the maximal fejér operator on real hardy spaces

Periodica Mathematica Hungarica Vol. 49 (1), 2004, pp. 15–25

THE MAXIMAL FEJER OPERATORON REAL HARDY SPACES

Gavin Brown (Sydney), Dai Feng (Sydney) andFerenc Moricz (Szeged)

[Communicated by: Gyorgy Petruska]

Abstract

We prove that the maximal Fejer operator is not bounded on the real Hardyspaces H1, which may be considered over T and R. We also draw corollaries for thecorresponding Hardy spaces over T

2 and R2.

1. Preliminaries: Real Hardy spaces

We shall briefly summarize the basic definitions and results. In the periodiccase we identify the unit circle (the so-called torus) with the interval T := [−π, π).Given a periodic function f integrable in Lebesgue’s sense on T, in symbol: f ∈L1(T), its conjugate function f is defined in terms of the following principal valueintegral:

f(x) := (P.V)1π

∫T

f(x − t)12

cott

2dt.

The real Hardy space H1(T) of real-valued functions f is defined by

H1(T) := {f ∈ L1(T) : f ∈ L1(T)}and endowed with the norm

‖f‖H1(T) := ‖f‖L1(T) + ‖f‖L1(T).

Mathematics subject classification number: Primary 42A24, 42A38, 42A50; Sec-ondary 42B08, 42B10.

Key words and phrases: conjugate function, Hilbert transform, real Hardy space,atomic Hardy space, weak-L1, Fourier series, conjugate series, Fourier transform, Fourierintegral, Fejer mean.

The authors thank the Australian Research Council for support of their collaboration. Thesecond author was partially supported also by NNSF of China under Grant # 1007 1007. Thethird author was partially supported also by the Hungarian National Foundation for ScientificResearch under Grants TS 044 782 and T 046 192.

0031-5303/2004/$20.00 Akademiai Kiado, Budapestc© Akademiai Kiado, Budapest Kluwer Academic Publishers, Dordrecht

16 g. brown, d. feng and f. moricz

A function a ∈ L∞(T) is called an atom if either a(x) ≡ 1 or if there exists asubinterval I of T such that

supp a ⊂ I, ‖a‖L∞(T) ≤ |I|−1, and∫

I

a(x)dx = 0.

The atomic Hardy space H1at(T) is defined to consist of all real-valued functions f

defined on T for which there exist a sequence {ak(x) : k = 1, 2, . . .} of atoms and asequence {λk : k = 1, 2, . . .} of real numbers such that

f(x) =∞∑

k=1

λkak(x) a.e and∞∑

k=1

|λk| < ∞. (1.1)

The norm in H1at(T) is defined by

‖f‖H1at(T) := inf

∞∑k=1

|λk|,

where the infimum is extended over all atomic representations (1.1) of f .It is well known (see [Be-Sh, pp. 372–373, Theorem 6.14]) that the spaces

H1(T) and H1at(T) coincide, while the norms ‖ ·‖H1(T) and ‖ ·‖H1

at(T) are equivalent.The treatment of the nonperiodic case is analogous to that of the periodic

case, except the fact that R := (−∞,∞) is a noncompact set. Given a real-valuedfunction f , integrable in Lebesgue’s sense on R, in symbol: f ∈ L1(R), its Hilberttransform f is defined by

f(x) := (P.V.)1π

∫R

f(x − t)t

dt.

The nonperiodic real Hardy space H1(R) as well as the atomic Hardy spaceH1

at(R) are defined analogously to the periodic case (with the exceptions that weconsider the Hilbert transform in place of the conjugate function and that the func-tion a(x) ≡ 1 is not an atom on R), and again these two spaces coincide.

2. Preliminaries: The maximal Fejer operator

Given a function f ∈ L1(T), its Fourier series is defined by

f(x) ∼∑k∈Z

f(k)eikx, x ∈ T; f(k) :=12π

∫T

f(t)e−iktdt, k ∈ Z. (2.1)

The Fejer mean of the series in (2.1) is defined by

σn(f, x) :=1

n + 1

n∑m=0

m∑|k|=0

f(k)eikx

=

∑|k|≤n

(1 − |k|

n + 1

)f(k)eikx, n ∈ N,

the maximal fejer operator on real hardy spaces 17

while the maximal Fejer operator σ∗(f) is defined by

σ∗(f, x) := sup{|σn(f, x)| : n ∈ N}. (2.2)

It is well known (see [Zy, Vol. 1, pp. 154–156, especially (7.1) Lemma]) that theoperator σ∗(f) is bounded from L1(T) to weak-L1(T), that is,

supα>0

α|{x ∈ T : σ∗(f, x) > α}| ≤ C‖f‖L1(T),

where by |E| we denote the Lebesgue measure of the measurable subset E of T, andthe constant C does not depend on f . Furthermore, it was proved in [Mo 2] thatσ∗(f) is bounded from H1(T) to L1(T).

We recall that the series∑k∈Z

(−i signk)f(k)eikx (2.3)

is called the conjugate series to the Fourier series in (2.1). Denote by σn(f, x) theFejer mean of series (2.3). The maximal conjugate Fejer operator is defined by (2.2)with σ in place of σ at both occurrences. It was proved in [We 1] that the operatorσ∗(f) is also bounded from L1(T) to weak-L1(T). Another proof of this statementwas independently given in [Mo 4], which did not rely on the theory of Schwartzdistributions.

It is well known that if f ∈ H1(T), then(f

)∧(k) = (−i signk)f(k), k ∈ Z.

By the uniqueness of Fourier coefficients it follows that(f

)∼(x) = −f(x) + f(0) (2.4)

at almost every x ∈ T. This implies that f ∈ H1(T) and∥∥f∥∥

H1(T)≤ 2‖f‖H1(T),

since it is clear that∣∣f(0)

∣∣ ≤ ‖f‖L1(T). Taking into account that

σ∗(f, x) = σ∗(f , x

), f ∈ H1(T),

we conclude that σ∗(f) is also bounded from H1(T) to L1(T).Given a function f ∈ L1(R), its Fourier transform is defined by

f(t) :=1√2π

∫R

f(x)e−itxdx, t ∈ R.

The nonperiodic counterpart of the Fourier series (2.1) is the Fourier integral given by

f(x) ∼ 1√2π

∫R

f(t)eixtdt, x ∈ R. (2.5)


The Fejer mean (sometimes called the Riesz mean of first order) of the integral in(2.5) is defined by

σT (f, x) :=1T

∫ T

0

{1√2π

∫|u|<t

f(u)eixudu

}dt

=1√2π

∫|u|≤T

(1 − |u|

T

)f(u)eixudu, T > 0;

(2.6)


σ∗(f, x) := sup{|σT (f, x)| : T > 0}. (2.7)

We recall that the conjugate Fourier integral to the integral in (2.5) is de-fined by

1√2π

∫R

(−i sign t)f(t)eixtdt.

We denote by σT (f, x) the Fejer mean of this integral (cf. (2.6)). The maximalconjugate Fejer operator σ∗(f) is defined analogously to (2.7).

We refer to [Ti, Chapters I-V] and [Mo 3] for more details.

3. The maximal Fejer operator is not bounded on H1

It was conjectured in [Mo 4] that neither σ∗(f) nor σ∗(f) are bounded onH1(T). In this section we shall prove that these conjectures are true.

Theorem 1. There exists a function f in H1(T) such that

‖σ∗(f)‖H1(T) = ∞. (3.1)

Taking into account (2.4), Theorem 1 remains valid with σ∗(f) in place ofσ∗(f) in (3.1).

For the proof of Theorem 1, we need three lemmas.

Lemma 1. If a function a(x) ∈ L∞(T) is such that

supp a ⊂ (−r, r), ‖a‖L∞(T) ≤ (2r)−1, and

∫ r

−r

a(x)dx = 0,

where r ∈ (0, 1/4), then there exists a constant C1 such that

σ∗(a, x) ≤{

C1/r for x ∈ T,

C1r/x2 for x ∈ T/(−2r, 2r).(3.2)

This lemma was essentially proved in [Mo 2]. Note that this a(x) is an atom.


Lemma 2. Under the conditions of Lemma 1, there exists a constant C2 suchthat

‖M(σ∗(a))‖L1(T) ≤ C2 log1r,

where by M(σ∗) we denote the Hardy–Littlewood maximal function applied to σ∗.

Proof. By a well-known inequality (see [Be-Sh, p. 250, Theorem 6.7]), wehave

‖M(σ∗(a))‖L1(T) ≤ C‖σ∗(a)‖L log+ L + C,

where C is an absolute constant. Making use of (3.2), we estimate as follows:∫T

σ∗(a, x) log+ σ∗(a, x)dx

≤ C

∫|x|<2r

1r

log1rdx + C

∫2r<|x|<π

r

x2log+ r

x2dx ≤ C2 log

1r. �

Lemma 3. If 0 < r < π/300 and

a(x) := (2r)−1{χ[0,r)(x) − χ[−r,0)(x)

}, x ∈ T, (3.3)

then there exist positive constants C3 and C0 = C0(r) ∈ (15, π/(20r)) such that

σ∗(a, x) ≥ C3

x2for x ∈

(C0r,

π

20

). (3.4)

Proof. We start with the representation

σn(a, x) =2π

∫T

a(x − t)Kn(t)dt, (3.5)

where

Kn(t) =2

n + 1

(sin n+1

2 t

2 sin t2

)2

, n = 0, 1, 2, . . . ,

is the familiar Fejer kernel. Set

Fn(y) :=∫ y

0

Kn(t)dt, y ∈ T.

By (3.3) and (3.5), we obtain

π

2rσn(a, x) = 2r

∫ r

−r

a(t)Kn(x − t)dt

=∫ x

x−r

Kn(t)dt −∫ x+r

x

Kn(t)dt

= −[Fn(x + r) + Fn(x − r) − 2Fn(x)]

(3.6)


= −F′′(θ)r2 = − (n + 1) sin(n + 1)θ

(sin θ

2

) − 2 sin2 n+12 θ

(cos θ

2

)4(n + 1) sin3 θ

2

r2,

where

θ = θ(x, r, n) ∈ (x − r, x + r) ⊂((

1 − 1C0

)x,

(1 +

1C0

)x

),

provided that x > C0r. We choose the integer n = n(x) so that

6π/5 < (n + 1)x ≤ 5π/4,

which is possible if x < π/20. Now, if C0 < π/(20r), then for x ∈ (C0r, π/20) wehave

π <6π

5

(1 − 1

C0

)< (n + 1)θ <

5π

4

(1 +

1C0

)<

4π

3,

whence

0 <3π

5(n + 1)

(1 − 1

C0

)<

θ

2≤ 5π

8(n + 1)

(1 +

1C0

)≤ π

3.

Now, it follows from (3.6) and the above inequalities that

rσn(x)(a, x) ≥ C4

sin2 n+12 θ

(cos θ

2

)(n + 1)θ3

r2 ≥ C5r2

(n + 1)θ3≥ C3r

2

x2,

where C4, C5 and C3 are positive absolute constants. This proves (3.4). �

Proof of Theorem 1. Let {rj : j = 1, 2, . . .} be a decreasing sequence ofpositive numbers with the following properties: 0 < r1 < π/(40C0), where C0 is theconstant from Lemma 3,

0 < rj+1 < rj

(log

1rj

)−1

, (3.7)

(log

1rj+1

)1/2

≥ 2C0C2

15

j∑i=1

(log

1ri

)1/2

, j = 1, 2, . . . , (3.8)

where the constants C0 and C2 are from Lemmas 3 and 2, respectively; and∞∑

j=1

λj < ∞, where λj :=(

log1rj

)−1/2

. (3.9)

Next, we define the sequence {aj(x) : j = 1, 2, . . .} of atoms by setting

aj(x) := (2rj)−1{χ[0,rj)(x) − χ[−rj ,0)(x)

}, x ∈ T, (3.10)

and the function

f(x) :=∞∑

j=1

λjaj(x). (3.11)


By (3.9), we have f ∈ H1(T).We shall apply the Hardy–Littlewood maximal function to our σ∗(f, x). To

this end, let x ∈ (C0rk, π/20), where k is temporarily fixed. By (3.10) and (3.11),while making use of Lemmas 1 and 3, we obtain

M(σ∗(f), x) ≥ 1x − C0rk

∫ x

C0rk

σ∗(f, t)dt

≥ λk

x − C0rk

∫ x

C0rk

σ∗(ak, t)dt −∑j �=k

λj

x − C0rk

∫ x

C0rk

σ∗(aj , t)dt

≥ 15λk

x − C0rk

∫ x

C0rk

rk

t2dt −

∞∑j=k+1

C1λj

x − C0rk

∫ x

C0rk

rj

t2dt

−k−1∑j=1

λjM(σ∗(aj), x)

≥ 15λk

C0(x − C0rk)−

∞∑j=k+1

C1λjrj

rk(x − C0rk)−

k−1∑j=1

λjM(σ∗(aj), x),

where the constants C1 and C0 are from Lemmas 1 and 3. By this, Lemma 2, (3.7)and (3.8), we conclude that∫ π/20

C0rk

M(σ∗(f), x)dx

≥ 15C0

(log

1rk

)1/2

− C1

∞∑j=k+1

λjrj

rklog

1rk

− C2

k−1∑j=1

(log

1rj

)1/2

≥ 152C0

(log

1rk

)1/2

− C1

∞∑j=k+1

λj → ∞ as k → ∞,

where the constant C2 is from Lemma 2. This implies that

‖M(σ∗(f))‖L1(T) = ∞. (3.12)

By virtue of a known result (see [Be-Sh, p. 250, Theorem 6.7]), it follows from(3.12) that

‖σ∗(f)‖L log+ L = ∞. (3.13)

It is also known (see [Ga, pp. 84–85]) that if a function is bounded from below,then it belongs to H1(T) if and only if it belongs to L log+ L(T). Since σ∗(f, x) ≥ 0,(3.13) implies (3.1) to be proved. �

The corresponding counterpart of Theorem 1 in the nonperiodic case is almosttrivial.


Theorem 2. If f ∈ L1(R) and σ∗(f) ∈ H1(R), then f(x) = 0 a.e.

Proof. It follows from the atomic decomposition of H1(R) that the integralof any function f ∈ H1(R) over R equals 0. In particular, if σ∗(f) ∈ H1(R), thenwe have ∫

R

σ∗(f, x)dx = 0.

Since σ∗(f, x) ≥ 0, this means that σ∗(f, x) = 0 a.e., that is, f(x) = 0 a.e., as wehave claimed. �

4. Consequencesfor the double maximal Fejer operator

Given a function f ∈ L1(T2), its (double) Fourier series is defined by

f(x, y) ∼∑ ∑(j,k)∈Z2

f(j, k)ei(jx+ky), (4.1)

where

f(j, k) :=1

4π2

∫ ∫T2

f(u, v)e−i(ju+kv)dudv, (j, k) ∈ Z2.

The Fejer mean of the series in (4.1) is defined by

σmn(f, x, y) :=∑

|j|≤m

∑|k|≤n

(1 − |j|

m + 1

) (1 − |k|

n + 1

)f(j, k)ei(jx+ky),

(m, n) ∈ N2;


σ∗(f, x, y) := sup{|σmn(f, x, y)| : (m, n) ∈ N

2}

. (4.2)

We recall that the series∑ ∑(j,k)∈Z2

(−i sign j)f(j, k)ei(jx+ky), (4.3)

∑ ∑(j,k)∈Z2

(−i signk)f(j, k)ei(jx+ky), (4.4)

and ∑ ∑(j,k)∈Z2

(−i sign j)(−i signk)f(j, k)ei(jx+ky) (4.5)

are called the conjugate series to the Fourier series in (4.1) with respect to thefirst variable, to the second variable, to both variables, respectively. We denoteby σ1,0

mn(f, x, y), σ0,1mn(f, x, y), and σ1,1

mn(f, x, y) the Fejer means of series (4.3), (4.4),


and (4.5), respectively. The maximal conjugate Fejer operators σ1,0∗ (f), σ0,1

∗ (f), andσ1,1∗ (f) with respect to the first variable, to the second variable, to both variables

are defined by (4.2) when σ is replaced by σ1,0, σ0,1, σ1,1 in it, respectively.Given a function f ∈ L1(T2), its conjugate function f1,0 with respect to the

first variable is defined by

f1,0(x, y) := (P.V.)1π

∫T

f(x − u, y)12

cotu

2du.

We recall that the hybrid Hardy space with respect to the first variable is defined by

H1,0(T

2)

:={f ∈ L1

(T

2)

: f1,0 ∈ L1(T

2)}

and endowed with the norm

‖f‖H1,0T2) := ‖f‖L1(T 2) +∥∥f1,0

∥∥L1(T2)

.

The conjugate function f0,1 and the hybrid Hardy space H0,1(T2) with respectto the second variable are defined analogously.

Given a function f ∈ H1,0(T2) ∩ H0,1(T2), its (double) conjugate functionf1,1 with respect to both variables is defined by means of iterated conjugations.Since this time f1,0, f0,1 ∈ L1

(T

2), it follows that the iterated conjugate functions(

f1,0)∼0,1(x, y) and

(f0,1

)∼1,0(x, y) exist and coincide at almost every point (x, y) ∈T

2. Now, the double conjugate function f1,1(x, y) is defined by this joint value. It isalso known that f1,1 can be represented in the form of the following double principalvalue integral:

f(x, y) := (P.V.)1π2

∫ ∫T2

f(x − u, y − v)(

12

cotu

2

) (12

cotv

2

)du dv.

We recall that the product (real) Hardy space H1,1(T

2)

is defined by

H1,1(T

2)

:={

f ∈ L1(T

2)

: f1,0, f0,1, f1,1 ∈ L1(T

2)}

and endowed with the norm

‖f‖H1,1(T2) := ‖f‖L1(T2) + ‖f1,0‖L1(T2) + ‖f0,1‖L1(T2) + ‖f1,1‖L1(T2).

We refer to [Gi-Mo] for more details on classical, hybrid, and product real Hardyspaces over T

2 as well as over R2.

It was proved in [Mo 1] that the operators σ∗(f) and σ1,0∗ (f) are bounded

from H1,0(T2) to weak-L1(T2); the operators σ∗(f) and σ0,1∗ (f) are bounded from

H0,1(T2) to weak-L1(T2); and it was proved in [We 2] that the operators σ∗(f),σ1,0∗ (f), σ0,1

∗ (f), and σ1,1∗ (f) are bounded from H1,1(T2) to L1(T2).

The following trivial observation gives rise to Corollary 1 of Theorem 1: If

g(x, y) = f1(x)f2(y), where f1, f2 ∈ L1(T),

then g ∈ L1(T

2)

and

g1,0(x, y) = f1(x)f2(y), g0,1(x, y) = f1(x)f2(y), g1,1(x, y) = f1(x)f2(y).


Corollary 1. There exists a function g in H1,1(T

2)

such that

‖σ∗(g)‖H1,0(T2) = ‖σ∗(g)‖H0,1(T2) = ∞.

This conclusion holds true with σ1,0∗ (g), σ0,1

∗ (g), and σ1,1∗ (g) in place of σ∗(g).

Given a function f ∈ L1(R2), its (double) Fourier transform is defined by

f(u, v) :=12π

∫ ∫R2

f(x, y)e−i(ux+vy)dx dy, (u, v) ∈ R2.

The nonperiodic counterpart of the Fourier series (4.1) is the Fourier integral de-fined by

f(x, y) ∼ 12π

∫ ∫R2

f(u, v)ei(xu+yv)du dv, (x, y) ∈ R2. (4.6)

The Fejer mean (sometimes called the Riesz mean of first order) of the integral in(4.6) is defined by

σT1,T2(f, x, y) :=12π

∫|u|≤T1

∫|v|≤T2

(1 − |u|

T1

) (1 − |v|

T2

)f(u, v)ei(xu+yv)du dv,

T1, T2 > 0;


σ∗(f, x, y) := sup{|σT1,T2(f, x, y)| : T1, T2 > 0}.

The corresponding counterpart of Corollary 1 in the nonperiodic case readsas follows.

Corollary 2. If f ∈L1(R2) and σ∗(f)∈H1,0(R2)∪H0,1(R2), then f(x, y)=0 for a.e. (x, y) ∈ R

2.

In fact, for definiteness assume that σ∗(f) ∈ H1,0(R2). Then we have

σ∗(f, ·, y) ∈ H1(R) for a.e y ∈ R.

By Theorem 2, we conclude that

σ∗(f, x, y) = 0 for a.e. (x, y) ∈ R2. (4.7)

It is well known that in the special case when T1 = T2 = T , the Fejer meanσT,T (f, x, y) converges to f(x, y) a.e. as T → ∞. From this and (4.7) it follows thatf(x, y) = 0 a.e.


References

[Be-Sh] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, NewYork, 1988.

[Ga] A. M. Garsia, Martingale inequalities, New York, 1973.[Gi-Mo] D. V. Giang and F. Moricz, Hardy spaces on the plane and double Fourier

transforms, J. Fourier Anal. Appl. 2 (1996), 487–505.[Mo 1] F. Moricz, On the maximal Fejer operator for double Fourier series of functions

in Hardy spaces, Studa Math. 116 (1995), 89–100.[Mo 2] F. Moricz, The maximal Fejer operator is bounded from H1(T) into L1(T), Anal-

ysis 16 (1996), 125–135.[Mo 3] F. Moricz, The maximal Fejer operator for Fourier transforms of functions in

Hardy spaces, Acta Sci. Math. (Szeged) 62 (1996), 537–555.[Mo 4] F. Moricz, The maximal conjugate Fejer operator is bounded from L1 to weak-L1,

Arch. Math. (Basel) 72 (1999), 118–126.[We 1] F. Weisz, The maximal Cesaro operator on Hardy spaces, Analysis 18 (1998),

157–166.[We 2] F. Weisz, Cesaro summability of two-parameter trigonometric Fourier series, J. Ap-

prox. Theory 90 (1997), 30–45.[Ti] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, 1937.[Zy] A. Zygmund, Trigonometric series, Cambridge Univ. Press, 1959.

(Received: July 17, 2003)

Gavin Brown

University of Sydney

Sydney, NSW 2006

Australia

E-mail: [email protected]

Dai Feng

Department of Mathematics

Beijing Normal University

Beijing 100 875

China

Current address:School of Mathematics and Statistics

University of Sydney

NSW 2006

Australia


Ferenc Moricz

University of Szeged

Bolyai Institute

Aradi Vertanuk tere 1

H-6720 Szeged

Hungary


the maximal fejér operator on real hardy spaces

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