet kernels have a closed form (see e.g. [SWS]): n p if x ∈ In , Dpψn (x) = Dpχn (x) = where x ∈ Gp . 0 if x 6∈ In ,
Ces` aro summability of the character system
5
We define the maximal operator σ ∗α f := sup |σnα f | n∈P
(f ∈ L1 (Gp )).
We say that the operator T : L1 → L0 is of type (q, q) if there exists a constant cq such that kT f kq ≤ cq kf kq
for all f ∈ Lq (Gp ) (1 ≤ q ≤ ∞).
T is said to be of type (H 1 , L1 ) if kT f k1 ≤ ckf kH 1
for all f ∈ H 1 (G).
We say that the operator T is of weak type (1, 1) if there exists a constant c > 0 such that for all λ > 0, f ∈ L1 (Gp ) the inequality µ({y ∈ G : T f (y) > λ}) ≤ ckf k1 /λ holds. Set S ∗α f := sup |Snα f | n∈P
for f ∈ L1 ,
where α is ψ or χ or any piecewise linear rearrangement of ψ (χ is of this kind) (for the notion of piecewise linear rearrangement see [SWS]). Then, for the case p = 2 (the Walsh case) it is known that S ∗ is of type (q, q) for all q ≥ 2 and for f ∈ Lq (q ≥ 2) it follows Sn f → f a.e. [SWS, Theorem 6.10]. Moreover, if α = χ (p = 2), f ∈ L1 (log+ L)2 (in particular if f ∈ Lq for any q > 1), then the Walsh–Kaczmarz–Fourier series of f converges to f a.e. on G (cf. Theorem 6.11 in [SWS]). Now, we turn our attention to the general case (p-series field). The main aim of this paper is to prove the following T h e o r e m 1. σnχ f → f (n → ∞) almost everywhere for all f ∈ L (Gp ). 1
T h e o r e m 2. The operator σ ∗χ is of type (q, q) for all 1 < q ≤ ∞, weak type (1, 1), and of type (H 1 , L1 ). These theorems in the in the Paley (not in the Kaczmarz) arrangement were studied by a number of authors. Fine [Fin] proved that every Walsh– Fourier series (in the Walsh case p = 2, for the Walsh–Paley system) is a.e. (C, α) summable for α > 1. Schipp [Sch3] gave a simpler proof for the case
6
G. G´ at and K. Nagy
α = 1. He proved that σ ∗ is of weak type (L1 , L1 ). Fujii [Fuj] discovered that σ ∗ is of type (H 1 , L1 ). The theorem of Schipp was generalized to the p-series fields by Taibleson [Tai], and later to bounded Vilenkin systems by ´l and Simon [PS]. Fujii’s result was extended to Hardy–Lorenz spaces by Pa Weisz [W], who proved that σ ∗ is of type (H p,q , Lp,q ) for every 12 < p < ∞ and 0 < q ≤ ∞.
2. The proofs In this paper c denotes an absolute constant which may not be the same at different occurrences. So does cq but it depends only on q (q ∈ R). In order to prove Theorems 1 and 2 we need some lemmas. L e m m a 3. Let k ∈ N. Then kKkχ (x) = 1 +
+p
|k|
k|k| −1
X l=1
|k|−1 p−1
X X
j=0 l=1
rjl (x)pj Kpψj (τj (x)) +
|k|−1
X
j=0
pj Dpψj (x)
p−1 l−1 XX l=1 i=0
k
ψ l r|k| (x)Kpψ|k| (τ|k| (x)) + r|k||k| (x)(k − k|k| p|k| )Kk−k
|k|
k|k| −1
X
+ (k − k|k| p )
i=0
i r|k| (x)Dpψ|k| (x)
+p
|k|
k|k| −1 j−1
X X
j=1 i=0
|k| |k| p
rji (x) +
(τ|k| (x)) +
i r|k| (x)Dpψ|k| (x).
P r o o f . k can be written in the form k = kn pn + · · · + k0 , Since
1 ≤ kn ≤ p − 1, ki ∈ N, i = 0, . . . , n.
k
χk (x) = r|k||k| (x)ψk−k|k| p|k| (τ|k|(x))
(k ∈ N, x ∈ Gp ),
we have for any l ∈ N, 0 < l < p − 1, x ∈ Gp , n
n
p X
χ (Dlp n +j (x) j=1
−
χ Dlp n (x))
and
n k−k np X
j=1
=
p X j=1
rnl (x)
j−1 X i=0
ψi (τn (x)) = rnl (x)pn Kpψn (τn (x))
(Dkχn pn +j (x) − Dkχn pn (x)) =
n
= rnkn (x)
k−k np X j=1
ψ Djψ (τn (x)) = rnkn (x)(k − kn pn )Kk−k n (τn (x)); np
Ces` aro summability of the character system
whence it follows that kKkχ (x)
n
=p
n
Kpχn (x)
+
p X
(Dpχn +j (x) − Dpχn (x)) + · · ·
j=1
pn X
χ χ + (D(k (x) − D(k n n (x)) + n −1)p +j n −1)p j=1
n k−k np X
j=1
+ (k − kn pn )Dkχn pn (x) + pn n X
=1+
7
(Dkχn pn +j (x) − Dkχn pn (x)) +
kX n −1 j=1
χ Djp n (x) =
(pj Kpχj (x) − pj−1 Kpχj−1 (x)) + pn Kpψn (τn (x))
kX n −1
j=1
l=1
ψ χ n n + rnkn (x)(k − kn pn )Kk−k n (τn (x)) + (k − kn p )Dk pn (x) + p np n
We
χ investigate Dlp n (x). n −1 p l−1 X X
χ Dlp n (x) =
rnl (x) +
kX n −1 j=1
χ Djp n (x).
For any 0 < l < p − 1, we have
χipn +j (x) = d
i=0 j=0
l−1 X i=0
rni (x)Dpψn (τn (x)) = d
l−1 X i=0
rni (x)Dpψn (x).
For any j ∈ N and x ∈ Gp , we have p
= =
j
Kpχj (x)
j−1 p−1 X pX
−p
j−1
l=1 i=1 p−1 X l rj−1 (x)pj−1 Kpψj−1 (τj−1 (x)) l=1
n p−1 X X j=1 l=1
+ pn
=
j−1 p−1 X pX
l=1 i=1
+p
l=1
j−1
p−1 X l=1
Dpψj−1 (x)
χ Dlp j−1 (x) = p−1 l−1 XX l=1 i=0
kKkχ (x) =
l rj−1 (x)pj−1 Kpψj−1 (τj−1 (x))+
kX n −1
χ Dlp j−1 +i (x) =
χ χ j−1 (Dlp j−1 +i (x) − Dlpj−1 (x)) + p
whence it follows that
= 1+
Kpχj−1 (x)
n X
j=1
pj−1 Dpψj−1 (x)
i rj−1 (x),
p−1 l−1 XX l=1 i=0
i rj−1 (x)+
ψ rnl (x)Kpψn (τn (x)) + rnkn (x)(k − kn pn )Kk−k n (τn (x)) + np
+ (k − kn pn )
kX n −1 i=0
rni (x)Dpψn (x) + pn
This completes the proof of Lemma 3.
kX n −1 j−1 X j=1 i=0
rni (x)Dpψn (x). u t
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G. G´ at and K. Nagy
L e m m a 4. For any f ∈ L1 (Gp ) set l f 4 := sup |EA (f rA )|. l,A∈N
4
Then the operator of type (H 1 , L1 ).
is of type (q, q) (1 < q ≤ ∞), of weak type (1, 1), and
P r o o f . Since f 4 ≤ |f |∗ , we have the operator 4 is of type (q, q) for any 1 < q ≤ ∞ and of weak type (1, 1). Thus the only thing we need to discuss is that the operator 4 is of type (H 1 , L1 ). To this end, we will show that the operator 4 is quasi-local (see [SWS]) (that is for any atom a ∈ H 1 we have Z |a4 | dµ ≤ c, Gp \Ik (xa )
a
where Ik (x ) is the correspondance interval relative to the atom a). Let a ∈ H 1 (Gp ) be an atom and y ∈ Ik (xa ). Then there exists a t ∈ N (t < k) for which y ∈ It (xa ) \ It+1 (xa ) holds. If A ≥ t + 1 and x ∈ Ik (xa ), then DpA (y − x) = 0. This implies that Z
l a(x)DpA (y − x)rA (x) dµ(x) = 0.
Ik (xa )
Z
If A ≤ t < k and x ∈ Ik (xa ), then DpA (y − x) = pA and
Ik (xa )
That is, Thus,
Z Gp
Z
=
Gp
l l a(x)DpA (y − x)rA (x) dµ(x) = pA rA (xa ) l EA (arA )(y) = 0
a4 (x) dµ(x) =
Ik (xa )
a(x) dµ(x) = 0.
for all A, l ∈ N and y ∈ Ik (xa ).
Z Ik
Z
(xa )
a4 (x) dµ(x) +
Z Ik
(xa )
a4 (x) dµ(x) =
s
1Ik (xa ) (x)a4 (x) dµ(x) ≤ k1Ik (xa ) k2 ka4 k2 ≤ c
1 pk
sZ Ik (xa )
p2k ≤ c.
For any f ∈ H 1 there exist λi ∈ C and ai ∈ H 1 atoms such that f = i=0 λi ai . Therefore,
P∞
kf 4 k1 ≤
∞ X i=0
|λi |ka4 i k1 ≤ ckf kH 1 .
u t
L e m m a 5. Let f ∈ L1 (Gp ), l ∈ N. Then the operator
Tl∗ f (y) := sup pA−l A≥l 1 1
Z
{x∈Gp :xl =···=xA−1 =0}
f (x + y) dµ(x)
(y ∈ Gp )
is of type (H , L ), of type (q, q) for all 1 < q ≤ ∞, and of weak type (1, 1) (uniformly in l).
Ces` aro summability of the character system
P r o o f . Set
X
g(z) := p−l
9
f (a0 e0 + · · · + al−1 el−1 + z),
z ∈ Gp .
ai ∈{0,1,...,p−1} i∈{0,1,...,l−1}
Then for n ∈ N, n > l, we get 1 En g(y) = µ(In (y)) =p
Z
X
n−l
ai ∈{0,1,...,p−1} i∈{0,1,...,l−1}
=p
n−l
In
Z In (y)
f (a0 e0 + · · · + al−1 el−1 + x + y) dµ(x) =
Z
{x∈Gp :xl =···=xn−1 =0}
Thus,
g(x) dµ(x) =
f (x + y) dµ(x).
Tl∗ f (y) = sup |En g(y)| ≤ g? (y). n≥l
Since the operator
f ? (y) := sup |En f (y)| n∈N
1
1
is of type (H , L ), of type (q, q) for all 1 < q ≤ ∞, and of weak type (1, 1) (see [SWS]), we have kTl∗ f kq ≤ kg? kq ≤ cq kgkq ≤ cq kf kq , kTl∗ f k1 ≤ kg? k1 ≤ ckgkH 1 ≤ ckf kH 1 , for all 1 < q ≤ ∞, and µ(Tl∗ f > λ) ≤ µ(g? > λ) ≤ ckgk1 /λ ≤ ckf k1 /λ. u t
This completes the proof of Lemma 5. For any l ∈ N, set Tl f (y) :=
sup
n,A∈N; A≥l
Z A−l p
{x∈Gp :xA−1 =···=xl =0}
f (x + y)
A Y j=0
n rj j (x) dµ(x) .
L e m m a 6. For any l ∈ N, the operator Tl is of type (q, q) for all 1 < q ≤ ∞, of weak type (1, 1), and of type (H 1 , L1 ). P r o o f . By virtue of Tl f (y) ≤ Tl∗ |f | and Lemma 5 the only thing we need to show is that the operator Tl is of type (H 1 , L1 ). Set t := t0 e0 + · · · + tl−1 el−1 ∈ Gp Then
for ti ∈ {0, . . . , p − 1} (0 ≤ i < l).
10
G. G´ at and K. Nagy
Z X ≤ p−l sup pA n,A∈N t
A≥l
= p−l
Tl f (y) ≤ {x∈Gp :xA−1 =···=x0 =0}
X
sup
Z A p
t A∈N;A≥l
≤ p−l
X
IA
f (x + y + t)
A Y j=0
n
rj j (x + t) dµ(x) =
nA f (x + y + t)rA (x + t) dµ(x) ≤
l sup |EA (f (y + t)rA (y + t))|.
t A,l∈N
By Lemma 4, we have kTl f k1 ≤ p−l
X t
kf 4 (· + t)k1 ≤ ckf (·)kH 1 . u t
This completes the proof of Lemma 6. L e m m a 7. Let f ∈ L1 (Gp ), l, t ∈ N, l < t. Then the operator Z := sup pA−l A≥t n,A∈N
Tl,t f (y) := {x∈Gp :xl =···=xt−1 =xt+1 =···=xA−1 =0,xt 6=0} A Q
n
rj j (x)
× dµ(x) 1 − rt (x) j=0
f (x + y) ×
(y ∈ Gp )
is of type (H 1 , L1 ), of type (q, q) for all 1 < q ≤ ∞, and of weak type (1, 1) (uniformly in l and t). P r o o f . We have
≤
p−1 X j=1
Tl,t f (y) ≤
Z sup pA−l A≥t
{x∈Gp :xl =···=xt−1 =xt+1 =···=xA−1 =0,xt =j}
n,A∈N
A Q
f (x + y) ×
rsns (x)
p−1 X × Tl f (y + jet ) dµ(x) ≤ c 1 − rt (x) j=1 s=0
for all y ∈ Gp . From this and Lemma 4 it follows that kTl,t f (·)kq ≤ c
p−1 X j=1
for each 1 < q ≤ ∞,
kTl f (· + jet )kq ≤
p−1 X j=1
cq kf (· + jet )kq = cq kf (·)kq
Ces` aro summability of the character system
kTl,t f (·)k1 ≤ c
p−1 X
kTl f (· + jet )k1 ≤
p−1 X
j=1
11
ckf (· + jet )kH 1 = ckf (·)kH 1 ,
j=1
and p−1 X
µ(Tl,t f > λ) ≤ µ c
Tl f (· + jet ) > λ ≤
j=1
≤
p−1 X
p−1 X
µ Tl f (· + jet ) >
j=1
c
j=1
cλ ≤ p−1
(p − 1)kf (· + jet )k1 kf (·)k1 =c . λ λ u t
The proof of Lemma 7 is complete. It is well known that [Vil, SWS] Dnψ (x) = ψn (x)
∞ X k=0
Let α Ka,b :=
Dpψk (x)
a+b−1 X j=a
p−1 X j=p−nk
Djα
and n(s) :=
∞ X
rkj (x)
(x ∈ Gp , n ∈ N).
(a, b ∈ N, α = χ, ψ),
ni pi (n, s ∈ N).
i=s
By a simple calculation, we get nKnα =
|n| ns −1 X X s=0 l=0
Knα(s+1) +lps ,ps + Dnα
(α = ψ, χ, n ∈ P).
L e m m a 8. Let s, t, n, l ∈ N, l < ns , and x ∈ It \ It+1 . If s ≤ t ≤ |n|, then
|Knψ(s+1) +lps ,ps (x)| ≤ cps+t .
On the other hand, if t < s ≤ |n|, then (
Knψ(s+1) +lps ,ps (x)
=
0 ps+t ψn(s+1) +lps (x) 1−r t (x)
if x − xt et ∈ 6 Is , if x − xt et ∈ Is .
P r o o f . If s ≤ t ≤ |n|, then for all k ∈ N we have |Dkψ (x)|
≤
t X j=0
p−1 X i p rj (x) ≤ cpt , j
i=p−kj
12
G. G´ at and K. Nagy
whence it follows that |Knψ(s+1) +lps ,ps (x)| ≤ cps+t . If |n| ≥ s > t, then t X
Dnψ(s+1) +lps +j (x) = ψn(s+1) +lps +j (x) t−1 X
= ψn(s+1) +lps +j (x)
p−1 X
pk
k=0
rki (x) + pt
i=p−jk
k=0
Dpψk (x)
p−1 X i=p−jk p−1 X
i=p−jt
rki (x) =
rti (x) .
Using this we have Knψ(s+1) +lps ,ps (x)
=
s pX −1
j=0
=
s pX −1
j=0
ψn(s+1) +lps +j (x)
s
+
pX −1 j=0
P
P
1
1
pk jk +
k=0
i=p−jt
rti (x) =
P
1
+
P
2.
= 0. In fact,
= ψn(s+1) +lps (x)
= ψn(s+1) +lps (x)
t−1 X
p−1 X
ψn(s+1) +lps +j (x)pt
First, we prove that
Dnψ(s+1) +lps +j (x) =
s pX −1
ψj (x)
j=0
t−1 X
pk jk =
k=0
p−1 X
t−1 X
pk jk
p−1 X
ψj (x) = 0,
jt =0
ji =0,i6=t,i=0,...,s−1 k=0
where we used the fact that (remember that x ∈ It \ It+1 , thus xt 6= 0) p−1 X
ψj (x) =
jt =0
= exp
2
exp
jt =0
2πi
p
s−1 X
xm jm
P
2.
2πi s−1 X
p
m=t
p−1 X
m=t+1
Second, we investigate P
p−1 X
exp
xm jm =
2πi
jt =0
p
xt jt = 0.
We have
= ψn(s+1) +lps (x)p
t
s −1 pX
j=0
ψj (x)
p−1 X i=p−jt
rti (x)
Ces` aro summability of the character system
and
s pX −1
j=0 s
=
pX −1 s−1 Y j=0
=
k=0 k6=t
p−1 X js−1 =0
=
s−1 Y p−1 X k=0 k6=t
rkjk (x)
jk =0
rtjt (x)
j
s−1 rs−1 (x) · · ·
p−1 X
ψj (x)
i=p−jt p−1 X
i=p−jt
rti (x)
k=0 k6=t
r jt (x) − 1
rkjk (x)
p−1
rt (x) − 1
1
s pX −1 s−1 Y
t
rt (x) − 1
=
p−1
X j rtjt (x) − 1 X jt−1 rt−1 (x) · · · r00 (x) = r (x) − 1 t jt =0 jt−1 =0 j0 =0
jt =0
Q
rti (x) =
j=0
p−1 X
p−1 X rtjt (x) − 1
rkjk (x)
=
13
=
0 ps−1
=:
p−1 j Q X rt t (x) − 1 1
jt =0
rt (x) − 1
=
Q
if x − xt et ∈ 6 Is , if x − xt et ∈ Is .
1
p , 1 − rt (x) u t
C o r o l l a r y 9. Let A, t ∈ N, A > t. If x ∈ It \ It+1 , then (
KpψA (x)
=
if x − xt et ∈ 6 IA , if x − xt et ∈ IA .
0
pt (1−rt (x))
If x ∈ IA , then KpψA (x) =
pA + 1 . 2
P r o o f . Let x ∈ IA , then Djψ (x) = j for all j ≤ pA . This implies that A
KpψA (x)
p 1 X 1 + pA = A j= . p j=1 2
If x ∈ It \ It+1 for some t < A, then by Lemma 8, we have DpψA (x) = 0 and A
p
A
KpψA (x)
=
)s −1 A (p X X s=0
l=0
ψ ψ ψ K(p A )(s+1) +lps ,ps (x) + DpA (x) = K(pA )(A+1) ,pA (x),
u t
so that Corollary 9 holds. L e m m a 10. Let
Z Lf (y) := sup l,A∈N
Gp
l f (x+ y)rA (x)KpψA (τA (x)) dµ(x)
(y ∈ Gp , f ∈ L1 (Gp )).
Then the operator L is of type (q, q) for all 1 < q ≤ ∞, of weak type (1, 1), and of type (H 1 , L1 ).
14
G. G´ at and K. Nagy
P r o o f . We have
Z Lf (y) ≤ sup
Z + sup l,A∈N
IA
l,A∈N
f (x +
Gp \IA
f (x +
l y)rA (x)KpψA (τA (x)) dµ(x)
l y)rA (x)KpψA (τA (x)) dµ(x)
=: R1 + R2 .
By Corollary 9, we have
Z
R1 = sup Z = sup l,A∈N
IA
IA
l,A∈N
+
l f (x + y)rA (x)KpψA (τA (x)) dµ(x) =
f (x +
p l y)rA (x)
A
+1 dµ(x) ≤ cf 4 (y). 2
First, we discuss R2 and decompose the sets Gp \ IA and It \ It+1 . Set ItT := {x ∈ Gp : xt 6= 0, xT 6= 0 and xi = 0 for i < T, i 6= t} for an integer T > t. Then Gp \ IA and It \ It+1 can be decomposed in the following way: Gp \IA =
A−1 [
(It \It+1 ),
t=0
T =t+1
Thus R2 ≤ sup
A−1 X Z
l,A∈N t=0
≤ ≤ = +
∞ X
∞ X
It \It+1
Z
l f (x + y)rA (x)KpψA (τA (x)) dµ(x) ≤
sup
T t=0 A>t,l∈N T =t+1 It
A−1 Z X sup
T t=0 A>t,l∈N T =t+1 It ∞ Z X
t=0 A>t,l∈N T =A
ItT
l f (x + y)rA (x)KpψA (τA (x)) dµ(x) ≤
sup
sup
ItT ∪{et ; 2et ; . . . ; (p−1)et }.
t=0 A>t,l∈N It \It+1 ∞ Z ∞ X X
∞ X
∞ [
where It \It+1 =
f (x +
f (x +
l y)rA (x)KpψA (τA (x)) dµ(x)
f (x +
l y)rA (x)KpψA (τA (x)) dµ(x)
=
l y)rA (x)KpψA (τA (x)) dµ(x)
=:
P
+
1
+
P
2.
If T < A and x ∈ ItT then τA (x)A−t−1 = 6 0, τA (x) 6= 0. ConseP A−T −1 quently τA (x) − τA (x)A−t−1 6∈ IA . Corollary 9 gives 1 = 0. P Second, we investigate 2 . P
2
=
∞ X
∞ Z X
sup
T t=0 A>t, l∈N T =A It
l f (x + y)rA (x)
pA−t−1 dµ(x) ≤ (1 − rA−t−1 (τA (x))
Ces` aro summability of the character system
≤c
∞ X
p−t
t=0
≤c
Z
sup pA Sp−1
A>t, l∈N
∞ X
p−t
t=0
j=1
p−1 X
sup
j=1 A>t, l∈N
≤c
∞ X
IA (jet )
Z A p
p−t
t=0
l f (x + y)rA (x)
15
1 dµ(x) ≤ 1 − rt (x)
IA (jet )
p−1 X
l f (x + y)rA (x) dµ(x) ≤
T0 f (y + jet ).
j=1
This together with Lemmas 4 and 6 gives for any 1 < p ≤ ∞ that kLf kp ≤ ckf 4 kp + c
∞ X
p−t
t=0
≤ cp kf kp + cp
∞ X
p−t
t=0
p−1 X
kT0 f (· + jet )kp ≤
j=1
p−1 X
kf (· + jet )kp ≤ cp kf kp .
j=1
On the other hand, by Lemmas 4 and 6 we have kLf k1 ≤ ckf 4 k1 + c
∞ X
p−t
t=0
≤ ckf kH 1 + c
∞ X
p−t
t=0
p−1 X j=1
p−1 X
kT0 f (· + jet )k1 ≤
j=1
kf (· + jet )kH 1 ≤ ckf kH 1 .
Finally, let λ > 0. Then µ(Lf > cλ) ≤ µ(f 4 > cλ) + µ
∞ X t=0
p−t
p−1 X
T0 f (· + jet ) > cλ ≤
j=1
∞ [ [ kf k1 {T0 f (· + jet ) > pt/2 cλ} ≤ +µ λ t=0 j=1 p−1
≤c
p−1
∞ X kf k1 X ≤c µ(T0 f (· + jet ) > pt/2 cλ) ≤ + λ t=0 j=1 p−1
≤c
∞ X X kf k1 kf (· + jet )k1 kf k1 p−t/2 +c ≤c . λ λ λ t=0 j=1
u t
The proof of Lemma 10 is complete. For any f ∈ L1 (Gp ), define the operator M as follows: M f (y) :=
sup
Z
k,n,A∈N;|n|
Gp
f (x +
k y)rA (x)Knψ (τA (x)) dµ(x)
(y ∈ Gp ).
16
G. G´ at and K. Nagy
L e m m a 11. The operator M is of type (q, q) for all 1 < q ≤ ∞, of weak type (1, 1), and of type (H 1 , L1 ). P r o o f . For any n ∈ P, we have nKnψ =
|n| ns −1 X X s=0 l=0
e ψ + Dψ . Knψ(s+1) +lps ,ps + Dnψ =: nK n n
From the fact that kDnψ k∞ ≤ n it follows that Z 1 k ψ sup f (x + y)rA (x)Dn (τA (x)) dµ(x) ≤ k,n,A∈N n Gp |n|
≤ sup
Z
n∈N Gp
and
Z M f (y) ≤ sup k,n,A∈N |n|
Gp
|f (x + y)|dµ(x) = kf k1
f (x +
k e ψ (τ (x)) dµ(x) y)rA (x)K n A
+ ckf k1 =
ff (y) + ckf k . =: M 1 f. For any A, t ∈ N, A > t ≥ 1, set Thus, we have to discuss the operator M
JtA := {x ∈ Gp : xA−1 = · · · = xA−t = 0; xA−t−1 6= 0} and
J0A := {x ∈ Gp : xA−1 6= 0}
for A ≥ 1.
Then Gp can be represented as the disjoint union Gp = IA ∪ 1 ≤ A ∈ N. If x ∈ IA and |n| < A, then τA (x) ∈ IA and e ψ (τ (x)) = nK e ψ (x) = nK n A n
Furthermore, we have ff (y) ≤ M
Z
sup k,n,A∈N |n|
n pA
sup k,n,A∈N |n|
n pA
IA
Gp \IA
Z
sup k,n,A∈N |n|
Gp \IA
n(n − 1) . 2
|n|
+
JtA , where
k e ψ (τA (x)) dµ(x) + f (x + y)rA (x)K n
k e ψ (τ (x)) dµ(x) ≤ f (x + y)rA (x)K n A
Z n(n − 1) ≤ sup k,n,A∈N 2pA
1 pA
t=0
Z
+
SA−1
f (x + y)
IA
k f (x + y)rA (x) dµ(x) +
|n| ns −1 X X s=0 l=0
k rA (x)Knψ(s+1) +lps ,ps (τA (x)) dµ(x)
≤
Ces` aro summability of the character system
17
≤ cf 4 (y) +
+
|n| ns −1 Z XX X 1 A−1 ψ k ≤ f (x + y)r (x)K (τ (x)) dµ(x) A (s+1) +lps ,ps A n A p t=0 s=0 l=0 JtA
sup k,n,A∈N |n|
≤ cf 4 (y) +
+
t nX s −1 Z XX 1 A−1 ψ k + f (x + y)r (x)K (τ (x)) dµ(x) A (s+1) +lps ,ps A n A p t=0 s=0 l=0 JtA
sup k,n,A∈N |n|
+ sup k,n,A∈N |n|
A nX s −1 Z X X 1 A−1 ψ k f (x+y)rA (x)Kn(s+1) +lps ,ps (τA (x)) dµ(x) =: A p t=0 s=t+1 l=0 JtA
=: cf 4 (y) + S 1 f (y) + S 2 f (y). First, we discuss S 1 f (y). If x ∈ JtA , then τA (x) ∈ It \ It+1 (s ≤ t < A). In case x ∈ It \ It+1 we have Knψ(s+1) +lps ,ps (x) = ps Dnψ(s+1) +lps (x) + t X
= ps ψn(s+1) +lps (x)
p−1 X
pk
k=0
ps (ps − 1) ψn(s+1) +lps (x) = 2
j=p−(n(s+1) +lps )k
rkj (x) +
ps − 1 , 2
for all x ∈ It \ It+1 (s ≤ t < A). For any x ∈ JtA and s ≤ t ≤ |n| < A, we have τA (x) = (0, 0, . . . , 0, xA−t−1 , xA−t−2 , . . . , x0 , xA , xA+1 , . . .) (the first t coordinates are zeros). This implies that n
A−1 ψn(s+1) +lps (τA (x)) = rtnt (xA−t−1 et ) · · · rA−1 (x0 eA−1 ) =
= exp for all x ∈ have 1
S f (y) ≤
JtA
2πi
p
(nA−1 x0 + · · · + nt xA−t−1 ) =:
i=0
ni rie (x)
(in case s = t we have nt = l). By this and l := A − t, we
sup k,n,A∈N |n|≤A
× ps l +
t−1 nX s −1 Z XX 1 A−1 k f (x + y)rA (x)ps ψn(s+1) +lps (τA (x)) × A p t=0 s=0 l=0 JtA t−1 X
sup k,n,A∈N |n|≤A
1 pA
p−1 X
pk n k + pt
k=s+1
+
A−t−1 Y
A−1 t −1 Z X nX t=0 l=0
j=p−nt
JtA
rtj (τA (x)) +
ps − 1 dµ(x)| + 2
k f (x + y)rA (x)pt ψn(t+1) +lpt (τA (x)) ×
18
G. G´ at and K. Nagy
× p
p−1 X
t
j=p−l
≤
k,n,A∈N |n|≤A
A−1 t−1 nX s −1 Z XX
1 pA
sup
t=0 s=0 l=0
t−1 X
s
× p l+
k
p nk + p
+
k,n,A∈N |n|≤A
p−1 X
t
j=p−l
k,n,A∈N |n|≤A
+ c sup k,n,A∈N |n|≤A
A−1 X t−1 X
1 pA
t=0 s=0
JtA
+ c sup k,n,A∈N |n|≤A
A−t−1 Y
1 pA
×
t=0 l=0
≤ c sup
A−t−1 Y i=0
k y)rA (x)
rini (x) dµ(x) +
A−t−1 Y i=0
rini (x) dµ(x)
+
j rini (x)rA−t−1 (x) dµ(x) +
t=0
A X l=1
Z
p2t
JtA
k f (x + y)rA (x) ×
j rini (x)rA−t−1 (x) dµ(x) +
Z
pt (pt − 1) 2
Z A−1 X 2t−A p
k,n,A∈N |n|≤A
f (x +
t=0 l=0 j=p−l
i=0
k,n,A∈N |n|≤A
i=0
ni rie (x) ×
k f (x + y)rA (x)
JtA
A−t−1 Y
pt − 1 + dµ(x) ≤ 2
i=0 A−1 t −1 p−1 X nX X
A−t−1 Y
A−1 t −1 X nX
≤ c sup
Z X t−1 X p−1 X 1 A−1 s+t k p f (x + y)rA (x) × A p t=0 s=0 j=p−nt JtA
×
k,n,A∈N |n|≤A
Z
p2s
t=0 s=0
k,n,A∈N |n|≤A
+ c sup
j rA−t−1 (x)
i=0
ni rie (x) ×
ps − 1 + dµ(x) + 2
j rA−t−1 (x)
k f (x + y)rA (x)pt
JtA
Z X t−1 X 1 A−1 s+t p A p
+ c sup
1 pA
t=0 l=0
× p ≤ c sup
p−1 X
t
A−t−1 Y
k f (x + y)rA (x)ps
JtA
A−1 t −1 Z X nX
1 pA
sup
j=p−nt
k=s+1
pt − 1 + dµ(x) ≤ 2
rtj (τA (x))
JtA
Z A−2l p
JtA
f (x +
A JA−l
f (x +
k y)rA (x)
k y)rA (x)
f (x +
A−t−1 Y i=0
k y)rA (x)
l−1 Y i=0
A−t−1 Y i=0
rini (x) dµ(x) ≤
e ni ri (x) dµ(x) ≤
rini (x) dµ(x) ≤
Ces` aro summability of the character system
≤ c sup
A X
A∈N l=1
p−l Tl f (y) ≤
∞ X
19
p−l Tl f (y).
l=1
Lemma 6 implies that the operator S 1 is of type (q, q) for all 1 < q ≤ ∞, of weak type (1, 1), and of type (H 1 , L1 ). Second, we investigate S 2 f (y). If x ∈ JtA , then τA (x) ∈ It \ It+1 (t < s ≤ A). We have two subcases: (i)
τA (x) − (τA (x))t et 6∈ Is ,
(ii)
τA (x) − (τA (x))t et ∈ Is . In case (i), Lemma 8 implies that Knψ(s+1) +lps ,ps (τA (x)) = 0.
Thus, it remains to discuss (ii). In this case τA (x) is of the form τA (x) = (xA−1 , xA−2 , . . . , xA−t , .., xA−s , . . . , x0 , xA , xA+1 , . . .) = = (0, 0, . . . , 0, xA−t−1 , 0, . . . , 0, xA−s−1 , . . . , x0 , xA , xA+1 , . . .) and n
n
s+1 A−1 ψn(s+1) +lps (τA (x)) = rsl (τA (x))rs+1 (τA (x)) · · · rA−1 (τA (x)) =
= exp
2πi
p
(x0 nA−1 + x1 nA−2 + · · · + xA−s−2 ns+1 + xA−s−1 l) = =
A−s−2 Y i=0
Set
ni l rie (x)rA−s−1 (x).
JtA,s := {x ∈ Gp : xA−1 = · · · = xA−s = 0, xA−t−1 6= 0},
for any A, s, t ∈ N, A ≥ s > t. This implies that S 2 f (y) = Z
k,n,A∈N |n|
JtA
f (x +
sup k,n,A∈N |n|
JtA,s
f (x +
A nX s −1 X X 1 A−1 pA t=0 s=t+1 l=0
k y)rA (x)Knψ(s+1) +lps ,ps (τA (x)) dµ(x) =
= Z
sup
p k y)rA (x)
A nX s −1 X X 1 A−1 pA t=0 s=t+1 l=0 s+t
ψn(s+1) +lps (τA (x)) dµ(x) ≤ 1 − rt (τA (x))
20
G. G´ at and K. Nagy
≤ Z
sup k,n,A∈N |n|
A nX s −1 X X 1 A−1 pA t=0 s=t+1 l=0
ps+t f (x + A,s
Jt
k y)rA (x)
A−s−2 Q i=0
ni l rie (x)rA−s−1 (x)
1 − rA−t−1 (x)
dµ(x) .
Set j := A − s, m := A − t. Then 2
S f (y) ≤ Z
≤c
A m−1 s −1 X X nX
sup k,n,A∈N |n|
m=1 j=0 l=0
pA−m−j A,A−j JA−m
p−1 X
sup
f (x +
k y)rA (x)
A m−1 X X
l=0 A∈N m=1 j=0
j−2 Q i=0
ni l rie (x)rj−1 (x)
1 − rm−1 (x)
p−m Tj,m−1 f (y) ≤ c
∞ m−1 X X
dµ(x) ≤
p−m Tj,m−1 f (y).
m=1 j=0
By Lemma 7, the operator Tl,m−1 is of type (q, q) for all 1 < q ≤ ∞ uniformly in l, m and of type (H 1 , L1 ). From this it follows that kS 2 kq ≤ c
∞ m−1 X X
p−m kTl,m−1 f kq ≤ cq kf kq
m=1 l=0
for all 1 < q ≤ ∞ and kS 2 k1 ≤ c
∞ m−1 X X m=1 l=0
p−m kTl,m−1 f k1 ≤ ckf kH 1 .
Moreover, ∞ m−1 X X
µ(S 2 > cλ) ≤ µ c
p−m Tl,m−1 f > cλ ≤
m=1 l=0
≤
∞ m−1 X X
µ(Tl,m−1 f > cλpm/2 ) ≤
m=1 l=0
∞ m−1 X X cp−m/2 m=1 l=0
λ
kf k1 ≤ ckf k1 /λ. u t
This completes the proof of Lemma 11. P r o o f o f T h e o r e m 2. Let 1 < q ≤ ∞. By Lemma 3, we have
Z
1 ∗χ
kσ f (·)kq ≤ sup f (x + ·) dµ(x)
+ n Gp n∈P q
|n|−1 j p−1 Z X p X
+ sup
n n∈P j=0
l=1
Gp
f (x +
·)rjl (x)Kpψj (τj (x)) dµ(x)
+ q
Ces` aro summability of the character system
+
sup
X pj p−1 X Z
|n|−1
n
n∈P j=0
Z
p|n|
+ sup n
l=1
Gp
X
f (x + ·)
l−1 X i=0
rji (x) dµ(x)
+ q
n|n| −1
Gp
n∈P
f (x + ·)Dpψj (x)
21
l r|n| (x)Kpψ|n| (τ|n| (x)) dµ(x)
q
l=1
+
Z
n|n|p|n| n|n| ψ
+ sup 1 − f (x + ·)r|n| (x)Kn−n p|n| (τ|n| (x)) dµ(x)
+ |n| n Gp n∈P q
n −1
Z |n| |n| X
n|n| p i + f (x + ·) r|n| (x)Dpψ|n| (x) dµ(x)
sup 1 −
+ n Gp
n∈P
Z
p|n|
+ sup n n∈P
Gp
q
i=0 n|n| −1 j−1
X X ψ i f (x + ·) r|n| (x)Dp|n| (x)
=: j=1
q
i=0
=: j1 + j2 + j3 + j4 + j5 + j6 + j7 . Clearly, j1 ≤ kf kq . The definition of the operator L, Lemma 10, and the fact that Z
Gp
f (x +
·)rjl (x)Kpψj (τj (x)) dµ(x)
≤ Lf (·)
give that
|n|−1
X pj p−1 X
j2 ≤ sup
n
n∈P j=0
and
Lf (·) ≤ cq kf kq q
l=1
n|n| −1
p|n| X
j4 ≤ sup Lf (·) ≤ cq kf kq . n∈P
n
l=1
By Lemma 4 and the fact that Z
Gp
q
f (x + ·)rjl (x)Dpψj (x) dµ(x) ≤ f 4 (·),
we have that |n|−1 j p−1 l−1
X p XX
j3 ≤ sup f 4 (·) ≤ cq kf kq , n∈P j=0
n
l=1 i=0
q
|n|−1
n|n| p|n| X 4
j6 ≤ sup 1 − f (·)
≤ cq kf kq , n n∈P q i=0
and
n|n| −1 j−1
p|n| X X 4
j7 ≤ sup f (·) ≤ cq kf kq . n∈P
n
j=1
i=0
q
22
G. G´ at and K. Nagy
Since
Z
Gp
we get
f (x +
n ψ ·)r|n||n| (x)Kn−n |n| (τ|n| (x)) dµ(x) |n| p
≤ M f (·),
n|n| p|n|
j5 ≤ sup 1 − M f (·)
≤ cq kf kq . n n∈P q
That is, kσ ∗χ f kq ≤ cq kf kq
for all f ∈ Lq (Gp ) and 1 < q ≤ ∞.
The same consideration as above gives that kσ ∗χ f k1 ≤ ckf kH 1
for all f ∈ L1 (Gp ).
On the other hand, we have
1 µ(σ f > cλ) ≤ µ sup n∈P n
∗χ
Z Gp
f > cλ +
+ µ(f 4 > cλ) + µ(Lf > cλ) + µ(M f > cλ) ≤ c
kf k1 . λ
This completes the proof of Theorem 2.
u t
P r o o f o f T h e o r e m 1. Taking into account that the maximal operator σ ∗χ is of weak type (1, 1), the Kaczmarz polynomials (which are of the form P =
k X
aj χj
for some a0 , . . . , ak ∈ C (k ∈ N))
j=0
are dense in L1 (Gp ), and the fact that lim σnχ P = P
n→∞
everywhere,
one can prove Theorem 1 by a standard argument (see, e.g., [SWS], [G´ at]).
References [Bal] L. A. Bala˘ sov, Series with respect to the Walsh system with monotone coefficients, Sibirsk Math. Z., 12(1971), 25–39. [Fin] N. J. Fine, Ces` aro summability of Walsh–Fourier series, Proc. Nat. Acad. Sci. U.S.A., 41(1955), 558–591. [Fuj] N. Fujii, A maximal inequality for H 1 functions on the generalized Walsh–Paley group, Proc. Amer. Math. Soc., 77(1979), 111–116. ´ t, On (C, 1) summability of integrable functions with respect to the Walsh– [G´ at] G. Ga Kaczmarz system, Studia Math., 130(1998), 135–148.
Ces` aro summability of the character system
23
´ l and P. Simon, On a generalization of the concept of derivative, Acta Math. [PS] J. Pa Acad. Sci. Hungar., 29(1977), 155–164. [Sch1] F. Schipp, Certain rearrangements of series in the Walsh series, Mat. Zametki, 18(1975), 193–201. [Sch2] F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Analysis Math., 2(1976), 63–75. ¨ [Sch3] F. Schipp, Uber gewiessen Maximaloperatoren, Ann. Univ. Sci. Budapest., Sectio Math., 18(1975), 189–195. [Sk1] V. A. Skvorcov, On Fourier series with respect to the Walsh–Kaczmarz system, Analysis Math., 7(1981), 141–150. [Sk2] V. A. Skvorcov, Convergence in L1 of Fourier series with respect to the Walsh– Kaczmarz system, Vestnik Mosk. Univ. Ser. Mat. Mekh., 6(1981), 3–6 (in Russian). ˘ [Sn] A. A. Sneider, On series with respect to the Walsh functions with monotone coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 12(1948), 179–192 (in Russian). ´ l, Walsh Series. An Introduction [SWS] F. Schipp, W. R. Wade, P. Simon, and J. Pa to Dyadic Harmonic Analysis, Adam Hilger (Bristol–New York, 1990). [Tai] M. H. Taibleson, Fourier series on the ring of integers in a p-series field, Bull. Amer. Math. Soc., 73(1967), 623–629. [Vil] N. Ya. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk. SSSR Ser. Math., 11(1947), 363–400 (in Russian). [Y] W. S. Young, On the a.e. convergence of Walsh–Kaczmarz–Fourier series, Proc. Amer. Mat. Soc., 44(1974), 353–358. [W] F. Weisz, Ces` aro summability of one- and two-dimensional Walsh–Fourier series, Analysis Math., 22(1996), 229–242.
Summiruemost~ po Qezaro sistemy harakterov pol p-r dov v numeracii Kaqmaa G. GAT i K. NAD^
Pust~ Gp — pole p-r dov. Dl integriruemyh funkci$ i f ∈ L1 (Gp ) dokazana shodimost~ σn f → f (n → ∞) poqti vsdu, gde σn f — (C, 1)-srednie f po sisteme harakterov v numeracii Kaqmaa. Dl maksimal~nogo operatora σ ∗ f := supn |σn f | dokazano, qto on imeet tip (q, q) dl vseh 1 < q ≤ ∞ i imeet slaby$ i tip (1, 1). Krome togo, dokazano, qto kσ ∗ f k1 ≤ ckf kH , gde H — prostranstvo Hardi na Gp . ´ COLLEGE OF NY´IREGYHAZA INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCES ´ 4400 NY´IREGYHAZA P.O.BOX 166. HUNGARY
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