I2 is given by In,m(x1, x2

24 (2008), 209–214 www.emis.de/journals ISSN 1786-0091

MAXIMAL (C, α, β) OPERATORS OF TWO-DIMENSIONAL WALSH-FOURIER SERIES

USHANGI GOGINAVA

Abstract. The main aim of this paper is to prove that for the boundedness of the maximal operator σ∗^α,β from the Hardy space H_p¡

I²¢

to the space Lp

¡I²¢

the assumptionp >max{1/(α+ 1),1/(β+ 1)}is essential.

We denote the set of non-negative integers byN. For a setX 6=∅letX² be its Cartesian productX×Xtaken with itself. By a dyadic interval inI := [0,1) we mean one of the form£

l2^−k,(l+ 1) 2^−k¢

for some k∈N, 0≤l <2^k. Given k ∈N and x∈[0,1), letI_k(x) denote the dyadic interval of length 2^−k which contains the point x. The Cartesian product of two dyadic intervals is said to be a rectangle. Clearly, the dyadic rectangle of area 2⁻ⁿ×2^−m containing (x¹, x²) ∈ I² is given by I_n,m(x¹, x²) := I_n(x¹)×I_m(x²). We also use the notation mes (A) for the Lebesgue measure of any measurable set A.

Letr₀(x) be a function defined by r₀(x) =

(

1, if x∈[0,1/2),

−1, if x∈[1/2,1), r0(x+ 1) =r0(x).

The Rademacher system is defined by

r_n(x) =r₀(2ⁿx), n≥1 and x∈[0,1).

Letw₀, w₁, . . . represent the Walsh functions, i.e. w₀(x) = 1 and if n= 2ⁿ¹ +· · ·+ 2^nr

is a positive integer with n1 > n2 >· · ·> nr then w_n(x) =r_n1(x)· · ·r_nr(x).

2000Mathematics Subject Classification. 42C10.

Key words and phrases. Walsh function, Hardy space, Maximal operator.

209

The Walsh-Dirichlet kernel is defined by D_n(x) =

Xn−1

k=0

w_k(x). Recall that

D₂ⁿ(x) =

(2ⁿ, if x∈[0,2⁻ⁿ), 0, if x∈[2⁻ⁿ,1).

The Kronecker product (w_n,m :n, m∈N) of two Walsh systems is said to be the two-dimensional Walsh system. Thus

wn,m

¡x¹, x²¢ :=wn

¡x¹¢ wm

¡x²¢ .

The partial sums of the two-dimensional Walsh-Fourier series are defined as follows:

S_n,mf¡ x¹, x²¢

= Xn−1

i=0 m−1X

j=0

fb(i, j)w_i,j¡ x¹, x²¢

, where the number

fb(i, j) = Z

I

f¡ u¹, u²¢

wi,j

¡u¹, u²¢

du¹du²

is said to be the (i, j)th Walsh-Fourier coefficient of the function f. The norm (or quasinorm) of the spaceL_p(I²) is defined by

kfk_p :=



 Z

I²

¯¯f¡

x¹, x²¢¯¯^pdx¹dx²





1/p

(0< p <+∞).

The σ-algebra generated by the dyadic rectangles {I_n,m(x¹, x²) :x, y ∈I}

will be denoted by F_n,m(n, m∈N), more precisely, F_n,m =σ©£

k2⁻ⁿ,(k+ 1) 2⁻ⁿ¢

×£

l2^−m,(l+ 1) 2^−m¢

: 0≤k <2ⁿ,0≤l <2^mª , where σ(A) denotes the σ-algebra generated by an arbitrary set system A.

Denote by f = ¡

f^(n,m), n∈N¢

two-parameter martingale with respect to (F_n,m, n, m∈N) (for details see, e.g. [6, 9]). The maximal function of a martingale f is defined by

f^∗ = sup

n,m∈N

¯¯f^(n,m)¯

¯.

In case f ∈L₁(I²), the maximal function can also be given by

f^∗¡ x¹, x²¢

= sup

n,m∈N

1

mes (In(x¹)×Im(x²))

¯¯

¯¯

¯¯

¯ Z

In(x¹)×Im(x²)

f(u, v)dudv

¯¯

¯¯

¯¯

¯ , (x¹, x²)∈I².

For 0 < p <∞ the Hardy martingale space H_p(I²) consists all martingales for which

kfk_Hp :=kf^∗k_p <∞.

Iff ∈L₁(I²) then it is easy to show that the sequence (S₂ⁿ_,2^m(f) :n, m∈N) is a martingale. If f is a martingale, that is f = (f^(n,m) :n, m∈ N) then the Walsh-Fourier coefficients must be defined in a little bit different way:

fb(i, j) = lim

k,l→∞

Z

I²

f^(k,l)¡ x¹, x²¢

w_i(x¹)w_j(x²)dx¹dx².

The Walsh-Fourier coefficients of the function f ∈ L1(I²) are the same as the ones of the martingale (S2ⁿ,2^m(f) :n, m∈N) obtained from the function f.

The (C, α, β) means of the two-dimensional Walsh-Fourier series of the mar- tingalef is given by

σ^α,β_n,m(f, x¹, x²) = 1 A^α_n−1

1 A^β_m−1

Xn

i=1

Xm

j=1

A^α−1_n−iA^β−1_m−jS_i,jf¡ x¹, x²¢

, where

A^α_n := (1 +α). . .(n+α) n!

for any n∈N, α6=−1,−2, . . .. It is known ([10]) thatA^α_n∼n^α. For the martingalef we consider the maximal operator

σ^α,β_∗ f = sup

n,m

|σ_n,m^α,β(f, x¹, x²)|.

The (C, α) kernel defined by K_n^α(x) := 1

A^α_n−1 Xn

k=1

A^α−1_n−jD_k(x).

In the one-dimensional case, Fine [1] proved that the (C, α) meansσ_n^αf of a functionf ∈L(I) converge a.e. tof asn → ∞. The maximal operatorσ_∗^αf :=

sup

n |σ_n^αf| (0< α <1) of the (C, α) means of the Walsh-Paley Fourier series was investigated by Weisz [8]. In his paper Weisz proved the boundedness of σ^α_∗ :Hp →Lp when p >1/(1 +α). The author [3] showed that in Theorem of Weisz the assumptionp >1/(α+ 1) is essential. In particular, we proved that the maximal operatorσ^α_∗ of the (C, α) means of the Walsh-Paley Fourier series is not bounded from the Hardy space H_1/(α+1)(I) to the space L_1/(α+1)(I).

For double Walsh-Fourier series it is known [5] that the (C, α, β) means σ^α,β_n,mf →f inL_pnorm asn, m→ ∞wheneverf ∈L_p(I²) for some 1≤p <∞.

On the other hand, in 1992 M´oricz, Schipp and Wade [4] proved with respect to the Walsh-Paley system that

σ_n,mf = 1 nm

Xn

i=1

Xm

k=1

S_i,k(f)→f

a.e. for each f ∈ Llog⁺L([0,1)²), when min{n, m} → ∞. In 2000 G´at proved [2] that the theorem of M´oricz, Schipp and Wade above can not be improved. Namely, let δ : [0,+∞) → [0,+∞) be a measurable function with property lim_t→∞δ(t) = 0. G´at proved [2] the existence of a function f ∈L¹(I²) such that f ∈Llog⁺Lδ(L), and σ_n,mf does not converge tof a.e.

as min{n, m} → ∞. That is, the maximal convergence space for the (C,1) means of two-dimensional partial sums is Llog⁺L(I²). Weisz [7] investigated the maximal operator of (C, α, β) means of two-dimensional Walsh-Fourier se- ries and proved that the maximal operator σ_∗^α,βf is bounded from H_p(I²) to L_p(I²) if 1/(1 +α),1/(1 +β) < p < ∞. In [7] Weisz conjectured that for the boundedness of the maximal operator σ^α,β_∗ from the Hardy space Hp(I) to the space Lp(I) the assumption p >1/(α+ 1),1/(1 +β) is essential. We give answer to the question and prove that the maximal operator σ_∗^α,β of the (C, α, β) (0 < α≤β ≤1) means of the two-dimensional Walsh-Fourier series is not bounded from the Hardy space H_1/(α+1)(I²) to the space L_1/(α+1)(I²).

The following is true.

Theorem 1. Let 0 < α ≤ β ≤ 1. Then the maximal operator σ_∗^α,β of the (C, α, β) means of the two-dimensional Walsh-Fourier series is not bounded from the Hardy space H_1/(α+1)(I²) to the space L_1/(α+1)(I²).

In order to prove Theorem 1 we need the following lemma.

Lemma 1. ([3]) Let n∈N and 0< α≤1. Then Z

I

1≤Nmax≤2ⁿ

¡A^α_N−1|K_N^α (x)|¢_1/(α+1)

dx≥c(α) n log (n+ 2). Proof of Theorem 1. Let

fn

¡x¹, x²¢ :=£

D₂ⁿ⁺¹¡ x¹¢

−D2ⁿ

¡x¹¢¤

w2ⁿ−1

¡x²¢ . Since

fb_n(ν, µ) = Z

I

£D₂ⁿ⁺¹¡ u¹¢

−D₂ⁿ¡ u¹¢¤

w_ν¡ u¹¢

du¹ Z

I

w₂ⁿ₋₁¡ u²¢

w_µ¡ u²¢

du²

=

(1, if ν = 2ⁿ, . . . ,2ⁿ⁺¹−1, µ= 2ⁿ−1, 0, otherwise,

we can write Si,jfn

¡x¹, x²¢

= Xi−1

ν=0

fbn(ν,2ⁿ−1)wν

¡x¹¢ w2ⁿ−1

¡x²¢ (1)

=







[D_i(x¹)−D₂ⁿ(x¹)]w₂ⁿ₋₁(x²), if i= 2ⁿ+ 1, . . . ,2ⁿ⁺¹−1, j ≥2ⁿ, fn(x¹, x²), if i≥2ⁿ⁺¹, j ≥2ⁿ,

0, otherwise.

We have

f_n^∗¡ x¹, x²¢

= sup

i,j

¯¯S₂ⁱ_,2^jf_n¡

x¹, x²¢¯¯=¯

¯f_n¡

x¹, x²¢¯¯,

(2) kf_nk_Hp =kf_n^∗k_p =kD₂ⁿk_p = 2^n(1−1/p). Let 1≤N <2ⁿ. Then from (1) we obtain

σ^α₂ⁿ_+N,2ⁿ⁺¹fn

¡x¹, x²¢

=

= 1

A^α₂n+N−1

1 A^β₂n+1−1

¯¯

¯¯

¯

2Xⁿ+N

i=1 2Xⁿ⁺¹

j=1

A^α−1₂n+N−iA^β−1₂n+1−jS_i,jf_n¡

x¹, x²¢¯

¯¯

¯¯

= 1

A^α₂n+N−1

1 A^β₂n+1−1

¯¯

¯¯

¯

2Xⁿ+N

i=2ⁿ+1 2Xⁿ⁺¹

j=2ⁿ

A^α−1₂n+N−iA^β−1₂n+1−jS_i,jf_n¡

x¹, x²¢¯

¯¯

¯¯

= 1

A^α₂n+N−1

1 A^β₂n+1−1

×

×

¯¯

¯¯

¯

2Xⁿ+N

i=2ⁿ+1 2Xⁿ⁺¹

j=2ⁿ

A^α−1₂n+N−iA^β−1₂n+1−j

£D_i¡ x¹¢

−D₂ⁿ¡ x¹¢¤

w₂ⁿ₋₁¡ x²¢¯

¯¯

¯¯

≥ c(α, β) 2^nα2^nβ

¯¯

¯¯

¯ XN

i=1

A^α−1_N−i£

D_i+2ⁿ¡ x¹¢

−D₂ⁿ¡ x¹¢¤¯

¯¯

¯¯

¯¯

¯¯

¯

2ⁿ

X

j=0

A^β−1₂n−j

¯¯

¯¯

¯

≥ c(α, β) 2^nα

¯¯

¯¯

¯ XN

i=1

A^α−1_N−i£

D_i+2ⁿ¡ x¹¢

−D₂ⁿ¡ x¹¢¤¯

¯¯

¯¯

= c(α, β) 2^nα

¯¯

¯¯

¯ XN

i=1

A^α−1_N−iD_i¡ x¹¢¯¯

¯¯

¯. Therefore,

σ_∗^α,βfn

¡x¹, x²¢

≥ sup

1≤N≤2ⁿ

¯¯σ^α₂n+N,2ⁿ⁺¹fn

¡x¹, x²¢¯¯

≥ c(α, β) 2^nα sup

1≤N≤2ⁿ

¯¯

¯¯

¯ XN

i=1

A^α−1_N−iD_i¡ x¹¢¯

¯¯

¯¯.

Then from Lemma 1 and (2) we get

°°σ^α,β_∗ fn

°°

1/(α+1)

kf_nk_H1/(α+1) ≥ c(α, β) 2^nα2^−nα



 Z

I

sup

1≤N≤2ⁿ

¡A^α_N−1|K_N^α (x)|¢_1/(α+1) dx





α+1

≥c(α, β)

µ n

log (n+ 2)

¶_α+1

→ ∞ asn → ∞.

Theorem 1 is proved. ¤

References

[1] N. J. Fine. Ces`aro summability of Walsh-Fourier series. Proc. Nat. Acad. Sci. U.S.A., 41:588–591, 1955.

[2] G. G´at. On the divergence of the (C,1) means of double Walsh-Fourier series. Proc.

Amer. Math. Soc., 128(6):1711–1720, 2000.

[3] U. Goginava. The maximal operator of the (C, α) means of the Walsh-Fourier series.

Ann. Univ. Sci. Budapest. Sect. Comput., 26:127–135, 2006.

[4] F. M´oricz, F. Schipp, and W. R. Wade. Ces`aro summability of double Walsh-Fourier series.Trans. Amer. Math. Soc., 329(1):131–140, 1992.

[5] W. R. Wade. A growth estimate for Ces`aro partial sums of multiple Walsh-Fourier series. In A. Haar memorial conference, Vol. I, II (Budapest, 1985), volume 49 of Colloq. Math. Soc. J´anos Bolyai, pages 975–991. North-Holland, Amsterdam, 1987.

[6] F. Weisz. Martingale Hardy spaces and their applications in Fourier analysis, volume 1568 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.

[7] F. Weisz. The maximal (C, α, β) operator of two-parameter Walsh-Fourier series. J.

Fourier Anal. Appl., 6(4):389–401, 2000.

[8] F. Weisz. (C, α) summability of Walsh-Fourier series.Anal. Math., 27(2):141–155, 2001.

[9] F. Weisz. Summability of multi-dimensional Fourier series and Hardy spaces, volume 541 ofMathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2002.

[10] A. Zygmund. Trigonometric series. 2nd ed. Vol. I. Cambridge University Press, New York, 1959.

Received August 28, 2007.

Institute of Mathematics,

Faculty of Exact and Natural Sciences, Tbilisi State University,

Chavchavadze str. 1, Tbilisi 0128,

Georgia

E-mail address: [email protected]