Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 579-582
579
FOURIER SERIES AND THE MAXIMAL OPERATOR ON THE
WEIGHTED SPECIAL ATOM SPACES
GERALDO SOARES DE SOUZA
Department of Mathematics
Auburn University
Auburn University, AL 36849-5310
(Received November 24, 1987 and in revised form July 5, 1988)
ABSTRACT.
For an interval
special atom looks like
I
.
in
b(t)
[0,2]
[(L(t)
P (I I
L
with halves
(R(t)],
R, a weighted
and
where
is a non-
p
ngative function satisfying some properties.
We consider the weighted special atom space
B(p)
*
formed by
combinations of these weighted atoms.
We showed that if
f z B(p)
linear
then its Fourier series converges almost
everywhere, using the Carleson-Hunt idea on their famous result about the
almost everywhere convergence on
KEY WORDS AND PHRASES.
L-spaces.
P
Maximal operator, Lorentz Spaces and Fourier Series.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
Primary 42A20.
INTRODUCTION.
In [2] the following space was introduced.
n
n-0
n-0
weighted special atom, that is, a real valued function
b(t)=--21’
which is either
interval in
III
[0,21"],
L
or
b(t)-
is the left ha
denotes the length of
I,
(|i|iPl
IL
or
[XR(t)- XL(t)]’
I
and
R
where
is a non-negative, real valued function which is increasing, and
is endowed with the norm
llfllB(@)m Inf n-07, ,,
ICnl
all possible representations of
is an
E
p(0)
p
and
0.
B(p)
where the infimum is taken over
is a Banach space.
For more details
B(p)
the reader is referred to [2].
a function
on the almost everywhere convergence of
theorem
The Carleson-Hunt
of f,
series
-, asserts that if f L then the Fourier
p
for
L
about this space
in
B(@)
f.
I
is the right half.
the characteristic function of
XE
[0,21.],
b, defined on
P
denoted by
<
<
S(f,x),
P
converges to
f
almost everywhere.
(See [I] and [5].)
Carleson-Hunt, we will prove that
In this note, using basically the idea of
B(@) then the operator
condition on p if f
with some additional
f
G.S. DE SOUZA
580
Tf(x)
.ISn(f’x)
sup
n
of
Sn(f,x)
where
is the nth partial sum of the Fourier series
L().
f, is bounded into some Banach space
We would like to point out that other direct proofs of the almost everywhere
B(p)
convergence for functions in
B(p)
any function in
For example, for some
are also available.
p
However we prefer this one,
satisfies Dini’s condition.
because it is a consequence of the boundedness of the maximal operator and this
boundedness could be useful in other contexts, for example in the interpolation of
operators.
One of the most important features of the space
F
functions
I
{z
D
on the disc
I
zl
C
F’(re
0
<
i]
is that under some
F
satisfying
<
dSdr
l-r
0
where the dash means the derivative of
For this
z.
with respect to
[2].
characterization see
2.
B(@)
p, it can be identified with the space of analytic
additional conditions on
PRELIMINARIES.
[0,-)
p: [0,-)
A function
DEFINITION 2.1.
is said to be in the class
b,
if it
satisfies
p(0)
i)
iv)
lh0
v)
h
0,
@(s)
ii)
ds
s
2 p(t)
<
p
Cp(h)
C
dt
C
p()
0
<
<
f
independent of
b,
We define the space
(t)(t)dt
measure and
<
(R)}
where
f*
non-negative, decreasing function.
Recall that if
wf(x)
t
a,
p.
and
h
p(t)
t
a
2e
log
t
{f:[0,2]
L()
by
R
is the decreasing rearrangement of
If(x)
>
t},
y]
f
where seas means Lebesgue
is a non-negative, decreasing function.
In [6], G. G. Lorentz showed that
and
p(t)
are
L(#)
Inf{y: seas {x;
f*(t)
defined by
C
with
I.
DEFINITION 2.2.
I1 il, s 0
is decreasing,
an absolute constant
Example of functions in the class
for
(t)
---[-
iii)
is non-decreasing,
lis sup
n- k>n
II
is a norm if and only if
L()
Also he proved that
is a
is a Banach space.
S (f x) is the nth partial sum of the Fourier series of f
n
then S (f,x) converges to f almost
ISk(f,x) S(f,x)
n
>n
everywhere if and only if
3.
wf(x)
0
almost everywhere.
MAIN RESULT
The main result can be stated as follows.
THEOREM.
Let
p
be a function in the class
operator defined by
boundedly, that is,
and S
(f,x)
n
Tf(x)
IITfll
sp
ISn(f,x)
M[IflIB(p),
where
b,.
maps
M
If
B(p)
B(p)
f
then the maximal
L()
into
for
(t)
is a positive, absolute constant
is the nth partial sum of the Fourier series of
f
p(t.__)
t
FOURIER SERIES AND THE MAXIMAL OPERATOR ON THE ATOM SPACES
PROOF.
[X
fh(t
(t)
[-h,0)
p(2h)
g(t)
estimate for
sin[n+1/2]t
t
2 sin
(t).
X
>
h
0
In fact let
_<
IS(g,x)
n
h-
<2C
; Dn(t)dt
_<
A
A
where
Sn(f,x)
<
Itl _< ,
<
0
dt
Ch
_<
<
x-h--
where
x
I
Dn (t)dt.
x-h
satisfied by the
Thus
h
2C
for
x
>
2h
x
x<-2h.
;; Dn(t)dt
Recall that
Therefore we have
I
,t,
x-h
for
-x
)D n(x-t)dt
is an absolute constant.
_<
Dn(t
x-h
and
C
where
n
S (g x)
n
IDn(t)l
We now use the elementary inequality
D
f(x-a)
which will follow from the
is the Dirichlet kernel.
Dirichlet kernels
fa(x)
where
[O,h]
[0,h]
D (t)
n
a
Consequently we just need to prove the result for
],
(t)
X
fa
T f
First of all we notice that the operator
maps B(p) into B(p) continuously.
581
is uniformly bounded in
x; that is,
and
n
(See [7], volume
is an absolute constant.
page 57.
Consequently we have
_<
I)
Tg(x)
2)
Tg(x)_< 2C
A
x
for ali
Ixl
for
It
We now evaluate
>
(t)
for
liTgll4,
2h.
-----.
p(t)
[ITgl[,- ;02n (Tg)*(x) dx- ;02h (Tg)*(x)
2h @(x)
2n @(x)
i A I
dx + 2Ch
dx.
0
12h
Now using the conditions on
llTgll _< AC
The constant
C
p
f(x)
p(2h)2h
llTfhll <
E
B(p)
f(t)
, l=nl < -.
Cnbn(t)
lITfll < Mn.0 Icnl
we get
If
B(@)
f
, Cnbn(X)
then
where
n-0
n-0
b
n
b (t)
n
where
which implies
complete.
f(x)
then
n-0
n-0
almost everywhere.
f
dx
Mp(2h)
M and so if
Let @ be in the class b,.
Let
PROOF.
12(Tg)*(x)0(X)2h
x
we get
p(2h) + 2Ch
(t)
XLn (t)] and
p(il n [(Rn
IITfll _< MIIflIB(p)- The proof is
to
dx +
may not be the same at every occurence.
Consequently
COROLLARY.
.
In fact, using I) and 2) we get
[S (f x)}
ICnl
<-
converges
and the
are weighted special atoms. Now observe that
wf(x)
lim sup
n
k>n
ISk(f,x)
S(f,x)
satisfies
wf(x) <
2Tf(x).
Therefore
>n
2n
0
w(x)
p(t) dt
other hand we see that
_<
[Tf(x)] *P(----!)
2
dt
0
wf
and so
wf
L().
k
wf_fk
where
fk(x)
n-0’ Cnbn(X)
and
On the
G.S. DE SOUZA
582
llf k
fllB(p)
0
as
k
-.
wf_fk(X)
wf(x)
Then
consequently the theorem, that is, the boundedness of
after definition 2.2 that
We emphasize that for
space
B
paper on
p
and
B(p)
L()
Sn(f,x)
t
I/p,
<
L(p,l)
is the space
p
in
and
T, implies that
f(x) almost everywhere.
p(t)
The proof is complete.
<
the space
[3],
and so the results in this
B(p)
is the
is a generalization of that result.
We also point out that for
f c B(p)
fk)(X)
2T(f
(t)
so that the Fourier series of
t, one can indeed show that there is a
f, diverges almost everywhere. (See [4]).
This answers a question asked of the author by Professor Guido Weiss concerning
almost everywhere convergence of functions in
B(p)
for
p(t)
t.
REFERENCES
[I]
CARLESON, L., On the convergence and growth of partial sums of Fourier
series, Acta. Math. 116, (1966) 135-157.
[2]
BLOOM, S. and DE SOUZA, G. S., Atomic decomposition of the generalized
Lipschitz spaces
To appear, Illinois Journal of Mathematics.
[3]
DE SOUZA, G. S., On the convergence of Fourier series, Internat. J.
[4]
DE SOUZA, G. S. and SAMPSON, G., A function in Dirichlet space
[5]
HUNT, R. A., On the convergence of Fourier series, Proc. Conference
Math. and Math. Sci.
7(4), (1984) 817-820.
that its Fourier series diverges almost everywhere.
[6]
[7]
B
such
Preprint.
Southern Illinois University, Edwardville, IIi. 1967.
LORENTZ, G. G., On the theory of spaces A, Pacific Journal of Math.
I(1951), 411-429.
ZYGMUND, A., Trigonometric Series, Cambridge University Press, London and
New York, 1959.
Division of Mathematics
Foundations, Analysis and Topology
Auburn University
Auburn, AL 36849-5310