I ntern. J. Math. & ,,ath. Si.
Vol. 5 No. 6 (1982) 715-720
715
ON THE HARDY-LITTLEWOOD MAXIMAL THEOREM
SHINJI YAMASHITA
Department of Mathematics
Tokyo Metropolitan University
Fukazawa, Setagaya, Tokyo 158
Japan
(Received May 5, 1981)
The Hardy-Littlewood maximal theorem is extended to functions of class
ABSTRACT.
PL
in the sense of E. F. Beckenbach and T.
the absolute constant in the inequality.
Rad,
with a more precise expression of
As applications we deduce some results on
hyperbolic Hardy classes in terms of the non-Euclidean hyperbolic distance in the
unit disk.
KEY WORDS AND PHRASES. Hardy-Littlewood’s Maximal Theorem, Subharmonic Functions of
Class PL, Hady Class, Hyperbolic Hardy Clas.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 30D55, 31A05.
i. INTRODUCTION.
Let
D.
{ Izl < i},
D
For a function
1
hereafter always
defined on
JUIp
T
on
g
[0,2),
T
let
T
u
be a function subharmonic in
we denote
I Ig Ip(t)dt ]I/p;
0
<
p
<
unless otherwise specified.
we customarily denote
lim
and let
SUPr
__> i-0
Urp’
Then, although
u
is not
S. YAMASHITA
716
u(reit), tT,
Ur(t)
where
S(t,R)
Let
D.
holomorphic in
0
<R<
MR(u)(t)
p
< co.
f(reit) at
limr 1-0
I
and the point
MR(U
u
of
e
it
(t
T);
T
is defined on
be the Hardy class consisting of all functions
llfll P
such that
[i,
IlfIlp =Ilf*llp
Each
e
Theorem
p
H
f
it
by
2.6,
LP(T).
f*
T, and
t
for a.e.
f
a(p,R)
We then observe that
p. 21].
llMR(Ifl)Ip/llfllp;
sup
< co
of class
a(p,R)
[4,
PL
f
log u
u
llul
i
H
p,
IlUllp <
p
Let
p,
[3,
where
0.
f
Theorem 27, p.
a*(p,R)
114]
then reads that
is defined in terms of functions
PL
D; we regard
u
flu*If p.
S
p
co, where
i
for
IfIPLp
if
f
PL
u
< a(p,R) < a*(p,R).
a(p,R)
a*(p,R)
D, and
such that
f
C Hp.
1.
Then
For
at
e
it
lull
P
It
( co.
for a.e.
t
Set
We first observe
a(1,R) 1/p.
be the family of subharmonic functions
p
0
D, the
holomorphic in
u*(t)
has the radial limit
Apparently,
PL, if u
u
as a subharmonic function.
-co
be the family of all
PLp
PL, or
is said to be of class
is subharmonic in
p
it follows that
THEOREM 1.
REMARK.
u
Let
will be observed that
and that
D
defined in
is subharmonic in
Ifl PL.
modulus
H
p. 9].
PL, the function
that
of order
The main purpose of the present paper is to prove
a(1,R) l/p,
a(p,R)
A function
and if
(p,R).
for each pair
a*(p,R)
that
Since
D
holomorphic in
f*(t)
has the radial limit
The celebrated Hardy-Littlewood theorem
T
f
by
a(p,R)
u
for
hereafter always
In the present paper we introduce the Hardy-Littlewood number
(p,R)
p
[u(z); z S(t,R)}.
sup
H
Let
<
R
The maximal function
i.
llflIp =II Ifl
be the domain consisting of the interior of the
zl
convex hull of the circle
For simplicity,
0 (r <i.
u
_>
0
in
D
such
HARDY-LITTLEWOOD MAXIMAL THEOREM
b(p,R)
llMR(U)Ilp/UIlp;
sup
p
is finite for
[3,
by
i
a*(p,R)<_b(p,R)
S p,
u
26,
Theorem
.
p
for
p.
717
0)
u
113].
Obviously,
H
To propose an application to the hyperbolic Hardy class
tanh-l(
(z,w)
z-wl)
wlllm
z
be the non-Euclidean hyperbolic distance between
(z,0)
log[(l +
D,
i, in
exp[p log (f)]
4],
Theorem
Iz)],
ll(f)|p <
with
f
follows from that of
(f)
PL
p
for all
H
with
f*(t)
at
it
f
H
(f(re it) ,f*(t) )Pdt -->
0
T.
The inequality
sup
(f)P(z);
z
T
holds for all
f
_
S(t,R)dt
Set (z)
holomorphic and bounded,
(f).
We thus define
(f) (or, (f) E PL)
H
E
f
H
If
as
(f)P
observed in
[6].
is bounded, f
has the radial
We then propose
r
-
(f)
is a member of
LP(T),
and
i-0.
a(l,R)I Y(f*)P(t)dt
T
HP-.
The first assertion, a consequence of the
of the F. Riesz theorem
2.
D.
The subharmonicity of
the function
as
f
in
is a semigroup with respect to the
Since each
t
is
w
and
A few modifications of the proof of [6
H
show that
for a.e.
For each
THEOREM 2.
IT
e
log
z
For
Ifl
o.
HP
f
multiplication, and is convex.
limit
D.
z
the hyperbolic counterpart of
the family of such
Therefore
zl)/(l
we let
[i,
Theorem
s@cond,
is the hyperbolic counterpart
2.6].
PROOFS.
For the proof of Theorem i it suffices to show that
a*(p,R) < a(1,R) 1/p < a(p,R).
S. YAMASHITA
718
a(p,R)
Since
<
a*(p,R),
the identities in Theorem i follow.
a(l,R)i/P.a, a(p,R)
To prove that
we let
f
I
and
IF, where
an inner-outer factorization, f
I
H
F
on
a e
T.
a.e. on
T
F
Since
D
is zero-free in
F I/P
g
Then
f admits
are an inner and an outer
I*l
function, respectively, such that the radial limits satisfy
f*l
0.
f
with
HP
I
and
so that
IF*I
f*l
g*l
p
Therefore,
IIMR(Ifl)II I < IIMR(IFI)III IIMR(
a(p,R)Pllgl[pp
Igl)ll p <
R)PflII
a(p
a(l,R) < a(p,R) p.
whence
a*(p,R) < a(l,R) I/p,
To prove that
p log v
and
(x)
e
X
one finds that
PL p
we let
v
p
(u).
v
with
Since
(u)
D, there exists a positive harmonic majorant of
majorant in
The F. Riesz decomposition then yields that
potential in
u
u
^
0.
v
P, where
u
u
admits a harmonic
D [5, p. 65].
in
P
Setting
>
is the Green
0
D, and
^ (z)= 2----
u
(
zl
z
)TIeit
(zD)
2
is the Poisson integral of the measure
The radial limit
,
dt.
d/As(t).
u’(t)dt +
d(t)
u*
of
LI(T)-
is of
u
It follows from a general theorem
ds(t) 0
that
u(z)
<_ h(z)
1
2--
a.e. on
I
T
it
2
(t)
db[
[2, Theorem],
and that
z2
1
and
is singular with respect, to
applied to the present
LI(T).
(u*)
(z
u*(t)dt
u
and
Consequently,
D),
and the Jensen inequality yields that
5o(u)
where
V
_<
)
_< v,
is the Poisson integral of
O(u*).
Set
f
e
h+ik
where
k
is a
conJu-
HARDY-LITTLEWOOD MAXIMAL THEOREM
gate of
h
in
v*p.
(u*)
D.
Then,
(h)V,
Ifl
On the other hand, vp
R(v)ll pP < MR(Ill) <
a(l,R)
719
fHI
so that
(u) <_ (h)
fl
If*
with
(h’)
D, whence
in
IlfllI
The Lebesgue dominated convergence theorem, together with
vP(t)r "< MR(v)P(t)
a,,
,R)I/P
Ilv*llp
Vlpl
IIfll
as
r-->
Therefore
i-0.
a*(p,R) <
follows from
We next prove Theorem 2.
((f)
Since
T),
p
p
IIVr Ilpp
yields that
(t
PL
p
Set
for all
f
H,
it follows that
a(p,R)
<_ a*(p,R)
a(l,R) I/p
so that
MR(
(f)*
As is observed in the proof of Theorem i,
g PLp,
LP(T).
and
(f)
p
( f*)p"
((f*)
T
a.e. on
because
Thus, the second assertion holds with
(f)
(f*)
The Lebesgue dominated convergence theorem with the estimate
(f(reit),f*(t)) p <_ 2p(f)p(re it)
+
2P((f*)P(t)
2P+IMR (((f)P)(t)
2P+IMR ((f))P(t),
again yields that
IT
((f(reit),f*(t))Pdt
REMARK.
I
ao_(p,R).
Since
(f)
- 0
i
r
as
for
f
i-0.
(e 2
1)/(e 2 + l)
p
H
0-’
it follows that
S. YAMASHITA
7_0
REFERENCES
i.
Duren, P. L.
Volume
2.
Grding,
(196)
p
of H Spaces, Academic Press
York and London, 1970.
Theor
38), New
(Pure
and Applied Mathematics,
L. and L. HSrmander. Strongly subharmonic functions, Math. Scand. 15
93-96; Correction, ibid. 18_(1966) 183.
3.
Hardy, G. H. and J. E. Littlewood. A maximal theorem with function-theoretic
applications, Acta Math. 5_(1930) 81-116.
4.
RadS,
5.
Yamashita, S. On some families of analytic functions on Riemann surfaces,Nagoya
Math. J. 31(1968) 57-68.
6.
Yamashita
T. Subharmonic Functions, Springer-Verlag (Ergebnisse der Mathematik und
ihrer Grenzgebiete, 5. Band), Berlin-GSttingen-Heidelberg, 1937.
S.
Hyperbolic Hardy class H
1
Math. Scand.
5(1979)
261-266.