FRITZ CARLSON'S INEQUALITY AND ITS APPLICATION

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 X¹

nˆ1

a_n< 

p X¹

nˆ1

a_n²

!¹₄ X¹

nˆ1

n²a_n²

!¹₄

holds for any positive sequence…an†¹_nˆ1 and not allan are 0. Let an:ˆbf…n†, for a periodic functionf. Then, there can be equality only iff is a multiple off⁰, and therefor an exponential functionC₀e^bx. This is plainly impossible, [7].

Note that the sumsP₁

nˆ1an2 andP₁

nˆ1n²an2 are supposed to be finite.

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

0 f…x†dx 

p Z ₁

0 f²…x†dx

¹₄ Z ₁

0 x²f²…x†dx

¹₄

: Here there is equality whenf…x†:ˆ_a‡bx¹ 2, for any positivea;b.

Forf 2A…T†andbf…0† ˆ0, the other expression of Carlson's inequality is kfk_A…T†C

kfk₂kf⁰k₂ ¹₂

: …1†

Herekfk_A…T†:ˆP

m2Zjbf…m†jandA…T†is the space of continuous functions on T having an absolutely convergent Fourier series. The variety of the constantCin…1†depends 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 idea¹ of Theorem 2.7.6. in [15], Carlson's inequality can be carried over fromRⁿ to Tⁿ. Our proof of the multi-dimensional case of …1†(for the case Rⁿsee [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_p0k kg _p …2†

holds for 1p2. Here and throughout the paper,p⁰is the dual exponent of p. Also, k kbg _p0:ˆ P

n2Zjbg…n†j^p0

¹

p0

and k kg _p:ˆ R

Tjg…x†j^pdx 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…2†is sharp and there is equality if and only ifgˆC0e^2mix form2Z.

For the space L^p… †Rⁿ and for the even integer p⁰, [2], the improvement is due to K.I. Babenko (1961) and for allp, [3], it is due to W. Beckner (1975).

That is

kbfk_p0 B_pⁿk kf _p …3†

holds for p2 ‰1;2Š. Bp:ˆ



p^1p p^0p10

s

is called the Babenko-Beckner constant.

bf…†:ˆR

Rⁿf…x†eÿ2i<;x>dx is the Fourier transform of f and h;xi:ˆ

P_n

ˆ1x.

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

B_p^dim…G†ÿm2:

HereG:ˆexp…g†andgis Lie algebras with the dual spacebg. dim…G†is the dimension of nilpotent Lie groups G and m is the dimension of generic coadjoint orbits ofGinbg.

For the even integerp⁰, [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…3†for functions in the spaceL^p… †, with small supports.T

The purpose of this paper is to prove Carlson's inequality of type…1†onn- 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 inRⁿ with components_kand_kin N₀ such that is equivalent to _kk for all 1kn. Define m:ˆQ_n

kˆ1m_k^k form2Zⁿand 0⁰:ˆ1.

Throughout this paper,jj:ˆP_n

kˆ1kand :ˆQ_n

kˆ1kk. The operator D:ˆQ_n

kˆ1 @^k

@xkk. Let also

Hp;a:ˆsup k kbg _p0

g

k k_p : g2L^p… †;Tⁿ suppgB…0;a†; k kg _p6ˆ0

( )

and defineH_p :ˆlim_a!0^‡H_p;a. Here and everywhere in the paperaobeys the restriction 0<a<¹₂ and B…0;a†is a closed ball of radius a, centered at the origin. Also,Tⁿ:ˆx2Rⁿ:jxj ¹₂; 1n

. Assume '…x†:ˆ 1 jxj ¹₂

0 jxj 1

such that '2C₀¹…Rⁿ†; 0'1 and 'a:ˆ'ÿ ^x_a

. Define …x†:ˆÿ

eÿ2i<b;x>ÿ1

'a…x†. Here b:ˆ…b1;b2; ;bn† andj j b_k ¹₂; 1kn.

With the previous notation, we prove the following:

Theorem 1 (Generalisation of Carlson's inequality). Let f 2A…Tⁿ† and bf…0† ˆ0. Let the absolute value of the multi-index be equal to the positive integersuch that1and >ⁿ_qwhere1<q2. Then we get

kfk_A…Tⁿ_†K_n;q^†kfk_q^1ÿqn X

jjˆ

kDfk_q 0

@

1 A

qn

:

In the casebf…0† 6ˆ0, we obtain

kfk_A…Tⁿ_† kfk₁‡K_n;q^†kfk_q^1ÿqn X

jjˆ

kDfk_q 0

@

1 A

qn

:

The positive constantKn;q^† depends only onn; andq.

Proof of Theorem1. The technique is analogous to the casenˆ1, due to Hardy, [7]. Letbf…0† ˆ0 andq⁰ be the dual exponent ofq. Define

102 amir kamaly

S:ˆ kbfk^q_q⁰0

T :ˆ X

jjˆ

kDdfk^q_q⁰0: Fort>0 we also define

P:ˆ X

jjˆ

…t‡ j…2m†j^q0†

ThenT P

jjˆkDdfk_q0

_q⁰

. By Ho«lder's inequality we get

kfk_A…Tⁿ_†ˆ X

jmj>0

jbf…m†jP^q10P^ÿq10 …4:1†

X

jmj>0

jbf…m†j^q0P 0

@

1 A

q10

X

jmj>0

P^ÿqq0 0

@

1 A

1q

t^ÿq10

tc_n;S‡T ¹

q0 X

jmj>0

1‡Cn;

t jmj^q0

_ÿ^q

q0

2 4

3 5

1q

: Because

X

jjˆ

t‡ j…2m†j^q0

ˆc_n;t‡X

jjˆ

j…2m†j^q0 t‡tC_n;jmj^q0: Here c_n;:ˆP

jjˆ1 and Ddf…m† ˆ …2im†bf…m†. The positive constant C_n; does depend onnand.

It is not hard to see that the sum P

jmj>0 1 1‡jmj^q0

ÿ ^q

q0

" #¹_q

is finite for >ⁿ_qand

Z ₁

0

dx

…1‡x^q0n†^qq0ˆÿ^n…qÿ1†_q

ÿ^{qÿ1†…qÿn†}_q

n…qÿ1†q ÿ…qÿ1† : …4:2†

Now, by…4:1†and…4:2†we obtain

kfk_A…Tⁿ_†c₀t^ÿq10

tc_n;S‡T ¹

q0 Z

Rⁿ

dx 1‡^Cn;_t jxj^q0

^q

q0

2 64

3 75

1q

ˆc0t^ÿq10 t Cn;

ⁿ

qq0

tcn;S‡T ¹

q0 Z

Rⁿ

dx …1‡ jxj^q0†^qq0

!¹_q

ˆc0t^ÿq10 t Cn;

ⁿ

qq0

tcn;S‡T ¹

q0 Z ₁

0

Z

fx2R^nÿ1:jxjˆ1g

r^nÿ1drdx 1‡r^q0 … †^qq0

!¹_q

ˆc0 wnÿ1

n

¹

qt^ÿq10 t C_n;

ⁿ

qq0

tcn;S‡T

ÿ ¹

q0

Z ₁

0

dx 1‡x^q0n

^q

q0

0 B@

1 CA

1q

ˆc0A^†_n;qt^nqq0 cn;S‡T t

¹

q0

; for a positive constantc0. Here

A^†_n;q :ˆ



…qÿ1†w_nÿ1ÿ^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 tˆ^S_T, then by using (two times) the classical Hausdorff-Young inequality…2†we get

kfk_A…Tⁿ_†c0A^†_n;q



cn;‡1 ÿ _qÿ1 qq

kbfk_q0

1ÿ_qⁿ X

jjˆ

kDdfk_q0

0

@

1 A

qn

c0A^†_n;q



cn;‡1 ÿ _qÿ1 qq

kfk_q^1ÿqn X

jjˆ

kDfk_q 0

@

1 A

qn

ˆK_n;q^†kfk_q^1ÿqn X

jjˆ

kDfk_q 0

@

1 A

qn

:

For the casebf…0† 6ˆ0 the proof is similar and we know thatjbf…0†j kfk₁:

104 amir kamaly

Application of Theorem 1 for estimating of theA…Tⁿ†-norm of andH_p;a Lemma (An upper bound for k k_A…Tⁿ_†). There exists a positive constant C0, does not depend on a, such that

k k_A…Tⁿ_†C0a:

Proof of Lemma. It is obvious that 2C₀¹…Rⁿ†and form2Zⁿ we get j …m†j ˆ jb

Z

jxja …x†eÿ2i<m;x>dxj aⁿ Z

jyj1jeÿ2ia<b;y>ÿ1jdy 

pn a^n‡1

Z

jyj1dxˆna^n‡1; because

jeÿ2ia<b;y>ÿ1j 2aj<b;y> 

pn a:

Heren:ˆ 

pn

wn andwnˆ_ÿ²nⁿ²

… †2 is the surface area of the unit sphere in Rⁿ.

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

X

jjˆ

kD k_q 

pn aX

jjˆ

kD'ak_q‡X

jjˆ

X

jj6ˆ0

^jjkD^ÿ'ak_q …5†



pn

a^1ÿ‡nq X

jjˆ

kD'k_q‡X

jjˆ

X

jj6ˆ0

^jja^jjÿjj‡nqkD^ÿ'k_q

a^1ÿ‡nq 

pn X

jjˆ

kD'k_q‡X

jjˆ

X

jj6ˆ0

^jjkD^ÿ'k_q 8>

<

>:

9>

=

>;

ˆAn;q;a^1ÿ‡nq;5 because

X

jjˆ

D'_a

qˆa^nqÿX

jjˆ

D' _q: Now, by Theorem 1 and invoking…5†, we get

kbk₁ˆ X

m2Zⁿ

jb…m†j na^n‡1‡ X

jmj>0

jb…m†j

na^n‡1‡K^†_n;qk k_q^1ÿqn X

jjˆ

kD k_q 0

@

1 A

qn

_na^n‡1‡ K_n;q^† 

pn k'k_{q 1ÿ}ⁿ

qA^qn;q;ⁿ

a n‡K_n;q^†An;q;^qn 

pn k'k_{q 1ÿq}ⁿ

a

ˆC0a;

because

k k_q 

pn

k'k_qa^1‡np:

Note that is the positive integer defined in Theorem 1 and kbk₁:ˆ k k_A…Tⁿ_†.

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

1‡C₀a

B_pⁿ; 1p2:

Proof of Theorem2. The technique is analogous to the casenˆ1, due to Y. Domar, [5]. Choosef 2L^p… †;Rⁿ g2L^p…Tⁿ†, such that f ˆg on the ball B…0;a†and zero outside of the ball. Defineg_b…x†:ˆeÿ2i<x;b>g…x†. Then

gbb…m† ˆbg…m‡b†

kfk_pˆ kgk_p fbb…m† ˆgbb…m†:

Also, we get

gbb…m† ÿbg…m† ˆ Z

e^ÿ2ihm;xiÿ

e^ÿ2ihb;xiÿ1 g…x†dx

Z

B…0;a† …x†g…x†e^ÿ2ihm;xidx

ˆ Z

B…0;a†g…x† X

m⁰2Zⁿ

…mb ⁰†e^2ihm0;xi

e^ÿ2ihm;xidx

ˆ X

m⁰2Zⁿ

…mb ⁰†bg…mÿm⁰†:

106 amir kamaly

Thus, we obtain b g_bÿbg

k k_p0 X

m⁰2Zⁿ

jb…m⁰†j X

m2Zⁿ

bg…m†

j j^{p0 1}

p0

ˆk kbg _p0k kb ₁: By triangle inequality we have

b g

k k_p0ÿk kgb_{b p}0kgb_bÿbgk_p0k kbg _p0k kb ₁: Similarly, fort2Rⁿ, we obtain

bg k k_p0

1ÿ k kb ₁

k kgbb _p0ˆ X

m2Zⁿ

gbb…m†

j j^{p0 1}

p0

ˆX

m

fbb…m†

^{p0 1}

p0

:

That is

bg k k_p0

ÿ1ÿ kbk₁

X

m

Z

fb:jbkj¹₂g

fb_b…m†

^p

0

db ¹

p0

ˆX

m

Z

ftÿm:jtkÿmkj¹₂g

bf…t†

^p

0

dt ¹

p0

ˆ kbfk_p0: Now, by Lemma we get

bg

k k_p0 kbfk_p0

1ÿ kbk₁ kbfk_p0

1ÿC0a: …6†

By…6†, thus, we obtain

Hp;a ˆsup

g

bg k k_p0

g

k k_p B_pⁿ 1ÿC0a: Because

supf

kbfk_p0

f

k k_pˆBpn;

(see [3], p. 160). Chooseasuch thatC₀a<¹₂, then we get 1

1ÿC0a1‡C₀a;

because_1ÿC¹_0aˆ1‡C0a‡O Cÿ 02a² . Hence H_p;aÿ

1‡C₀a B_pⁿ:

Remark. The arguments in this proof can be used to prove that the quo- tient of the normsbgandbf inl^p0 andL^p0, respectively, is 1‡O…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.

REFERENCES

1. M.E. Andersson, Local variants of the Hausdorff-Young inequality, Gyllenberg, Persson (Eds.) Analysis, Algebra and Computers in Mathematical Research: Proceedings of the 21st Nordic congress of mathematicians (1994), Marcel-Dekker, New York.

2. K.I. Babenko,An inequality in theory of Fourier integrals,Amer. Math. Soc. Transl. (2) 44 (1965), 115^128.

3. W. Beckner,Inequalities in Fourier integrals, Ann. of Math. 102 (1975), 159^182.

4. F. Carlson,Une ine¨galite¨,Arkiv fo«r matematik, astronomi och fysik 25 B, No. 1 (1934).

5. Y. Domar, private communication with P. Sjo«lin.

6. J.J. Fournier,Sharpness in Young's inequality for convolution, Pacific J. Math. 72, No. 2 (1974), 293^397.

7. G.H. Hardy,A note on two inequalities,J. London Math.Soc. 11 (1936), 167^170.

8. G.H. Hardy, J.E. Littlewood, G. Po¨lya,Inequalities,Cambridge University Press, London J.

London (1934).

9. J. Inoue, L^p-Fourier transforms on nilpotent Lie groups and solvable Lie groups action on Siegel domains,Pacific J. Math. 155, No. 2 (1992), 295^318.

10. A. Kamaly,Some new inequalities for integral and discrete norms,Unpubished (1996).

11. B. Kjellberg,Ein Momentenproblem,Arkiv fo«r matematik, astronomi och fysik 29 A, No. 2 (1942).

12. N.Y. Krugljak, L. Maligranda and L.E. Persson,A Carlson type inequality with blocks and interpolation,Studia Math. 104, No. 2 (1993), 161^180.

13. R.A. Kunze,L^p-Fourier transforms on locally copmact unimodular groups, Trans. Amer.

Math. Soc. 89 (1958), 519^540.

14. D. Mu«ller,On the Spectral Synthesis Problem for Hypersurfaces ofRⁿ,J. Funct. Anal. 47 (1982), 247^280.

15. W. Rudin,Fourier Analysis on Groups,Interscience Publishers, London (1962).

16. B. Russo,The norm of the L^p-Fourier transform on unimodular groups,Trans. Amer. Math.

Soc. 192 (1974), 293^318.

17. P. Sjo«lin,A remark on the Hausdorff-Young inequality,Proc. Amer. Math. Soc. 123 (Oct.

1995), 3085^3088.

18. E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals(1937), Oxford, at Clar- endon Press.

ROSA VILLAN REDUTTV. 13 SE-187 68 TØBY SWEDEN

108 amir kamaly