C O L L O Q U I U M M A T H E M A T I C U M VOL. 82 1999 NO. 2

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 82 1999 NO. 2

FEJ ´ ER MEANS OF TWO-DIMENSIONAL FOURIER TRANSFORMS ON H

(R × R)

BY

FERENC W E I S Z (BUDAPEST)

Abstract. The two-dimensional classical Hardy spaces Hp(R×R) are introduced and it is shown that the maximal operator of the Fej´er means of a tempered distribution is bounded from Hp(R×R) to Lp(R²) (1/2 < p ≤ ∞) and is of weak type (H₁^♯(R×R), L1(R²)) where the Hardy space H₁^♯(R × R) is defined by the hybrid maximal function. As a con- sequence we deduce that the Fej´er means of a function f ∈ H₁^♯(R × R) ⊃ L log L(R²) converge to f a.e. Moreover, we prove that the Fej´er means are uniformly bounded on Hp(R×R) whenever 1/2 < p < ∞. Thus, in case f ∈ Hp(R×R), the Fej´er means converge to f in Hp(R × R) norm (1/2 < p < ∞). The same results are proved for the conjugate Fej´er means.

1. Introduction. The Hardy–Lorentz spaces H

(R × R) of tempered distributions are endowed with the L

(R

) Lorentz norms of the non- tangential maximal function. Clearly, H

(R × R) = H

(R × R) are the usual Hardy spaces (0 < p ≤ ∞).

In Zygmund [22] (Vol. II, p. 246) it is shown that the Fej´er means σ

f of a one-dimensional function f ∈ L

(R) converge to f a.e. as T → ∞.

Moreover, the maximal operator of the Fej´er means, σ

:= sup

|σ

|, is of weak type (1, 1), i.e.

sup

γ>0

γλ(σ

f > γ) ≤ Ckf k

(f ∈ L

(R))

(see Zygmund [22], Vol. I, p. 154 and M´oricz [14]). M´oricz [14] also verified that σ

is bounded from H

(R) to L

(R). The author [19] proved that σ

is also bounded from H

(R) to L

(R) whenever 1/2 < p < ∞, 0 < q ≤ ∞.

In [16] we investigated the Fej´er means of two-parameter Fourier series and proved that σ

:= sup

|σ

| is bounded from H

(T × T) to

1991 Mathematics Subject Classification: Primary 42B08; Secondary 42B30.

Key words and phrases: Hardy spaces, p-atom, atomic decomposition, interpolation, Fej´er means.

This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.

[155]

L

(T

) (3/4 < p ≤ ∞, 0 < q ≤ ∞) and is of weak type (H

(T × T), L

(T

)), i.e.

sup

γ>0

γλ(σ

f | > γ) ≤ Ckf k

1(T×T)

(f ∈ H

(T × T)).

Moreover, the Fej´er means σ

f converge to f a.e. as n, m → ∞ whenever f ∈ H

(T × T) ⊃ L log L(T

) (see Weisz [15], [16] and Zygmund [22] for L log L(T

)).

In this paper we sharpen and generalize these results for the Fej´er means of two-dimensional Fourier transforms.

We show that the maximal operator σ

is bounded from H

(R × R) to L

(R

) whenever 1/2 < p < ∞, 0<q ≤∞, and is of weak type (H

(R × R), L

(R

)). We introduce the conjugate distributions e f

, the conjugate Fej´er means e σ

and the conjugate maximal operators e σ

(i, j = 0, 1). We prove that the operator e σ

is also of type (H

(R × R), L

(R

)) (1/2 <

p < ∞, 0 < q ≤ ∞) and of weak type (H

(R × R), L

(R

)).

A usual density argument then implies that the Fej´er means σ

f con- verge to f a.e. and the conjugate Fej´er means e σ

f converge to e f

(i, j = 0, 1) a.e. as T, U → ∞ provided that f ∈ H

(R × R). Note that e f

is not necessarily in H

(R × R) whenever f is.

We also prove that the operators σ

and e σ

(T, U ∈ R) are uniformly bounded from H

(R×R) to H

(R×R) if 1/2 < p < ∞, 0 < q ≤ ∞. From this it follows that σ

f → f and e σ

f → e f (i, j = 0, 1) in H

(R × R) norm as T, U → ∞ whenever f ∈ H

(R×R) and 1/2 < p < ∞, 0 < q ≤ ∞.

2. Hardy spaces and conjugate functions. Let R denote the real numbers, R

the positive real numbers and let λ be the 2-dimensional Lebesgue measure. We also use the notation |I| for the Lebesgue measure of the set I. We briefly write L

for the real L

(R

, λ) space; the norm (or quasinorm) in this space is defined by kf k

:= (

T

R²

|f |

dλ)

(0 < p ≤ ∞).

The distribution function of a Lebesgue-measurable function f is defined by

λ({|f | > ̺}) := λ({x : |f (x)| > ̺}) (̺ ≥ 0).

The weak L

space L

(0 < p < ∞) consists of all measurable functions f for which

kf k

:= sup

̺>0

̺λ({|f | > ̺})

< ∞

and we set L

= L

.

The spaces L

are special cases of the more general, Lorentz spaces L

. In their definition another concept is used. For a measurable function f the non-increasing rearrangement is defined by

f (t) := inf{̺ : λ({|f | > ̺}) ≤ t}. e

The Lorentz space L

is defined as follows: for 0 < p < ∞, 0 < q < ∞, kf k

:=



0

f (t) e

t

dt t



, while for 0 < p ≤ ∞,

kf k

:= sup

t>0

t

f (t). e Let

L

:= L

(R

, λ) := {f : kf k

< ∞}.

One can show the following equalities:

L

= L

, L

= L

(0 < p ≤ ∞) (see e.g. Bennett–Sharpley [1] or Bergh–L¨ofstr¨ om [2]).

Let f be a tempered distribution on C

(R

) (briefly f ∈ S

(R

) = S

).

The Fourier transform of f is denoted by b f . In the special case when f is an integrable function,

f (t, u) = b 1 2π

\

R

\

R

f (x, y)e

e

dx dy (t, u ∈ R) where ı = √

−1.

For f ∈ S

and t, u > 0 let

F (x, y; t, u) := (f ∗ P

× P

)(x, y) where ∗ denotes convolution and

P

(x) := ct

t

+ x

(x ∈ R) is the Poisson kernel.

For α > 0 let

Γ

:= {(x, t) : |x| < αt},

a cone with vertex at the origin. We denote by Γ

(x) (x ∈ R) the translate of Γ

with vertex at x. The non-tangential maximal function is defined by

F

(x, y) := sup

(x^′,t)∈Γα(x), (y^′,u)∈Γβ(y)

|F (x

, y

; t, u)| (α, β > 0).

For 0 < p, q ≤ ∞ the Hardy–Lorentz space H

(R × R) = H

consists of all tempered distributions f for which F

∈ L

; we set

kf k

:= kF

k

.

For 0 < p < ∞, 0 < q ≤ ∞ Chang and Fefferman [3] and Lin [12] proved the equivalence kF

k

∼ kF

k

(α, β > 0). It is known that if f ∈ H

(0 < p < ∞) then f (x, y) = lim

F (x, y; t, u) in the sense of distributions (see Gundy–Stein [11], Chang–Fefferman [3]).

Let us introduce the hybrid Hardy spaces. For f ∈ L

and t > 0 let G(x, y; t) := 1

√ 2π

\

R

f (v, y)P

(x − v) dv and

G

(x, y) := sup

(x^′,t)∈Γα(x)

|G(x

, y; t)| (0 < α < 1).

We say that f ∈ L

is in the hybrid Hardy–Lorentz space H

(R×R) = H

if

kf k

:= kG

k

< ∞.

The equivalences kG

k

∼ kG

k

(α > 0, 0 < p < ∞, 0 < q ≤ ∞) and H

∼ H

∼ L

(1 < p < ∞, 0 < q ≤ ∞)

were proved in Fefferman–Stein [7], Gundy–Stein [11] and Lin [12]. Note that for p = q the usual definitions of the Hardy spaces H

= H

and H

= H

are obtained.

The following interpolation result concerning Hardy–Lorentz spaces will be used several times in this paper (see Lin [12] and also Weisz [17]).

Theorem A. If a sublinear (resp. linear ) operator V is bounded from H

to L

(resp. to H

) and from L

to L

(p

≤ 1 < p

< ∞) then it is also bounded from H

to L

(resp. to H

) if p

< p < p

and 0 < q ≤ ∞.

In this paper the constants C are absolute, while C

(resp. C

) depend only on p (resp. p and q) and may be different in different contexts.

One can prove similarly to the discrete case (see Weisz [16]) that L log L := L log L(R

) ⊂ H

⊂ H

, more exactly,

(1) kf k

= sup

̺>0

̺λ(F

> ̺) ≤ Ckf k

(f ∈ H

) and

kf k

≤ C + Ck|f | log

|f |k

(f ∈ L log L) where log

u = 1

log u.

For a tempered distribution f ∈ H

(0 < p < ∞) the Hilbert transforms or conjugate distributions e f

, e f

and e f

are defined by

( e f

)

(t, u) := (−ı sign t) b f (t, u) (t, u ∈ R)

(conjugate with respect to the first variable),

( e f

)

(t, u) := (−ı sign u) b f (t, u) (t, u ∈ R) (conjugate with respect to the second variable) and

( e f

)

(t, u) := (− sign(tu)) b f (t, u) (t, u ∈ R)

(conjugate with respect to both variables). We use the notation e f

:= f . Gundy and Stein [10], [11] verified that if f ∈ H

(0 < p < ∞) then all conjugate distributions are also in H

and

(2) kf k

= k e f

k

(i, j = 0, 1).

Furthermore (see also Chang and Fefferman [3], Frazier [9], Duren [5]), (3) kf k

∼ kf k

+ k e f

k

+ k e f

k

+ k e f

k

.

As is well known, if f is an integrable function then f e

(x, y) = p.v. 1

π

\

R

f (x − t, y)

t dt := lim

ε→0

1 π

\

ε<|t|

f (x − t, y)

t dt,

f e

(x, y) = p.v. 1 π

\

R

f (x, y − u)

u du,

f e

(x, y) = p.v. 1 π

\

R

\

R

f (x − t, y − u)

tu dt du.

Moreover, the conjugate functions e f

, e f

and e f

exist almost everywhere, but they are not integrable in general. Similarly, if f ∈ H

then e f

and e f

are not necessarily in H

.

3. Fej´ er means. Suppose first that f ∈ L

for some 1 ≤ p ≤ 2. It is known that under certain conditions

f (x, y) = 1 2π

\

R

\

R

f (t, u)e b

e

dt du (x, y ∈ R).

This motivates the definition of the Dirichlet integral s

f : s

f (x, y) := 1

2π

t

\

−t u

\

−u

f (v, w)e b

e

dv dw (t, u > 0).

The conjugate Dirichlet integrals are introduced by es

f (x, y) := 1

2π

t

\

−t u

\

−u

(−ı sign v) b f (v, w)e

e

dv dw (t, u > 0),

es

f (x, y) := 1 2π

t

\

−t u\

−u

(−ı sign w) b f (v, w)e

e

dv dw (t, u > 0)

and

es

f (x, y) := 1 2π

t

\

−t u

\

−u

(− sign(vw)) b f (v, w)e

e

dv dw (t, u > 0).

The Fej´er and conjugate Fej´er means are defined by e

σ

f (x, y) := 1 T U

T

\

0 U

\

0

es

f (x, y) dt du (T, U > 0; i, j = 0, 1).

We write s

f =: es

f and σ

f := e σ

f . It is easy to see that s

f (x, y) :=

\

R

\

R

f (x − v, y − w) sin tv

πv · sin uw πw dv dw and

σ

f (x, y) :=

\

R

\

R

f (x − t, y − u)K

(t)K

(u) dt du where

K

(t) := 2

π · sin

(T t/2) T t

is the Fej´er kernel. Note that

(4)

\

R

K

(t) dt = 1 (T > 0) (see Zygmund [22], Vol. II, pp. 250–251).

We extend the definition of the Fej´er means and conjugate Fej´er means to tempered distributions as follows:

e

σ

f := e f

∗ (K

× K

) (T, U > 0; i, j = 0, 1).

One can show that e σ

f is well defined for all tempered distributions f ∈ H

(0 < p ≤ ∞) and for all functions f ∈ L

(1 ≤ p ≤ ∞) (cf.

Fefferman–Stein [7]).

The maximal and maximal conjugate Fej´er operators are defined by e

σ

f := sup

T,U >0

|e σ

f | (i, j = 0, 1).

We again write σ

f := e σ

f .

4. The boundedness of the maximal Fej´ er operator. A function a ∈ L

is called a rectangle p-atom if there exists a rectangle R ⊂ R

such that

(i) supp a ⊂ R,

(ii) kak

≤ |R|

,

(iii) for all x, y ∈ R and all N ≤ [2/p − 3/2],

\

R

a(x, y)x

dx =

\

R

a(x, y)y

dy = 0.

If I is an interval then let rI be the interval with the same center as I and with length r|I| (r ∈ N). For a rectangle R = I × J let rR = rI × rJ.

An operator V which maps the set of tempered distributions into the collection of measurable functions will be called p-quasi-local if there exist a constant C

> 0 and η > 0 such that for every rectangle p-atom a supported on the rectangle R and for every r ≥ 2 one has

\

R²\2^rR

|T a|

dλ ≤ C

2

.

Although H

cannot be decomposed into rectangle p-atoms, in the next theorem it is enough to take such atoms (see Weisz [16], Fefferman [8]).

Theorem B. Suppose that the operator V is sublinear and p-quasi-local for some 0 < p ≤ 1. If V is bounded from L

to L

then

kV f k

≤ C

kf k

(f ∈ H

).

Since the Fej´er kernel is positive, we can prove the following inequality in the same way as in the discrete case (see Weisz [18]):

(5) kσ

f k

≤ C

kf k

(1 < p ≤ ∞).

Now we can formulate our main result.

Theorem 1. We have

(6) kσ

f k

≤ C

kf k

(f ∈ H

)

for every 1/2 < p < ∞ and 0 < q ≤ ∞. In particular, if f ∈ H

then

(7) λ(σ

f > ̺) ≤ C

̺ kf k

1

(̺ > 0).

P r o o f. First we will show that the operator σ

is p-quasi-local for each 1/2 < p ≤ 1. To this end let a be an arbitrary rectangle p-atom with support R = I × J and

2

< |I| ≤ 2

, 2

< |J| ≤ 2

(K, L ∈ Z).

We can suppose that the center of R is zero. In this case [−2

, 2

] ⊂ I ⊂ [−2

, 2

] and

[−2

, 2

] ⊂ J ⊂ [−2

, 2

].

To prove the p-quasi-locality of the operator σ

we have to integrate |σ

a|

over

R

\ 2

R = (R \ 2

I) × J ∪ (R \ 2

I) × (R \ J)

∪ I × (R \ 2

J) ∪ (R \ I) × (R \ 2

J) where r ≥ 2 is an arbitrary integer.

First we integrate over (R \ 2

I) × J. Obviously,

\

R\2^rI

\

J

|σ

a(x, y)|

dx dy ≤ X

|i|=2^r−2

(i+1)2^K

\

i2^K

\

J

|σ

a(x, y)|

dx dy.

For x, y ∈ R let A

(x, y) :=

x

\

−∞

a(t, y) dt, A

(x, y) :=

y

\

−∞

a(x, u) du and

A

(x, y) :=

x

\

−∞

y

\

−∞

a(t, y) dt du.

By (iii) of the definition of the rectangle atom we can show that supp A

⊂ R and A

is zero at the vertices of R (k, l = 0, 1). Moreover, using (ii) we can compute that

(8) kA

k

≤ |I|

|J|

(|I| · |J|)

(k, l = 0, 1).

Integrating by parts we can see that

|σ

a(x, y)| =   

\

I

\

J

A

(t, u)K

(x − t)K

(y − u) dt du   

≤

\

I

\

J

A

(t, u)K

(y − u) du   |K

(x − t)| dt.

Using the inequality

|K

(t)| ≤ C/t

(T ∈ R

) we get

|σ

a(x, y)| ≤

\

I

\

J

A

(t, u)K

(y − u) du    C

|x − t|

dt

≤ C2

i

\

I

\

J

A

(t, u)K

(y − u) du    dt

for x ∈ [i2

, (i + 1)2

). H¨older’s inequality, the one-dimensional version of

(5) and (8) imply

\

J

|σ

a(x, y)|

dy

≤ C

2

i

|J|



I

\

J

sup

U ∈R+

\

J

A

(t, u)K

(y − u) du    dy dt 

≤ C

2

|J|

i



I



R

sup

U ∈R+

\

J

A

(t, u)K

(y − u) du   

2

dy 

dt 

≤ C

2

|J|

i



I



J

|A

(t, y)|

dy 

dt 

≤ C

2

|I|

|J|

i



I

\

J

|A

(t, y)|

dy dt 

≤ C

2

|I|

i

.

Hence

\

R\2^rI

\

J

|σ

a(x, y)|

dx dy ≤ C

X

2

2

i

≤ C

2

. Next we integrate over (R \ 2

I) × (R \ J):

\

R\2^rI

\

R\J

|σ

a(x, y)|

dx dy ≤ X

|i|=2^r−2

X

|j|=1 (i+1)2^K

\

i2^K

(j+1)2^L

\

j2^L

|σ

a(x, y)|

dx dy.

Integrating by parts we obtain, for x ∈[i2

, (i+1)2

) and y ∈[i2

, (i+1)2

),

|σ

a(x, y)| =   

\

I

\

J

A

(t, u)K

(x − t)K

(y − u) dt du   

≤ C2

2

i

j

\

I

\

J

|A

(t, u)| dt du

≤ C2

2

|I|

|J|

i

j

.

Thus

\

R\2^rI

\

R\J

|σ

a(x, y)|

dx dy ≤ C

X

|i|=2^r−2

X

|j|=1

2

2

2

i

j

≤ C

2

.

The integrations over I × (R \ 2

J) and over (R \ I) × (R \ 2

J) are

similar. Hence σ

is p-quasi-local. Theorem B implies (6) for p = q. Applying

Theorem A and (5) we obtain (6).

Let us single out this result for p = 1 and q = ∞. If f ∈ H

then (1) implies

kσ

f k

= sup

̺>0

γλ(σ

f > ̺) ≤ Ckf k

≤ Ckf k

, which shows (7). The proof of the theorem is complete.

Note that Theorem 1 was proved for Fourier series and for 3/4 < p < ∞ by the author [16] with another method.

We can state the same for the maximal conjugate Fej´er operator.

Theorem 2. For i, j = 0, 1 we have

ke σ

f k

≤ C

kf k

(f ∈ H

)

for every 1/2 < p < ∞ and 0 < q ≤ ∞. In particular, if f ∈ H

then λ(e σ

f > ̺) ≤ C

̺ kf k

(̺ > 0).

P r o o f. By Theorem 1 for p = q and (2) we obtain

ke σ

f k

= kσ

f e

k

≤ C

k e f

k

= C

kf k

(f ∈ H

) for every 1/2 < p < ∞. Now Theorem 2 follows from Theorem A and (1).

Since the set of those functions f ∈ L

whose Fourier transform has a compact support is dense in H

(see Wiener [20]), the weak type inequalities of Theorems 1 and 2 and the usual density argument (see Marcinkiewicz–

Zygmund [13]) imply

Corollary 1. If f ∈ H

(⊃ L log L) and i, j = 0, 1 then e

σ

f → e f

a .e. as T, U → ∞.

Note that e f

is not necessarily in H

whenever f is.

Now we consider the norm convergence of σ

f . It follows from (5) that σ

f → f in L

norm as T, U → ∞ if f ∈ L

(1 < p < ∞). We are going to generalize this result.

Theorem 3. Assume that T, U ∈ R

and i, j = 0, 1. Then ke σ

f k

≤ C

kf k

(f ∈ H

) for every 1/2 < p < ∞ and 0 < q ≤ ∞.

P r o o f. Since (σ

f )

= e σ

f , by Theorem 2 we have k(σ

f )

k

≤ C

kf k

(f ∈ H

) for all T, U ∈ R

and i, j = 0, 1. (3) implies that

kσ

f k

≤ C

kf k

(f ∈ H

; T, U ∈ R

).

Hence, for i, j = 0, 1,

ke σ

f k

≤ C

kf k

(f ∈ H

; T, U ∈ R

).

which together with Theorem A implies Theorem 3.

Corollary 2. Suppose that 1/2 < p < ∞, 0 < q ≤ ∞ and i, j = 0, 1. If f ∈ H

then

e

σ

f → e f

in H

norm as T, U → ∞.

We suspect that Theorems 1, 2 and 3 are not true for p ≤ 1/2 though we could not find any counterexample.

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Department of Numerical Analysis E¨otv¨os L. University

P´azm´any P. s´et´any 1/D H-1117 Budapest, Hungary E-mail: weisz@ludens.elte.hu

Received 17 June 1998