### FRITZ CARLSON'S INEQUALITY AND ITS APPLICATION

AMIR KAMALY

Abstract

A Carlson-type inequality is proved and it is applied to show a Babenko-Beckner type of the Hausdorff-Young inequality onn-dimensional torus.

Introduction

Fritz Carlson's inequality (1934) states, [4], that
X^{1}

n1

a_{n}<

p X^{1}

n1

a_{n}^{2}

!^{1}_{4}
X^{1}

n1

n^{2}a_{n}^{2}

!^{1}_{4}

holds for any positive sequence
an^{1}_{n1} and not allan are 0. Let an:bf
n,
for a periodic functionf. Then, there can be equality only iff is a multiple
off^{0}, and therefor an exponential functionC_{0}e^{bx}. This is plainly impossible,
[7].

Note that the sumsP_{1}

n1an2 andP_{1}

n1n^{2}an2 are supposed to be finite.

The corresponding integral inequality, [4], [7], is
Z _{1}

0 f xdx

p Z _{1}

0 f^{2}
xdx

^{1}_{4} Z _{1}

0 x^{2}f^{2}
xdx

^{1}_{4}

:
Here there is equality whenf
x:_{abx}^{1} 2, for any positivea;b.

Forf 2A
Tandbf
0 0, the other expression of Carlson's inequality is
kfk_{A
T}C

kfk_{2}kf^{0}k_{2}
^{1}_{2}

: 1

Herekfk_{A
T}:P

m2Zjbf mjandA Tis the space of continuous functions on T having an absolutely convergent Fourier series. The variety of the constantCin 1depends on the definitions ofTand the Fourier series off.

MATH. SCAND. 86 (2000), 100^108

Received May 20, 1997.

B. Kjellberg, [11], and D. Mu«ller, [14] (Lemma 3.1) proved a multi-
dimensional extension of Carlson's inequality of the integral type. By using
the idea^{1} of Theorem 2.7.6. in [15], Carlson's inequality can be carried over
fromR^{n} to T^{n}. Our proof of the multi-dimensional case of
1(for the case
R^{n}see [10]) is new and more direct.

The well-known classical Hausdorff-Young inequality (1912^1923) states
that, for any complex-valued functiongin the Banach spaceL^{p}
,T

b g

k k_{p}0k kg _{p}
2

holds for 1p2. Here and throughout the paper,p^{0}is the dual exponent
of p. Also, k kbg _{p}0: P

n2Zjbg
nj^{p}^{0}

^{1}

p0

and k kg _{p}: R

Tjg
xj^{p}dx
ÿ ^{1}_{p}

are supposed to be finite.

Titchmarsh, [18], proved
2 for the space L^{p}
R in 1924. In fact,
2 is
true for locally compact unimodular groups , [13]. The result is due to R.A.

Kunze (1957). Hardy and Littlewood, [8], showed that
2is sharp and there
is equality if and only ifgC0e^{2mix} form2Z.

For the space L^{p}
R^{n} and for the even integer p^{0}, [2], the improvement is
due to K.I. Babenko (1961) and for allp, [3], it is due to W. Beckner (1975).

That is

kbfk_{p}0 B_{p}^{n}k kf _{p}
3

holds for p2 1;2. Bp:

p^{1}^{p}
p^{0}^{p}^{1}^{0}

s

is called the Babenko-Beckner constant.

bf :R

R^{n}f
xeÿ2i<;x>dx is the Fourier transform of f and h;xi:

P_{n}

1_{}x.

B. Russo (1974), [16], and J.J.F. Fournier (1977), [6], proved 3 for cer- tain classes of locally compact unimodular groups.

The extension of 3is due to J. Inoue (1992), [9]. For certain classes of nilpotent Lie groups he improved 3and obtained the constant

B_{p}^{dim
Gÿ}^{m}^{2}:

HereG:exp gandgis Lie algebras with the dual spacebg. dim Gis the dimension of nilpotent Lie groups G and m is the dimension of generic coadjoint orbits ofGinbg.

For the even integerp^{0}, [1], M.E. Andersson (1994) and for all p, [17], P.

1The referee made kindly this idea clear to me. He also informed me of the references [11] and [14] and gave me valuable comments on this paper (see the remark).

Sjo«lin (1995) proved a Babenko-Beckner type inequality
3for functions in
the spaceL^{p}
, with small supports.T

The purpose of this paper is to prove Carlson's inequality of type 1onn- dimensional torus and applying it to prove a Babenko-Beckner type of the Hausdorff-Young inequality for periodic functions with small supports.

Theorems and Proofs

Let the multi-indices andbe vectors inR^{n} with components_{k}and_{k}in
N_{0} such that is equivalent to _{k}_{k} for all 1kn. Define
m^{}:Q_{n}

k1m_{k}^{}^{k} form2Z^{n}and 0^{0}:1.

Throughout this paper,jj:P_{n}

k1kand :Q_{n}

k1kk. The operator
D^{}:Q_{n}

k1 @^{}^{k}

@xkk. Let also

Hp;a:sup k kbg _{p}0

g

k k_{p} : g2L^{p}
;T^{n} suppgB
0;a; k kg _{p}60

( )

and defineH_{p} :lim_{a!0}^{}H_{p;a}. Here and everywhere in the paperaobeys the
restriction 0<a<^{1}_{2} and B
0;ais a closed ball of radius a, centered at the
origin. Also,T^{n}:x2R^{n}:jx_{}j ^{1}_{2}; 1n

.
Assume '
x: 1 jxj ^{1}_{2}

0 jxj 1

such that '2C_{0}^{1}
R^{n}; 0'1 and
'a:'ÿ ^{x}_{a}

. Define x:ÿ

eÿ2i<b;x>ÿ1

'a
x. Here b:
b1;b2; ;bn
andj j b_{k} ^{1}_{2}; 1kn.

With the previous notation, we prove the following:

Theorem 1 (Generalisation of Carlson's inequality). Let f 2A
T^{n} and
bf
0 0. Let the absolute value of the multi-index be equal to the positive
integersuch that1and >^{n}_{q}where1<q2. Then we get

kfk_{A
T}^{n}_{}K_{n;q}^{
}kfk_{q}^{1ÿ}^{q}^{n} X

jj

kD^{}fk_{q}
0

@

1 A

qn

:

In the casebf 0 60, we obtain

kfk_{A
T}^{n}_{} kfk_{1}K_{n;q}^{
}kfk_{q}^{1ÿ}^{q}^{n} X

jj

kD^{}fk_{q}
0

@

1 A

qn

:

The positive constantKn;q^{
} depends only onn; andq.

Proof of Theorem1. The technique is analogous to the casen1, due to
Hardy, [7]. Letbf
0 0 andq^{0} be the dual exponent ofq. Define

102 amir kamaly

S: kbfk^{q}_{q}^{0}0

T : X

jj

kDd^{}fk^{q}_{q}^{0}0:
Fort>0 we also define

P: X

jj

t j
2m^{}j^{q}^{0}

ThenT P

jjkDd^{}fk_{q}0

_{q}^{0}

. By Ho«lder's inequality we get

kfk_{A
T}^{n}_{} X

jmj>0

jbf
mjP^{q}^{1}^{0}P^{ÿ}^{q}^{1}^{0}
4:1

X

jmj>0

jbf
mj^{q}^{0}P
0

@

1 A

q10

X

jmj>0

P^{ÿ}^{q}^{q}^{0}
0

@

1 A

1q

t^{ÿ}^{q}^{1}^{0}

tc_{n;}ST
^{1}

q0 X

jmj>0

1Cn;

t jmj^{q}^{0}^{}

_{ÿ}^{q}

q0

2 4

3 5

1q

: Because

X

jj

t j
2m^{}j^{q}^{0}

c_{n;}tX

jj

j
2m^{}j^{q}^{0} ttC_{n;}jmj^{q}^{0}^{}:
Here c_{n;}:P

jj1 and Dd^{}f
m
2im^{}bf
m. The positive constant
C_{n;} does depend onnand.

It is not hard to see that the sum P

jmj>0 1
1jmj^{q}^{0}^{}

ÿ ^{q}

q0

" #^{1}_{q}

is finite for >^{n}_{q}and

Z _{1}

0

dx

1x^{q}^{0}^{n}^{}^{q}^{q}^{0}ÿ^{n
qÿ1}_{q}

ÿ^{
qÿ1
qÿn}_{q}

n qÿ1q ÿ qÿ1 : 4:2

Now, by 4:1and 4:2we obtain

kfk_{A
T}^{n}_{}c_{0}t^{ÿ}^{q}^{1}^{0}

tc_{n;}ST
^{1}

q0 Z

R^{n}

dx
1^{C}^{n;}_{t} jxj^{q}^{0}^{}

^{q}

q0

2 64

3 75

1q

c0t^{ÿ}^{q}^{1}^{0} t
Cn;

^{n}

qq0

tcn;ST
^{1}

q0 Z

R^{n}

dx
1 jxj^{q}^{0}^{}^{q}^{q}^{0}

!^{1}_{q}

c0t^{ÿ}^{q}^{1}^{0} t
Cn;

^{n}

qq0

tcn;ST
^{1}

q0 Z _{1}

0

Z

fx2R^{nÿ1}:jxj1g

r^{nÿ1}drdx
1r^{q}^{0}^{}
^{q}^{q}^{0}

!^{1}_{q}

c0 wnÿ1

n

^{1}

qt^{ÿ}^{q}^{1}^{0} t
C_{n;}

^{n}

qq0

tcn;ST

ÿ ^{1}

q0

Z _{1}

0

dx
1x^{q}^{0}^{n}^{}

^{q}

q0

0 B@

1 CA

1q

c0A^{
}_{n;q}t^{nq}^{q}^{0} cn;ST
t

^{1}

q0

; for a positive constantc0. Here

A^{
}_{n;q} :

qÿ1w_{nÿ1}ÿ^{n
qÿ1}_{q}

ÿ^{
qÿ1
qÿn}_{q}

qÿ
qÿ1 ÿCn;^{n
1ÿq}_{q}

q

vu ut

;
andwnÿ1 is the surface area of the unit sphere inR^{nÿ1}.

Choose t^{S}_{T}, then by using (two times) the classical Hausdorff-Young
inequality
2we get

kfk_{A
T}^{n}_{}c0A^{
}_{n;q}

cn;1
ÿ _{qÿ1}
qq

kbfk_{q}0

1ÿ_{q}^{n} X

jj

kDd^{}fk_{q}0

0

@

1 A

qn

c0A^{
}_{n;q}

cn;1
ÿ _{qÿ1}
qq

kfk_{q}^{1ÿ}^{q}^{n} X

jj

kD^{}fk_{q}
0

@

1 A

qn

K_{n;q}^{
}kfk_{q}^{1ÿ}^{q}^{n} X

jj

kD^{}fk_{q}
0

@

1 A

qn

:

For the casebf
0 60 the proof is similar and we know thatjbf
0j kfk_{1}:

104 amir kamaly

Application of Theorem 1 for estimating of theA
T^{n}-norm of andH_{p;a}
Lemma (An upper bound for k k_{A
T}^{n}_{}). There exists a positive constant C0,
does not depend on a, such that

k k_{A
T}^{n}_{}C0a:

Proof of Lemma. It is obvious that 2C_{0}^{1}
R^{n}and form2Z^{n} we get
j
mj jb

Z

jxja
xeÿ2i<m;x>dxj a^{n}
Z

jyj1jeÿ2ia<b;y>ÿ1jdy

pn
a^{n1}

Z

jyj1dxna^{n1};
because

jeÿ2ia<b;y>ÿ1j 2aj<b;y>

pn a:

Heren:

pn

wn andwn_{ÿ}^{2}n^{n}^{2}

2 is the surface area of the unit sphere in
R^{n}.

Furthermore, by Leibniz's formula, together with Minkowski's inequality we obtain

X

jj

kD^{} k_{q}

pn aX

jj

kD^{}'ak_{q}X

jj

X

jj60

^{jj}kD^{ÿ}'ak_{q}
5

pn

a^{1ÿ}^{n}^{q} X

jj

kD^{}'k_{q}X

jj

X

jj60

^{jj}a^{jjÿjj}^{n}^{q}kD^{ÿ}'k_{q}

a^{1ÿ}^{n}^{q}

pn X

jj

kD^{}'k_{q}X

jj

X

jj60

^{jj}kD^{ÿ}'k_{q}
8>

<

>:

9>

=

>;

An;q;a^{1ÿ}^{n}^{q};5
because

X

jj

D^{}'_{a}

qa^{n}^{q}^{ÿ}X

jj

D^{}' _{q}:
Now, by Theorem 1 and invoking
5, we get

kbk_{1} X

m2Z^{n}

jb
mj na^{n1} X

jmj>0

jb mj

na^{n1}K^{
}_{n;q}k k_{q}^{1ÿ}^{q}^{n} X

jj

kD^{} k_{q}
0

@

1 A

qn

_{n}a^{n1} K_{n;q}^{
}

pn
k'k_{q}
_{1ÿ}^{n}

qA^{q}n;q;^{n}

a
nK_{n;q}^{
}An;q;^{q}^{n}

pn
k'k_{q}
_{1ÿ}_{q}^{n}

a

C0a;

because

k k_{q}

pn

k'k_{q}a^{1}^{n}^{p}:

Note that is the positive integer defined in Theorem 1 and
kbk_{1}: k k_{A
T}^{n}_{}.

Theorem 2 (An upper bound for Hp;a). For a fixed n2N, there exists a positive constant C0which does not depend on a, such that

H_{p;a}

1C_{0}a

B_{p}^{n}; 1p2:

Proof of Theorem2. The technique is analogous to the casen1, due to
Y. Domar, [5]. Choosef 2L^{p}
;R^{n} g2L^{p}
T^{n}, such that f g on the ball
B
0;aand zero outside of the ball. Defineg_{b}
x:eÿ2i<x;b>g
x. Then

gbb m bg mb

kfk_{p} kgk_{p}
fbb
m gbb
m:

Also, we get

gbb m ÿbg m Z

e^{ÿ2ihm;xi}ÿ

e^{ÿ2ihb;xi}ÿ1
g
xdx

Z

B
0;a
xg
xe^{ÿ2ihm;xi}dx

Z

B 0;ag x X

m^{0}2Z^{n}

mb ^{0}e^{2ihm}^{0}^{;xi}

e^{ÿ2ihm;xi}dx

X

m^{0}2Z^{n}

mb ^{0}bg
mÿm^{0}:

106 amir kamaly

Thus, we obtain
b
g_{b}ÿbg

k k_{p}0 X

m^{0}2Z^{n}

jb
m^{0}j X

m2Z^{n}

bg m

j j^{p}^{0}
^{1}

p0

k kbg _{p}0k kb _{1}:
By triangle inequality we have

b g

k k_{p}0ÿk kgb_{b} _{p}0kgb_{b}ÿbgk_{p}0k kbg _{p}0k kb _{1}:
Similarly, fort2R^{n}, we obtain

bg
k k_{p}0

1ÿ k kb _{1}

k kgbb _{p}0 X

m2Z^{n}

gbb m

j j^{p}^{0}
^{1}

p0

X

m

fbb m

^{p}^{0}
^{1}

p0

:

That is

bg
k k_{p}0

ÿ1ÿ kbk_{1}

X

m

Z

fb:jbkj^{1}_{2}g

fb_{b}
m

^{p}

0

db
^{1}

p0

X

m

Z

ftÿm:jtkÿmkj^{1}_{2}g

bf t

^{p}

0

dt
^{1}

p0

kbfk_{p}0:
Now, by Lemma we get

bg

k k_{p}0 kbfk_{p}0

1ÿ kbk_{1} kbfk_{p}0

1ÿC0a: 6

By 6, thus, we obtain

Hp;a sup

g

bg
k k_{p}0

g

k k_{p} B_{p}^{n}
1ÿC0a:
Because

supf

kbfk_{p}0

f

k k_{p}Bpn;

(see [3], p. 160). Chooseasuch thatC_{0}a<^{1}_{2}, then we get
1

1ÿC0a1C_{0}a;

because_{1ÿC}^{1}_{0}_{a}1C0aO Cÿ 02a^{2}
. Hence
H_{p;a}ÿ

1C_{0}a
B_{p}^{n}:

Remark. The arguments in this proof can be used to prove that the quo-
tient of the normsbgandbf inl^{p}^{0} andL^{p}^{0}, respectively, is 1O
a, asa!0^{}.
The Babenko-Beckner type of the Hausdorff-Young inequality for periodic
functions with small supports

Theorem3 HpBpn.

The proof is a consequence of Theorem 2.

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ROSA VILLAN REDUTTV. 13 SE-187 68 TØBY SWEDEN

108 amir kamaly