AN EXTENSION OF THE HARDY-LITTLEWOOD-P´OLYA INEQUALITY

Acta Mathematica Scientia 2011,31B6:2285–2288http://actams.wipm.ac.cn AN EXTENSION OF THE Dedicated to Professor Peter D.. In addition, we derive an accurate estimate for the best consta

Acta Mathematica Scientia 2011,31B(6):2285–2288

http://actams.wipm.ac.cn

AN EXTENSION OF THE

Dedicated to Professor Peter D Lax on the occasion of his 85th birthday

Congming Li John Villavert

Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309 USA

E-mail: congming.li@colorado.edu; john.villavert@colorado.edu

Abstract The Hardy-Littlewood-P´olya (HLP) inequality [1] states that if a ∈ lp, b ∈ lq

and

p > 1, q > 1, 1

p+

1

q > 1, λ = 2 −

 1

p+

1 q

 , then

X

r 6=s

arbs

|r − s|λ ≤ Ckakpkbkq

In this article, we prove the HLP inequality in the case where λ = 1, p = q = 2 with a logarithm correction, as conjectured by Ding [2]:

X

r 6=s,1≤r,s≤N

arbs

|r − s|λ ≤ (2 ln N + 1)kak2kbk2

In addition, we derive an accurate estimate for the best constant for this inequality

Key words Hardy-Littlewood-P´olya inequality; logarithm correction

2000 MR Subject Classification 46A45; 46B45; 49J40

1 Introduction

The well-known Hardy-Littlewood-Sobolev (HLS) inequality states that

Z

R n

Z

R n

f(x)g(y)

|x − y|λdxdy ≤ Cr,λ,nkf krkgks (1) for any f ∈ Lr(Rn) and g ∈ Ls(Rn) provided that

0 < λ < n, 1 < r, s < ∞ with 1

r +1

s+λ

n = 2

Hardy and Littlewood also introduced a double weighted inequality which was later gen-eralized by Stein and Weiss [3]:

Z

R n

Z

R n

f(x)g(y)

|x|α|x − y|λ|y|βdxdy ≤Cα,β,r,λ,nkf krkgks, (2)

∗ Received September 27, 2011 Research supported by the NSF grants DMS-0908097 and EAR-0934647.

2286 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B

where 1 < r, s < ∞, 0 < λ < n, α + β ≥ 0,

1 −1

r −λ

n<

α

n <1 − 1

r+1

s +λ+ α + β

To obtain the best constant in the weighted Hardy-Littlewood-Sobolev (WHLS) inequality (2), one can maximize the functional

J(f, g) =

Z

R n

Z

R n

f(x)g(y)

|x|α|x − y|λ|y|βdxdy with the constraints kf kr= kgks= 1 On the other hand, the Hardy-Littlewood-P´olya (HLP) inequality [1, inequality 381, p.288] [4]–a discrete analogue of the HLS inequality– is provided

in the setting of lp-spaces More precisely, the HLP inequality states that if a ∈ lp, b ∈ lq and

p >1, q > 1, 1

p+1

q >1, λ = 2 − 1

p+1 q

 ,

then

X

r 6=s

arbs

where the constant C depends on p and q only

The following theorem was conjectured by X Ding [2] It can be regarded as an extension

of the well-known HLP inequality in the case p = q = 2 and λ = 1 with a logarithm correction: Theorem 1 Let p = q = 2 and λ = 2 −1

p−1

q = 1 If a, b ∈ lp, then X

r 6=s,1≤r,s≤N

arbs

|r − s|≤ 2(ln N + 1)kak2kbk2. (4)

In fact we shall prove instead the following theorem in which Theorem 1 is a consequence Theorem 2 Let

P a 2

r =P b 2

r =1

X

r 6=s,1≤r,s≤N

arbs

then

2 ln N − 2 ≤ λN ≤ 2 ln N + 2(1 − ln 2)

Consequently we have:

λN <2 ln N + 1

2 Proof of Theorem 2

We prove Theorem 2 in three main steps

In step 1, we choose ar= br= √1

N and calculate that X

r 6=s,1≤r,s≤N

arbs

|r − s|≥ 2 ln N − 2.

This shows that λN ≥ 2 ln N − 2

In step 2, we derive the Euler-Lagrange equations for the maximizers a and b

No.6 C.M Li & J Villavert: AN EXTENSION OF THE HARDY-LITTLEWOOD-P ´ OLYA 2287

In step 3, we use the Euler-Lagrange equations to show that

λN ≤ 2 ln N + 2(1 − ln 2), thus completing the proof The calculations in steps 1 and 3 will make use of the following inequalities For a positive integer M , we have that

ln(M + 1) ≤

M

X

l=1

1

l ≤ 1 + ln M and

M

X

l=1

ln l ≥ M ln M − M + 1

Step1 Let ar= br= √1

N, then P a2

r=P b2

r= 1 where the summation is from 1 to N

It follows that

X

r 6=s,1≤r,s≤N

arbs

|r − s| =

1 N X

r 6=s,1≤r,s≤N

1

r− s =

2 N

N −1

X

s=1

N

X

r=s+1

1

r− s

N

N −1

X

s=1

N −s

X

l=1

1

l ≥ 2 N

N −1

X

s=1

ln(N − s + 1) = 2

N

N

X

l=1

ln l

≥ 2

N(N ln N − N + 1) ≥ 2(ln N − 1)

Using the definition of λN along with the preceding calculations, we arrive with the following estimate:

Step 2 We derive the Euler-Lagrange equations for the maximizers of (5) Let

JN(a, b) = X

r 6=s,1≤r,s≤N

arbs

|r − s|− λN

s X

1≤r≤N

a2 r

X

1≤s≤N

b2

Then by our definition of λN, we have JN(a, b) ≤ 0, and by compactness, there exist elements

aand b with kak2= kbk2= 1 such that

JN(a, b) = 0

Thus, we must have 0 = d

da rJN(a, b)

(a=a,b=b) Taking the derivative directly in (7) about ar, we obtain:

X

s 6=r,1≤s≤N

bs

|r − s|− λNar= 0.

Similarly, taking the derivative about ¯bs, we obtain:

X

r 6=s,1≤r≤N

ar

|r − s|− λNbs= 0.

Combining the above two equations together, we obtain the Euler-Lagrange equations:











λNar= X

s 6=r,1≤s≤N

bs

|r − s|

λNbs= X

r 6=s,1≤r≤N

ar

|r − s|.

(8)

2288 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B

Step 3 Here we will show that λN ≤ 2 ln N + 2(1 − ln 2)

With a change of sign if necessary, we may assume that

ar 0 = max{|ar|, |bs| : 1 ≤ r, s ≤ N } > 0

In fact, we may assume that all components are non-negative (and consequently positive by (8)), and ar 0 is the maximum for some r0 Then

λN =

N

X

s 6=r 0 ,s=1

bs

a0|r0− s| ≤

N

X

s 6=r 0 ,s=1

|bs|

|a0||r0− s| ≤

N

X

s 6=r 0 ,s=1

1

|r0− s|

=

r 0 −1

X

s=1

1

r0− s+

N

X

s=r 0 +1

1

s− r0

=

r 0 −1

X

l=1

1

l +

N −r 0

X

l=1

1 l

≤ 2 + ln(r0− 1) + ln(N − r0) = 2 + ln((r0− 1)(N − r0))

≤ 2 + ln N − 1

2

2

≤ 2 + ln N

2

2

= 2 + 2(ln N − ln 2)

Hence

Combining the estimates (6) and (9) yields

2 ln N − 2 ≤ λN ≤ 2 ln N + 2(1 − ln 2)

References

[1] Hardy G H, Littlewood J E P´ olya G Inequalities, Volume 2 Cambridge University Press, 1952

[2] Ding X Private Communication

[3] Stein E B, Weiss G Fractional integrals in n-dimensional Euclidean space J Math Mech, 1958, 7(4): 503–513

[4] Hardy G H, Littlewood J E, P´ olya G The maximum of a certain bilinear form Proc London Math Soc,

1926, 25(2): 265–282

[5] Stein E B, Weiss G Introduction to Fourier Analysis on Euclidean Spaces Princeton: Princeton University Press, 1971

[6] Lieb E Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities Ann Math, 1983, 118: 349–374

[7] Chen W, Li C Classification solutions of some nonlinear elliptic equations Duke Math J, 1991, 63: 615–622

[8] Li C, Chen W, Ou B Classification of solutions for an integral equation Comm Pure and Appl Math,

2006, 59: 330–343

[9] Chen W, Li C The best constant in some weighted Hardy-Littlewood-Sobolev inequality Proc Amer Math Soc, 2008, 136: 955–962

of the well-known HLP inequality in the case p = q = and λ = with a logarithm correction: Theorem Let p = q = and λ = −1...

Using the definition of λN along with the preceding calculations, we arrive with the following estimate:

Step We derive the Euler-Lagrange equations for the maximizers of (5)