ON MULTILINEAR FRACTIONAL INTEGRALS. Loukas Grafakos Yale University

Transcription

1 ON MULTILINEAR FRACTIONAL INTEGRALS Loukas Grafakos Yale Uiversity Abstract. I R, we prove L p 1 L p K boudedess for the multiliear fractioal itegrals I α (f 1,...,f K )(x) = R f 1 (x θ 1 y)...f K (x θ K y) y α dy where the θ j s are ozero ad distict. We also prove multiliear versios of two iequalities about fractioal itegrals ad a multiliear Lebesgue differetiatio theorem. 1. Itroductio. Although it is ot kow whether the bi(sub)liear maximal fuctio 1 M(f,g)(x) = sup N>0 2N or the biliear Hilbert trasform H(f,g)(x) = p.v. N N f(x + t)g(x t) dt f(x + t)g(x t) dt t map L p (R 1 ) L p (R 1 ) L 1 (R 1 ) boudedess ito L 1 for the correspodig multiliear fractioal itegrals ca be obtaied. Throughout this ote, K will deote a iteger 2 ad θ j, j =1,...,K will be fixed, distict ad ozero real umbers. We are goig to be workig i R ad α will be a fixed real umber umber stricly betwee 0 ad. We deote by f the K-tuple (f 1,...,f K ) ad by I α the K-liear fractioal itegral operator defied as follows: I α (f)(x) = f 1 (x θ 1 y)...f K (x θ K y) y α dy. Whe K = 1 the operators I α are the usual fractioal itegrals as studied i [ST]. We also deote by M(f) the K-subliear maximal fuctio M(f)(x) = sup(ω N ) 1 f 1 (x θ 1 y)... f K (x θ K y) dy N>0 1 Typeset by AMS-TEX

2 where Ω is the volume of the uit ball i R. It is trivial to check that for ay positive p 1,...,p K with harmoic mea s>1, M maps L p 1 L p K ito L s. If we deote by f the Hardy-Littlewood maximal fuctio of f, the M(f) is domiated by the product C θk ((f p 1/s 1 ) ) s/p 1...((f p K/s 1 ) ) s/p K ad hece its boudedess follows from Hölder s iequality ad the L s boudedess of the Hardy-Littlewood maximal fuctio. This argumet breaks dow whe s = 1 but a slight modificatio of it gives that M maps ito weak L 1 i this edpoit case. It is coceivable however that M map ito L 1 sice it carries K-tuples of compactly supported fuctios ito compactly supported fuctios. This problem remais uresolved. The L p L q L r boudedess of the biliear Hilbert trasform H(f,g) is more subtle ad it remais uresolved eve i the case r>1. I this ote, we study the easier problem of the multiliear fractioal itegrals. Our first result cocers the L p 1 L p K L r boudedess of I α for r 1. Theorem 1. Let s be the harmoic mea of p 1,...,p K > 1 ad let r be such that the idetity 1/r + α/ =1/s holds. The I α maps L p 1 L p K ito L r for /( + α) s < /α (equivaletly 1 r< ). Note that i the case K = 1, the correspodig rage of s is the smaller iterval 1 <s</α(equivaletly /( α) <r< ). Whe K = 1, the followig theorem has bee proved by Hirschma [HI] for periodic fuctios ad by Hedberg [HE] for positive fuctios. Theorem 2. Let p j be positive real umbers ad let s>1 be their harmoic mea. The for q, r > 1 ad 0 <θ<1, the followig iequality is true: I αθ (f) L r C I α (f) θ L q k f k 1 θ L p k where 1 r = θ q + 1 θ. s I the edpoit case s = /α, Trudiger [T] for α = 1, ad Strichartz [STR] for other α proved expoetial itegrability of I α whe K = 1. Hedberg [HE] gave a simpler proof of theorem 3 below whe K =1. By ω 1 we deote the area of the uit sphere S 1. The factor L i the expoet below is a ormalizig factor ad should be there by homogeeity. Theorem 3. Let s = /α be the harmoic mea of p 1,...,p K > 1. Let B beaballof radius R i R ad let f j L p j (B) be supported i the ball B. The for ay γ<1, there exists a costat C 0 (γ) depedig oly o, oα, o the θ j s ad o γ, such that the followig iequality is true: (1.1) e LIα(f ω γ 1,...,f K ) /( α) 1 f 1 L p 1... f K L p K dx C 0 (γ)r where L = θ k /p k. B k 2

3 All the commets i this paragraph refer to the case K = 1. [HMT] (for α = 1) ad later Adams [A] (for all α) showed that iequality (1.1) caot hold if γ>1. Moser [M] showed expoetial itegrability of ω 1/ 1 ( ) / 1 1 φ(x) / φ L suggestig that theorem 3 be true i the edpoit case γ = 1. (Use formula (18) page 125 i [ST] to show that Moser s result follows from a improved theorem 3 with γ = 1.) I fact, Adams [A] proved iequality (1.1) i the edpoit case γ = 1 ad also deduced the sharp costats for Moser s expoetial iequality for higher order derivatives. Chag ad Marshall [CM] proved a similar sharp expoetial iequality cocerig the Dirichlet itegral. 2. Proof of theorem 1. We deote by B the measure of the set B ad by χ A the characteristic fuctio of the set A. We also use the otatio s = s/(s 1) for s 1. We cosider first the case s 1. We will show that I α maps L p 1 L p K L r,. The required result whe s>1 is goig to follow from a applicatio of the Marcikiewitz iterpolatio theorem. Without loss of geerality we ca assume that f j 0 ad that f j Lpj = 1. Fix a λ>0 ad defie µ>0byl 1 ω ( 1 (α )s + )1/s µ /r = λ 2 where ω 1 ad L are as i theorem 3. Hölder s iequality ad our choice of µ give that Iα (f)(x) = f 1 (x θ 1 y)...f K (x θ K y) y α dy y >µ (2.1) f k (x θ k y) L s (y) y α χ y >µ L s f k (x θ k y) L p k (y) ( ω 1 (α )s + ) 1/s µ (α/ 1+1/s ) = λ/2 Let I 0 α(f)(x) = y µ f 1(x θ 1 y)...f K (x θ K y) y α dy. We compute its L s orm: (2.2) I 0 α(f) L s ( ) ) 1/s ( 1/s s y ( α fk χ y µ dy 1 s y α χ y µ dy) ) 1/s ( ( ) s y Cµ α/s α fk χ y µ dxdy ( Cµ α/s fk s L p k y α dy ) 1/s = Cµ α/s µ α/s = Cµ α y µ L s By (2.1) the set {x : Iα (f)(x) >λ/2} is empty. This fact together with Chebychev s iequality ad (2.2) gives us the followig iequality: {x : I α (f)(x) >λ} {x : I 0 α(f)(x) >λ/2} 2 s λ s I 0 α s L s Cλ s µ sα = C θk λ r which is the required weak type estimate for I α. 3

4 We ow do the case /( + α) s 1. The correspodig rage of r s is 1 r /( α). Assume that K = 2 ad that p 1 p 2 > 1. Also assume that r = 1 first. Sice s < /α we must have that p 2 < /α. We get I α (f 1,f 2 ) L 1 = f 1 (x θ 1 y)f 2 (x θ 2 y) y α dxdy = f 1 (x) f 2 (x (θ 2 θ 1 )y) y α dy dx (2.3) = θ 2 θ 1 α f 1 (x)i α (f 2 )(x)dx C θ1,θ 2 f 1 L p 1 I α (f 2 ) p. L 1 Sice 1 <p 2 < /α, we ca apply theorem 1, Ch. V o fractioal itegrals i [ST] to boud (2.3) by C θ2,θ 1 f 1 L p 1 f 2 L p 2. The case for a geeral r>1follows by iterpolatig betwee the edpoit case r = 1 ad the case r close to. Suppose ow that the theorem is true for K 1, K 3. We will show that it true for K. Agai we first do the case r = 1. We may assume without loss of geerality that p 1 p K > 1. I α (f) L 1 = f 1 (x θ 1 y)...f K (x θ K y) y α dxdy = f 1 (x) f 2 (x (θ 2 θ 1 )y)...f K (x (θ K θ 1 )y) y α dy dx = (2.4) θ k θ 1 α f 1 (x)i α (f 2,...,f K )(x)dx C θk f 1 L p 1 I α (f 2,...,f K ) p L 1 k 1 Defie s 1 by 1/s 1 =1/s 1/p 1. Sice r = 1, we have that 1/p 1 + α/ =1/s 1. We ca apply the iductio hypothesis oly if we have that /( + α) s 1 < /α. This iequality follows from the idetity 1 + α/ =1/s which relates s ad r = 1. From our iductio hypothesis we get that (2.4) is bouded by C θj fk L p k The case r 1 follows by iterpolatio. 3. Proof of theorem 2. As i the proof of theorem 1, fix f j 0 such that f j L p j = 1. Like [HE], split I αθ (f)(x) = fk (x θ k y) y αθ dy + fk (x θ k y) y αθ dy m=1 y <δ y δ fk (x θ k y) y αθ dy + fk (x θ k y) y α y (θ 1)α dy y δ2 m y δ (δ2 m ) αθ fk (x θ k y) y dy + δ (θ 1)α fk (x θ k y) y α dy m=1 y δ2 m 4 y δ

5 Cδ α(θ ɛ) M(f)(x)+δ α(θ 1) I α (f)(x). Now choose δ =(I α (f)(x)/m (f)(x)) 1/α to get I αθ (f)(x) C(I α (f)(x)) θ (M(f)(x)) 1 θ. Hölder s iequality with expoets 1/r = 1/( s 1 θ )+1/( q θ ) will give I αθ(f) L r C Iα(f) θ L q/θ M(f) L s/(1 θ) = C I α (f) θ L q M(f) 1 θ L C I s α (f) θ Lq by the boudedess of the maximal fuctio M o L q. This cocludes the proof of theorem Proof of theorem 3. A simple dilatio argumet shows that if we kow theorem 3 for a specific value of R = R 0 with a costat C 0(γ) o the right had side of (1.1), the we also kow it for all other values of R with costat C 0(γ)(R/R 0 ). We select R 0 =1/P where P = 2 mi θ k 1 ad we will assume that the radius of B is R 0. Furthermore, we ca assume that the f j s satisfy f j 0 ad f j L p j =1. Nowfixx B. The same argumet as i theorem 2 with θ = 1 gives that (4.1) I α (f)(x) Cδ α M(f)(x)+ fk (x θ k y) y α dy. y δ Sice all f k are supported i the ball B ad x B the itegral i (4.1) is over the set {y : δ y PR 0 =1}. Hölder s iequality with expoets p 1,...,p K ad α gives fk (x θ k y) y α dy (4.2) δ y 1 fk (x θ k y) L pk (y) ( δ y 1 y dy) ( α)/ = L 1( ω 1 l 1 δ ) ( α)/. Combiig (4.1) ad (4.2) we get: (4.3) I α (f)(x) Cδ α M(f)(x)+L 1 ( ω 1 l ( 1 δ ) ) ( α)/. The choice δ = 1 gives I α (f)(x) CM(f)(x) for all x B ad therefore the selectio δ = δ(x) =ɛ ( I α (f)(x)(cm(f)(x)) 1) 1/α will satisfy δ 1 for all ɛ 1. (4.3) ow implies I α (f)(x) ɛ α I α (f)(x)+l 1 ( ω 1 l ( (CM(f)(x)) /α ɛ I α (f)(x) /α ) ) ( α)/. Algebraic maipulatio of the above gives: (4.4) ω 1 γ ( LI α (f)(x) ) /( α) ( l (CM(f)(x)) /α ) ɛ I α (f)(x) /α 5

6 where we set γ =(1 ɛ α ) /( α). We expoetiate (4.4) ad we itegrate over the set B 1 = {x B : I α (f)(x) 1} to obtai B 1 e ( /( α) ω γ LI α 1 (f)(x)) dx 1 ɛ (CM(f)(x)) B1 /α I α (f)(x) dx C 1 /α ɛ M(f)(x) /α dx C 2 ɛ. The last iequality follows from the boudedess of the maximal fuctio of f o L /α. The itegral of the same expoetial over the set B 2 = B B 1 is estimated trivially by ( /( α) ω γ LI α 1 (f)(x)) dx e ω 1 L/( α) B 2 C 3 Ω R0 = C 4. B 2 e Addig the itegrals above over B 1 ad B 2 we obtai the required iequality with a costat C 0(γ) = max(c 2,C 4 )(1 +(1 γ ( α)/ ) /α ). The costat C 0 (γ) i the statemet of theorem 3 is the C 0(γ)R 0 = C 0(γ)P. We obtai the followig Corollary. Let B, f k, p k, ad s as i theorem 3. The I α (f 1,...,f K ) is i L q (B) for every q>0. I fact the followig iequality is true: I α (f 1,...,f K ) L q (B) C k f k L p k for some costat C depedig oly o q o o α ad o the θ j s. The corollary follows sice expoetial itegrability of I α implies itegrability to ay power q. (Here γ is fixed < 1.) 5. A multiliear differetiatio theorem. We ed this ote by provig the followig multiliear Lebesgue differetiatio theorem. Let f j L p j (R ) ad suppose that their harmoic mea is s 1. The lim T 1 ɛ(f)(x) = lim ɛ 0 ɛ 0 Ω ɛ f 1 (x θ 1 y)...f K (x θ K y)dy = f 1 (x)...f K (x) a.e. y ɛ The case s = 1 is a cosequece of the weak type iequality {x R : M(f)(x) > λ} C λ f 1 L p 1... f 1 L p K which is easily obtaied from {x R : M(f)(x) >λ} K j=1 {x R :(f j ) (x) > (ɛ j 1 /ɛ j ) p j } C K j=1 (ɛ j 1/ɛ j ) p j f j L p j after miimizig over all ɛ 1,...,ɛ K > 0. (Take ɛ 0 = λ.) The stadard argumet preseted i [SWE], page 61, will prove that the sequece {T ɛ (f)(x)} ɛ>0 is Cauchy for almost all x ad therefore it coverges. Sice for cotiuous f 1,...,f K it coverges to the value of their product at the 6

7 poit x R, to deduce the geeral case it will suffice to show that {T ɛ (f)} ɛ>0 coverges to the product of the f j s i the L s orm as ɛ 0. (The some subsequece will coverge to the product a.e.) Deotig by (τ y f)(x) =f(x y) the traslatio of f by y, we get T ɛ (f) f 1...f K L s 1 Ω ɛ 1 Ω ɛ y ɛ j=1 y ɛ j τ θj yf j j f j L sdy K τ θj yf j f j L p j f j L p k dy 0 as y 0 sice the last itegrad is a cotious fuctio of y which vaishes at the origi. The last iequality above follows by addig ad subtractig 2K 2 suitable terms ad applyig Hölder s iequality K times. k j refereces [A] D. Adams, A sharp iequality of J. Moser for higher order derivatives, Aals of Math. 128 (1988), [CM] S.-Y. A. Chag ad D. E. Marshall, O a sharp iequality cocerig the Dirichlet itegral, Amer. J. Math. 107 (1985), [CG] R. R. Coifma ad L. Grafakos, Hardy space estimates for multiliear operators I, to appear i Revista Matematica Iberoamericaa. [G] L. Grafakos, Hardy space estimates for multiliear operators II, to appear i Revista Matematica Iberoamericaa. [HI] I. I. Hirschma Jr., A covexity theorem for certai groups of trasformatios, J. Aalyse Math. 2 (1953), [HE] L. I. Hedberg, O certai Covolutio Iequalities, Proceedigs of the AMS, 36 (1972), [HMT] J. A. Hempel, G. R. Morris ad N. S. Trudiger, O the sharpess of a limitig case of the Sobolev embeddig theorem, Bull. Austr. Math. Soc. 3 (1970), [M] J. Moser, A sharp form of a iequality by N. Trudiger, Idiaa Uiv. Math. J. 20 (1971), [ST] E. M. Stei, Sigular Itegrals ad Differetiablity Properties of Fuctios, Priceto Uiv. Press, Priceto, NJ [SWE] E. M. Stei ad G. Weiss, A itroductio to Fourier Aalysis o Euclidea spaces, Priceto Uiv. Press, Priceto, NJ [STR] R. S. Strichartz, A ote o Trudiger s extesio of Sobolev s iequalities, Idiaa Uiv. Math. J. 21 (1972), [T] N. S. Trudiger, O imbeddigs ito Orlicz spaces ad some applicatios, J. Math. Mech. 17 (1967),

8 Departmet of Mathematics, Yale Uiversity, Box 2155 Yale Statio, New Have, CT