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

Acta Mathematica Scientia 2011,31B(6):2285–2288 http://actams.wipm.ac.cn AN EXTENSION OF THE ´ HARDY-LITTLEWOOD-POLYA INEQUALITY∗ 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 and The Hardy-Littlewood-P´ olya (HLP) inequality [1] states that if a ∈ lp , b ∈ lq 1 + > 1, λ = − p q p > 1, q > 1, then r=s a r bs ≤C a |r − s|λ p 1 + p q , b q In this article, we prove the HLP inequality in the case where λ = 1, p = q = with a logarithm correction, as conjectured by Ding [2]: r=s,1≤r,s≤N a r bs ≤ (2 ln N + 1) a |r − s|λ b 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 Introduction The well-known Hardy-Littlewood-Sobolev (HLS) inequality states that Rn Rn f (x)g(y) dxdy ≤ Cr,λ,n f |x − y|λ r g (1) s for any f ∈ Lr (Rn ) and g ∈ Ls (Rn ) provided that < λ < n, < r, s < ∞ with 1 λ + + = r s n Hardy and Littlewood also introduced a double weighted inequality which was later generalized by Stein and Weiss [3]: Rn ∗ Received Rn f (x)g(y) dxdy ≤ Cα,β,r,λ,n f |x|α |x − y|λ |y|β r g s, (2) September 27, 2011 Research supported by the NSF grants DMS-0908097 and EAR-0934647 2286 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B where < r, s < ∞, < λ < n, α + β ≥ 0, 1− λ α − < 1, q > 1, then r=s 1 + > 1, λ = − p q ar b s ≤C a |r − s|λ p 1 + p q , (3) b q, 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 = and λ = with a logarithm correction: Theorem Let p = q = and λ = − p1 − q1 = If a, b ∈ lp , then r=s,1≤r,s≤N ar b s ≤ 2(ln N + 1) a |r − s| b (4) In fact we shall prove instead the following theorem in which Theorem is a consequence Theorem Let ar b s λN = max , (5) ar = br =1 |r − s| r=s,1≤r,s≤N then ln N − ≤ λN ≤ ln N + 2(1 − ln 2) Consequently we have: λN < ln N + Proof of Theorem We prove Theorem in three main steps In step 1, we choose ar = br = √1N and calculate that r=s,1≤r,s≤N ar b s ≥ ln N − |r − s| This shows that λN ≥ ln N − 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-POLYA 2287 In step 3, we use the Euler-Lagrange equations to show that λN ≤ ln N + 2(1 − ln 2), thus completing the proof The calculations in steps and will make use of the following inequalities For a positive integer M , we have that M ln(M + 1) ≤ l=1 Step1 Let ar = br = It follows that r=s,1≤r,s≤N ≤ + ln M and l √1 , N ar b s = |r − s| N N = ≥ a2r = then r=s,1≤r,s≤N N −1 N −s s=1 l=1 M ln l ≥ M ln M − M + l=1 b2r = where the summation is from to N = r−s N ≥ l N N −1 N −1 s=1 N r − s r=s+1 ln(N − s + 1) = s=1 N N ln l l=1 (N ln N − N + 1) ≥ 2(ln N − 1) N Using the definition of λN along with the preceding calculations, we arrive with the following estimate: λN ≥ ln N − (6) Step We derive the Euler-Lagrange equations for the maximizers of (5) Let JN (a, b) = r=s,1≤r,s≤N ar b s − λN |r − s| a2r 1≤r≤N b2s (7) 1≤s≤N Then by our definition of λN , we have JN (a, b) ≤ 0, and by compactness, there exist elements a and b with a = b = such that JN (a, b) = Thus, we must have = dadr JN (a, b) (a=a,b=b) Taking the derivative directly in (7) about ar , we obtain: s=r,1≤s≤N bs − λN ar = |r − s| Similarly, taking the derivative about ¯bs , we obtain: r=s,1≤r≤N ar − λN bs = |r − s| Combining the above two equations together, we obtain the Euler-Lagrange equations:   bs   λ a =   N r |r − s| s=r,1≤s≤N (8) ar    λ b = N s   |r − s| r=s,1≤r≤N 2288 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B Step Here we will show that λN ≤ ln N + 2(1 − ln 2) With a change of sign if necessary, we may assume that ar0 = max{|ar |, |bs | : ≤ r, s ≤ N } > In fact, we may assume that all components are non-negative (and consequently positive by (8)), and ar0 is the maximum for some r0 Then N λN = s=r0 ,s=1 = r0 −1 s=1 bs ≤ a0 |r0 − s| N + r0 − s s=r N s=r0 ,s=1 = s − r0 +1 |bs | ≤ |a0 ||r0 − s| r0 −1 l=1 + l N −r0 l=1 N s=r0 ,s=1 |r0 − s| l ≤ + ln(r0 − 1) + ln(N − r0 ) = + ln((r0 − 1)(N − r0 )) ≤ + ln N −1 2 ≤ + ln N 2 = + 2(ln N − ln 2) Hence λN ≤ ln N + 2(1 − ln 2) (9) Combining the estimates (6) and (9) yields ln N − ≤ λN ≤ ln N + 2(1 − ln 2) References [1] Hardy G H, Littlewood J E P´ olya G Inequalities, Volume 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