Notes on the spaces of bilinear multipliers

NOTES ON THE SPACES OF BILINEAR MULTIPLIERS

arXiv:0905.4151v1 [math.CA] 26 May 2009

OSCAR BLASCO

Abstract. A locally integrable function m(ξ, η) defined on Rn × Rn is said to be a bilinear multiplier on Rn of type (p1 , p2 , p3 ) if Z Z fˆ(ξ)ˆ g (η)m(ξ, η)e2πi(hξ+η,xi dξdη Bm (f, g)(x) = Rn

Rn

defines a bounded bilinear operator from Lp1 (Rn ) × Lp2 (Rn ) to Lp3 (Rn ). The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where m(ξ, η) = M (ξ − η) for a given M defined on Rn is also addressed.

1. Introduction. Throughout the paper C00 (Rn ) denotes the space of continuous functions defined in Rn with compact support, S(Rn ) denotes the Schwartz class on Rn , i.e. f : Rn → C such that f ∈ C ∞ (Rn ) and xα α

∂ |β| f (x) β n ∂x1 1 ...∂xβ n αn 1 xα ...x n 1 n

is bounded for any β = (β1 , ..., βn )

and α = (α1 , ..., αn ) where x = and |β| = β1 + ... + βn and P(Rn ) the set of functions in S(R ) such that fˆ ∈ C00 (Rn ) where fˆ(ξ) = Rstands for−2πihx,ξi f (x)e dx. Rn ˜ p,q (Rn )), for 1 ≤ p, q ≤ ∞, for We shall use the notation Mp,q (Rn ) (respect. M ′ n the space of distributions u ∈ S (R ) such that u ∗ φ ∈ Lq (Rn ) for all φ ∈ Lp (Rn ) (respect. for the space of bounded functions m such that Tm defines a bounded ˆ operator from Lp (Rn ) to Lq (Rn ) where T\ m (φ)(ξ) = m(ξ)f (ξ).) We endow the n ˜ space Mp,q (R ) with the “norm” of the operator Tm , that is kmkp,q = kTm k. Let us start off by mentioning some well known properties of the space of linear multipliers (see [1, 14]): Mp,q (Rn ) = {0} whenever q < p, Mp,q (Rn ) = Mq′ ,p′ (Rn ) for 1 < p ≤ q < ∞ and for 1 ≤ p ≤ 2, M1,1 (Rn ) ⊂ Mp,p (Rn ) ⊂ M2,2 (Rn ). We also have the identifications ˜ 2,2 (Rn ) = L∞ (Rn ), M M1,q (Rn ) = {u ∈ S ′ (Rn ) : u ∈ Lq (Rn )}, 1 < q < ∞, M1,1 (Rn ) = {u ∈ S ′ (Rn ) : u = µ ∈ M (Rn )}. In this paper we shall be dealing with their bilinear analogues. Definition 1.1. Let 1 ≤ p1 , p2 ≤ ∞ and 0 < p3 ≤ ∞ and let m(ξ, η) be a locally integrable function on Rn × Rn . Define Z Z Bm (f, g)(x) = fˆ(ξ)ˆ g (η)m(ξ, η)e2πi(hξ+η,xi dξdη Rn

Rn

Key words and phrases. spaces of bilinear multipliers, bilinear Hilbert transform, bilinear fractional transform. Partially supported by Proyecto MTM2008-04594/MTM. 1

2

OSCAR BLASCO

for f, g ∈ P(Rn ). m is said to be a bilinear multiplier on Rn of type (p1 , p2 , p3 ) if there exists C > 0 such that kBm (f, g)kp3 ≤ Ckf kp1 kgkp2 for any f, g ∈ P(Rn ), i.e. Bm extends to a bounded bilinear operator from Lp1 (Rn )× Lp2 (Rn ) to Lp3 (Rn ) (where we replace L∞ (Rn ) for C0 (Rn ) in the case pi = ∞ for i = 1, 2). We write BM(p1 ,p2 ,p3 ) (Rn ) for the space of bilinear multipliers of type (p1 , p2 , p3 ) and kmkp1 ,p2 ,p3 = kBm k. The study of bilinear multipliers for smooth symbols (where m(ξ, η) is a “nice” regular function) goes back to the work by R.R. Coifman and Y. Meyer in [6]. Particularly simple examples are the following bilinear convolution-type operators: For a given K ∈ L1loc (Rn ) we define Z (1) CK (f, g)(x) = f (x − y)g(x + y)K(y)dy R

for f and g belonging to C00 (Rn ). ˆ − η) defines a multiplier in BM(p ,p ,p ) (Rn ) If K ∈ L1 (Rn ) then m(ξ, η) = K(ξ 1 2 3 for 1/p1 + 1/p2 = 1/p3 if p3 ≥ 1 and kmkp1 ,p2 ,p3 ≤ kKk1. R Indeed, for f and g ∈ S(R), one has f (x − y) = Rn fˆ(ξ)e2πihx−y,ξi dξ and R g(x + y) = Rn gˆ(η)e2πihx+y,ηi dη. Hence we have Z f (x − y)g(x + y)K(y)dy CK (f, g)(x) = n ZR Z Z = fˆ(ξ)ˆ g (η)K(y)e2πihx−y,ξi e2πihx+y,ηi dξdηdy Rn Rn Rn Z Z Z = K(y)e−2πihξ−η,yi dy)e2πihξ+η,xi dξdη fˆ(ξ)ˆ g (η)( Rn Rn Rn Z Z ˆ − η)e2πihξ+η,xi dξdη. = gˆ(η)fˆ(ξ)K(ξ Rn

Rn

This motivates the introduction of the following class of multipliers. ˜ (p ,p ,p ) (Rn ) Definition 1.2. Let 1 ≤ p1 , p2 ≤ ∞ and 0 < p3 ≤ ∞. We denote by M 1 2 3 n the space of measurable functions M : R → C such that m(ξ, η) = M (ξ − η) ∈ BM(p1 ,p2 ,p3 ) (R), that is to say Z Z BM (f, g)(x) = fˆ(ξ)ˆ g (η)M (ξ − η)e2πihξ+η,xi dξdη Rn

Rn

extends to a bounded bilinear map from Lp1 (Rn ) × Lp2 (Rn ) into Lp3 (Rn ). We keep the notation kM kp1 ,p2 ,p3 = kBM k. 1 It was only in the last decade that the cases M0 (x) = |x|1−α were shown to define bilinear multipliers of type (p1 , p2 , p3 ) for 1/p3 = 1/p1 + 1/p2 − α for 1 < p1 , p2 < ∞ and 0 < α < 1/p1 + 1/p2 (see (3) in Theorem 1.3) and, in the case n = 1, M1 (x) = −isign(x) was shown to define a bilinear multiplier of type (p1 , p2 , p3 ) for 1/p3 = 1/p1 + 1/p2 for 1 < p1 , p2 < ∞ and p3 > 2/3 (see (2) in Theorem 1.3). These two main examples correspond to the following bilinear operators: the bilinear fractional integral defined by Z f (x − y)g(x + y) dy, 0 < α < 1 Iα (f, g)(x) = |y|1−α R


and the bilinear Hilbert transform defined by Z f (x − y)g(x + y) 1 dy H(f, g)(x) = lim ε→0 π |y|>ε y respectively. Let us collect the results about their boundedness which are known nowadays. Theorem 1.3. Let 1 < p1 , p2 < ∞, 0 < α < 1/p1 + 1/p2 , 1/q = 1/p1 + 1/p2 − α, 1/p3 = 1/p1 + 1/p2 and 2/3 < p3 < ∞. Then there exist constants A and B such that (2)

kH(f, g)kp3 ≤ Akf kp1 kgkp2 (Lacey-Thiele, [12, 13]),

(3)

kIα (f, g)kq ≤ Bkf kp1 kgkp2 . (Kenig-Stein [11], Grafakos-Kalton [10] ).

Our objective is to study the basic properties of the classes BM(p1 ,p2 ,p3 ) (R) and Mp1 ,p2 ,p3 (R), to find examples of bilinear multipliers in these classes and to get methods to produce new ones. As usual, if f ∈ L1 (Rn ) we denote by τx , Mx and Dtp the translation τx f (y) = f (y − x) for x ∈ Rn , the modulation Mx f (y) = e2πihx,yi f (y) and the dilation Dtp f (x) = t−n/p f ( xt ) for 0 < p, t < ∞. With this notation out of the way one has, for 1 ≤ p ≤ ∞ and 1/p + 1/p′ = 1, (4)

[ ˆ (τ x f )(ξ) = M−x f (ξ),

\ ˆ (M x f )(ξ) = τx f (ξ),

p p′ ˆ \ (D t f )(ξ) = Dt−1 f (ξ).

Clearly τx , Mx and Dtp are isometries on Lp (Rn ) for any 0 < p ≤ ∞. Although most of the results presented in what follows have a formulation in n ≥ 1 we shall restrict ourselves to the case n = 1 for simplicity. The reader is referred to [2, 3, 4, 5, 7] for several similar results on other groups, and to find same methods of transference.

2. Bilinear multipliers: The basics Let us start by pointing out a characterization, for p3 ≥ 1, in terms of the duality, whose elementary proof is left to the reader. Proposition 2.1. Let 1 ≤ p3 ≤ ∞. Then m ∈ BM(p1 ,p2 ,p3 ) (R) if and only if there exists C > 0 such that Z ˆ + η)m(ξ, η)dξdη| ≤ Ckf kp kgkp khkp′ fˆ(ξ)ˆ g (η)h(ξ | 2 1 3 R2

for all f, g, h ∈ P(R). We now present a basic example R of a bilinear multiplier. For a Borel regular measure in R µ we denote µ ˆ (ξ) = R e−2πixξ dµ(x) its Fourier transform.

ˆ(αξ + βη) Proposition 2.2. Let p3 ≥ 1 and 1/p1 + 1/p2 = 1/p3 and let m(ξ, η) = µ where µ is a Borel regular measure in R and (α, β) ∈ R2 . Then m ∈ BM(p1 ,p2 ,p3 ) (R) and kmkp1 ,p2 ,p3 ≤ kµk1 .

3

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OSCAR BLASCO

Proof. Let us first rewrite the value Bm (f, g) as follows: Z Bm (f, g)(x) = fˆ(ξ)ˆ g (η)ˆ µ(αξ + βη)e2πi(ξ+η)x dξdη R2 Z Z ˆ = f (ξ)ˆ g (η)( e−2πi(αξ+βη)t dµ(t))e2πi(ξ+η)x dξdη R2 R Z Z fˆ(ξ)ˆ g (η)e2πi(x−αt)ξ e2πi(x−βt)η dξdη)dµ(t) ( = 2 ZR R f (x − αt)g(x − βt)dµ(t). = R

Hence, using Minkowski’s inequality, one has Z kf (· − αt)g(· − βt)kp3 d|µ|(t) kBm (f, g)kp3 ≤ R Z kf (· − αt)kp1 kg(· − βt)kp2 d|µ|(t) ≤ R Z d|µ|(t) = kµk1 kf kp1 kgkp2 . = kf kp1 kgkp2 R

Let us start with some elementary properties of the bilinear multipliers when composing with translations, modulations and dilations. Proposition 2.3. Let m ∈ BM(p1 ,p2 ,p3 ) (R). ˜ s2 ,p2 (R) then m1 (ξ)m(ξ, η)m2 (η) ∈ BM(s ,s ,p ) (R). ˜ s1 ,p1 (R) and m2 ∈ M (a) If m1 ∈ M 1 2 3 Moreover km1 mm2 ks1 ,s2 ,p3 ≤ km1 ks1 ,p1 kmkp1 ,p2 ,p3 km2 ks2 ,p2 (b) τ(ξ0 ,η0 ) m ∈ BM(p1 ,p2 ,p3 ) (R) for each (ξ0 , η0 ) ∈ R2 and kτ(ξ0 ,η0 ) mkp1 ,p2 ,p3 = kmkp1 ,p2 ,p3 . (c) M(ξ0 ,η0 ) m ∈ BM(p1 ,p2 ,p3 ) (R) for each (ξ0 , η0 ) ∈ R2 and kM(ξ0 ,η0 ) mkp1 ,p2 ,p3 = kmkp1 ,p2 ,p3 (d) If

2 q

=

1 p1

+

1 p2

−

1 p3

and 0 < t < ∞ then Dtq m ∈ BM(p1 ,p2 ,p3 ) (R) and kDtq mkp1 ,p2 ,p3 = kmkp1 ,p2 ,p3 .

Proof. Use (4) to deduce the following formulas (5)

Bm1 mm2 (f, g) = Bm (Tm1 f, Tm2 g).

(6)

Bτ(ξ0 ,η0 ) m (f, g) = Mξ0 +η0 Bm (M−ξ0 f, M−η0 g).

(7)

BM(ξ0 ,η0 ) m (f, g) = Bm (τ−ξ0 f, τ−η0 g).

(8)

Bm (Dtp1 f, Dtp2 g) = Dtp3 BDtq m (f, g).


5

Let us check only the validity of last one. The other ones follow easily from the previous facts. Z 1 1 ′ ′ p1 p2 t p1 fˆ(tξ)t p2 gˆ(tη)m(ξ, η)e2πi(ξ+η)x dξdη Bm (Dt f, Dt g)(x) = 2 ZR 1 1 x ξ η ′ ′ t p1 fˆ(ξ)t p2 gˆ(η)m( , )e2πi(ξ+η) t t−2 dξdη = t t 2 R Z x ξ η − 1 − 1 + 1 − p1 fˆ(ξ)ˆ g (η)t p1 p2 p3 m( , )e2πi(ξ+η) t dξdη = t 3 t t R2 = Dtp3 BDtq m (f, g)(x). From (8) we can see that the condition 1/p1 + 1/p2 = 1/p3 is also connected to the homogeneity of the symbol. Proposition 2.4. Let m ∈ BM(p1 ,p2 ,p3 ) (R) such that m(tξ, tη) = m(ξ, η) for any t > 0. Then p11 + p12 = p13 . Proof. From assumption Dt∞ m = m. Using (8) we have Bm (Dtp1 f, Dtp2 g) = t1/p3 −(1/p1 +1/p2 ) Dtp3 Bm (f, g) and therefore kBm (f, g)kp3

= kDtp3 Bm (f, g)kp3 = t−1/p3 +(1/p1 +1/p2 ) kBm (Dp1 f, Dtp2 g)kp3 ≤ t−1/p3 +(1/p1 +1/p2 ) kBm kkf kp1 kgkp2 .

For this to hold for any 0 < t < ∞ one needs 1/p1 + 1/p2 = 1/p3 .

Let us combine the previous results to get new bilinear multipliers from a given one. Proposition 2.5. Let p3 ≥ 1 and m ∈ BM(p1 ,p2 ,p3 ) (R). (a) If Q = [a, b] × [c, d] and 1 < p1 , p2 < ∞ then mχQ ∈ BM(p1 ,p2 ,p3 ) (R) and kmχQ kp1 ,p2 ,p3 ≤ Ckmkp1 ,p2 ,p3 . (b) If Φ ∈ L1 (R2 ) then Φ∗m ∈ BM(p1 ,p2 ,p3 ) (R) and kΦ∗mkp1 ,p2 ,p3 ≤ kΦk1 kmkp1 ,p2 ,p3 . ˆ ∈ BM(p ,p ,p ) (R) and kΦmk ˆ p1 ,p2 ,p3 ≤ kΦk1 kmkp1 ,p2 ,p3 . (c) If Φ ∈ L1 (R2 ) then Φm 1 2 3 R∞ 1 1 1 1 + p3 −( p1 + p2 (d) If ψ ∈ L (R , t ) then mψ (ξ, η) = 0 m(tξ, tη)ψ(t)dt ∈ BM(p1 ,p2 ,p3 ) (R). Moreover kmψ kp1 ,p2 ,p3 ≤ kψk1 kmkp1 ,p2 ,p3 . ˜ p2 ,p2 for ˜ p1 ,p1 for 1 < p1 < ∞ and χ[c,d] ∈ M Proof. (a) Use that χ[a,b] ∈ M 1 < p2 < ∞ together with Proposition 2.3 part (a). (b) Note that Z Z ˆ m(ξ − u, η − v)Φ(u, v)dudv)e2πi(ξ+η)x dξdη f (ξ)ˆ g (η)( BΦ∗m (f, g)(x) = 2 2 R ZR Z = fˆ(ξ)ˆ g (η)m(ξ − u, η − v)e2πi(ξ+η)x dξdη Φ(u, v)dudv 2 R2 ZR = Bτ(u,v) m (f, g)(x)Φ(u, v)dudv. R2

6

OSCAR BLASCO

From the vector-valued Minkowski inequality and Proposition 2.3 part (b), we have Z kBτ(u,v) m (f, g)kp3 |Φ(u, v)|dudv kBΦ∗m (f, g)kp3 ≤ R2

≤ (c) Observe that BΦm ˆ (f, g)(x)

=

Z

R2

=

Z

R2

kmkp1 ,p2 ,p3 kf kp1 kgkp2 kΦk1 .

Z fˆ(ξ)ˆ g (η)(

R2

M(−u,−v) m(ξ, η)Φ(u, v)dudv)e2πi(ξ+η)x dξdη

BM(−u,−v) m (f, g)(x)Φ(u, v)dudv.

Argue as above, using now Proposition 2.3 part (c), to conclude the result. (d) Use now Proposition 2.3 part (d), for p13 − ( p11 + p12 ) = − q2 , Z Z ∞ ˆ Bmψ (f, g)(x) = f (ξ)ˆ g (η)( Dtq−1 m(ξ, η)t−2/q ψ(t)dt)e2πi(ξ+η)x dξdη 2 0 ZR∞ BDq−1 m (f, g)(x)t−2/q ψ(t)dt. = 0

t

With all these procedures we have several useful methods to produce multipliers in BM(p1 ,p2 ,p3 ) (R). Let us mention one application of each of them. ˜ (p ,p ) and m2 ∈ M ˜ (p ,p ) then Example 2.6. (1) If p11 + p12 = p13 , m1 ∈ M 1 1 2 2 m(ξ, η) = m1 (ξ)m2 (η) ∈ BMp1 ,p2 ,p3 . (2) If m ∈ BM(p1 ,p2 ,p3 ) (R), p3 ≥ 1 and Q1 , Q2 are bounded measurable sets in R then Z 1 m(ξ + u, η + v)dudv ∈ BM(p1 ,p2 ,p3 ) (R). |Q1 ||Q2 | Q1 ×Q2 ˆ ∈ BM(p ,p ,p ) (R) for 1 + 1 = 1 , p3 ≥ 1. (3) If Φ ∈ L1 (R2 ) then Φ 1 2 3 p1 p2 p3 (4) If m ∈ BM(p1 ,p2 ,p3 ) (R), | p11 + p12 − p13 | < 1 then Z ∞ dt m1 (ξ, η) = m(tξ, tη) ∈ BM(p1 ,p2 ,p3 ) (R). 1 + t2 0 A combination of the previous results gives the following examples of bilinear multipliers in BM (1,1,p3 ) (R) whose proof is left to the reader. Corollary 2.7. Let Φ ∈ L1 (R2 ), ψ1 ∈ Lp1 (R) and ψ2 ∈ Lp2 (R) and 1 p3 ≤ 1 then ˆ η)ψˆ2 (η) ∈ BM (1,1,p ) (R). m(ξ, η) = ψˆ1 (ξ)Φ(ξ,

1 p1

+

1 p2

=

3

Let us use Proposition 2.1 and interpolation to get a sufficient integrability condition to guarantee that m ∈ BM(p1 ,p2 ,p3 ) (R). Theorem 2.8. Let 1 ≤ p1 , p2 ≤ p ≤ 2 and p3 ≥ p′ such that m ∈ Lp (R2 ) then m ∈ BM(p1 ,p2 ,p3 ) (R).

1 p1

+

1 p2

−

2 p

=

1 p3 .

If


7

Proof. Let us show first that m ∈ BM(p,p,∞) (R). Let f, g ∈ Lp (R) and h ∈ L1 (R). Using H¨ older and Hausdorff-Young’s inequalities one gets Z ˆ + η)m(ξ, η)dξdη| ≤ kmkLp(R2 ) khk ˆ ∞ kfˆkp′ kˆ | fˆ(ξ)ˆ g (η)h(ξ gkp′ R2

≤

kmkLp(R2 ) khk1 kf kp kgkp.

Similarly, changing the variables ξ + η = u, ξ = −v, one has Z Z ˆ + η)m(ξ, η)dξdη = ˆ fˆ(ξ)ˆ g (η)h(ξ fˆ(−v)ˆ g(u + v)h(u)m(−v, u + v)dvdu. R2

R2

An argument as above gives also the estimate Z ˆ | fˆ(−v)ˆ g (u + v)h(u)m(−v, u + v)dvdu| ≤ kmkLp(R2 ) kgk1 kf kp khkp . R2

This shows that m ∈ BM(p,1,p′ ) (R). A similar argument shows also that m ∈ BM(1,p,p′ ) (R). Given 1 ≤ p˜1 ≤ p and p′ ≤ p˜3 ≤ ∞ with p˜11 − p˜13 = p1 we have 0 ≤ θ ≤ 1 such that 1 θ 1 θ 1−θ 1−θ p˜1 = p + 1 and p˜3 = ∞ + p′ . Hence, by interpolation, m ∈ BM(p˜1 ,p,p˜3 ) (R). Similarly m ∈ BM(p,p˜2 ,˜q3 ) (R) whenever 1 ≤ p˜2 ≤ p and p′ ≤ q˜3 ≤ ∞ with 1 1 1 p˜2 − q˜3 = p . To finish the proof we observe that if 1 < p1 < p and 1 < p2 < p then for each 0 < θ < 1 there exist 1 ≤ p˜1 ≤ p1 < p and 1 ≤ p˜2 ≤ p2 < p such that 1 1 1 1 − = (1 − θ)( − ), p1 p p˜1 p Denoting p˜3 , q˜3 the values such that (1 − θ) θ 1 = + , p1 p˜1 p

1 p˜2

−

1 p

1 1 1 1 − = θ( − ). p2 p p˜2 p =

1 p˜3

and

(1 − θ) 1 θ = + , p2 p p˜1

1 p˜2

− p1 =

1 q˜3

one obtains that

1 (1 − θ) θ = + . p3 p˜3 q˜3

Hence the result follows again from interpolation between the last ones.

3. Bilinear multipliers defined by functions in one variable Let us restrict ourselves to a smaller family of multipliers where m(ξ, η) = M (ξ − η) for some M defined in R. These multipliers satisfy (9)

Bm (Mx f, Mx g) = M2x Bm (f, g).

˜ p1 ,p2 ,p3 (R) for the space of functions As in the introduction we use the notation M M : R → C such that m(ξ, η) = M (ξ − η) ∈ BM(p1 ,p2 ,p3 ) (R), that is to say Z fˆ(ξ)ˆ g (η)M (ξ − η)e2πi(ξ+η)x dξdη, BM (f, g)(x) = R2

defined for fˆ and gˆ compactly supported, extends to a bounded bilinear map from Lp1 (R) × Lp2 (R) into Lp3 (R). We keep the notation kM kp1 ,p2 ,p3 = kBM k. The reader should be aware that the starting assumption on the function M is only relevant for the definition of the bilinear mapping to make sense when acting on certain classes of “nice” functions. Then a density argument allows to extend functions belonging to Lebesgue spaces. We would like to point out the following observation.

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OSCAR BLASCO

˜ (p ,p ,p ) (R) are functions such that Mn (x) → M (x) a.e Remark 3.1. If Mn ∈ M 1 2 3 ˜ (p ,p ,p ) (R) and kM kp1 ,p2 ,p3 ≤ supn kMn kp1 ,p2 ,p3 . and supn kMn k < ∞ then M ∈ M 1 2 3 Indeed, this fact follows from Fatou’s lemma, since kBM (f, g)kp3 ≤ lim inf kBMn (f, g)kp3 ≤ sup kMn kp1 ,p2 ,p3 kf kp1 kgkp2 . n

1

Remark 3.2. The case M (x) = |x|1−α (and even the n-dimensional case) corresponds to the bilinear fractional integral. This was first shown by C. Kenig and E. ˜ (p ,p ,p ) (R) for any 1 < p1 , p2 < ∞, 0 < α < 1/p1 +1/p2 Stein in [11] to belong to M 1 2 3 and 1/p1 + 1/p2 = 1/p3 − α. Another very important and non trivial example is the bilinear Hilbert transform, given by M (x) = −isign(x), which was shown by M. ˜ (p ,p ,p ) (R) for any 1 < p1 , p2 < ∞, Lacey and C.Thiele in [12, 13, ?] to belong to M 1 2 3 1/p1 + 1/p2 = 1/p3 and p3 > 2/3. These results were extended to other cases in [10] and [8, 9] respectively. We start reformulating the definition of this class of bilinear multipliers. Proposition 3.3. Let M ∈ L1loc (R), f, g ∈ P(R). Then Z 1 u−v u+v (10) BM (f, g)(x) = )ˆ g( )M (v)e2πiux dudv fˆ( 2 R2 2 2 (11)

(12)

BM (f, g)(−x) = \ BM (f, g)(x) =

Z

R

(d τx g ∗ M )(ξ)τd x f (ξ)dξ.

1 1 1 \ \ CM (D 1/2 f , D1/2 g)(x). 2

Proof. (10) follows changing variables. To show (11) observe that Z BM (f, g)(−x) = τd τx g(η)M (ξ − η)dξdη x f (ξ)d R2 Z Z τx g(η)M (ξ − η)dη)τd ( d = x f (ξ)dξ R R Z (d τx g ∗ M )(ξ)τd = x f (ξ)dξ R

Finally, using (10), we have Z Z 1 u+v u−v BM (f, g)(x) = fˆ( )ˆ g( )M (v)dv e2πiux dv 2 R 2 2 R Z 1 ∞ ∞ ˆ CM (D1/2 f , D1/2 gˆ)(u)e2πiux du. = 2 R

This implies (12).

For symbols M which are integrable we can write BM in terms CK for some kernel K. ˆ (−t). Then BM = CK , i.e Proposition 3.4. Let M ∈ L1 (R) and set K(t) = M Z f (x − t)g(x + t)K(t)dt BM (f, g) = R


Proof.

9

Z

ˆ (−t)dt f (x − t)g(x + t)M ZR Z ˆ (−t)dt fˆ(ξ)ˆ g (η)e2πi(x−t)ξ e2πi(x+t)η dξdη)M ( = R R2 Z Z ˆ ˆ (t)e2πi(ξ−η)t dt)e2πi(ξ+η)x dξdη f (ξ)ˆ g (η)( M =

CK (f, g)(x)

=

R2

R

= BM (f, g)(x). This class does have much richer properties than BM(p1 ,p2 ,p3 ) (R). As above use the notation ft (x) = Dt1 f (x) = 1t f ( xt ) for a function f defined in R. The following facts are immediate. (13)

τy BM (f, g) = BM (τy f, τy g), y ∈ R.

(14)

M2y BM (f, g) = BM (My f, My g), y ∈ R.

(15)

(BM (f, g))t = BD1−1 M (ft , gt ), t > 0. t

When specializing the properties obtained for m(ξ, η) to the case M (ξ − η) we get the following facts: BM (τ−y f, τy g) = BMy M (f, g), y ∈ R.

(16)

BM (My f, M−y g) = Bτ2y M (f, g), y ∈ R.

(17) For (18)

1 q

=

1 p1

+

1 p2

−

1 p3

we have

BM (Dtp1 f, Dtp2 g) = Dtp3 BDtq M (f, g), t > 0.

˜ (p ,p ,p ) (R). As in the previous section we can generate new multipliers in M 1 2 3 ˜ (p ,p ,p ) (R). Then Proposition 3.5. Let p3 ≥ 1, φ ∈ L1 (R) and M ∈ M 1 2 3 ˜ (p ,p ,p ) (R) and kφ ∗ M kp1 ,p2 ,p3 ≤ kφk1 kM kp1 ,p2 ,p3 . (a) φ ∗ M ∈ M 1 2 3 ˆ ∈M ˆ kp ,p ,p ≤ kφk1 kM kp ,p ,p . ˜ (p ,p ,p ) (R) and kφM (b) φM 1 2 3 1 2 3 1 2 3 R∞ 1 1 1 1 + p3 −( p1 + p2 ) ˜ (p ,p ,p ) (R). (c) If ψ ∈ L (R , t ) then Mψ (ξ) = 0 M (tξ)ψ(t)dt ∈ M 1 2 3 Moreover kMψ kp1 ,p2 ,p3 ≤ kψk1 kM kp1 ,p2 ,p3 . Proof. (a) Apply Minkowski’s inequality to the following fact: Z Z ˆ f (ξ)ˆ g (η)( M (ξ − η − u)φ(u)du)e2πi(ξ+η)x dξdη Bφ∗M (f, g)(x) = 2 R ZR Z \ = ( M g (η)M (ξ − η)e2πi(ξ+η)x dξdη)e2πiux φ(u)du −u f (ξ)ˆ R R2 Z = Mu BM (M−u f, g)(x)φ(u)du. R

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OSCAR BLASCO

(b) Observe that Bφm ˆ (f, g)(x)

Z

=

ZR

=

2

R2

Z fˆ(ξ)ˆ g (η)( (M−u m)(ξ − η)φ(u)du)e2πi(ξ+η)x dξdη R

BM−u m (f, g)(x)φ(u)du.

Use now Minkowski’s again and (16). (c) Write p13 − ( p11 + p12 ) = − 1q , Z Z ∞ BMψ (f, g)(x) = fˆ(ξ)ˆ g (η)( Dtq−1 M (ξ)t−1/q ψ(t)dt)e2πi(ξ+η)x dξdη 2 0 ZR∞ BDq−1 M (f, g)(x)t−1/q ψ(t)dt. = 0

t

The result follows from (18) and Minkowski’s again.

˜ (p ,p ,p ) (R). Then m(ξ, η) = Proposition 3.6. Let p3 ≥ 1, φ ∈ L1 (R) and M ∈ M 1 2 3 ˆ + η) ∈ BM(p ,p ,p ) (R) and kmkp1 ,p2 ,p3 ≤ kφk1 kM kp1 ,p2 ,p3 . M (ξ − η)φ(ξ 1 2 3 Proof. Apply Young’s inequality to the following fact: Z Z fˆ(ξ)ˆ g (η)M (ξ − η)( φ(y)e−2πi(ξ+η)y dy)e2πi(ξ+η)x dξdη Bm (f, g)(x) = 2 R ZR Z ˆ f (ξ)ˆ g (η)M (ξ − η)e2πi(ξ+η)(x−y) dξdη)φ(y)dy ( = R

=

R2

φ ∗ BM (f, g)(x).

˜ (p ,p ,p ) (R) are reduced to {0} for some values Let us show that the classes M 1 2 3 of the parameters. Theorem 3.7. Let p3 ≥ 1 such that

1 p1

+

1 p2

0 we get |M M = 0. 1 p1

Corollary 3.10. (see [16, Prop 3.1]) Let p3 ≥ 1 such that 1 ˜ p3 + 1. Then M(p1 ,p2 ,p3 ) (R) = {0}.

+ p12

˜ (p ,p ,p ) (R). From Proposition 3.5 we have that φ ∗ M ∈ Proof. Let M ∈ M 1 2 3 ˜ (p ,p ,p ) (R) for any φ compactly supported and continuous. Now use Theorem M 1 2 3 3.9 to conclude that φ ∗ M = 0 for any compactly supported and continuous φ. This implies that M = 0. Let us now use some interpolation methods to get more examples of multipliers ˜ (p ,p ,p ) (R). First note that, selecting α = 1 and β = −1 in Proposition 2.2 in M 1 2 3 we obtain the following simple example. ˜ (p ,p ,p ) (R) for 1 + 1 = 1 ≤ 1 Proposition 3.11. If µ ∈ M (R) then M = µ ˆ∈M 1

2

3

p1

p2

p3

and kM k ≤ kµk1 . ˆ Theorem 3.12. Let p13 ≤ p11 + p12 ≤ min{2, p13 + 1}. If M ∈ L1 (R) and M = K 1 1 1 1 q ˜ (p ,p ,p ) (R) with for some K ∈ L (R) where p1 + p2 − p3 = 1 − q then M ∈ M 1 2 3 kM kp1 ,p2 ,p3 ≤ CkKkq . Proof. Consider the trilinear form T (K, f, g) =

Z

f (x − t)g(x + t)K(t)dt. R

ˆ Now use PropoFrom Proposition 3.4 we have BM (f, g) = T (K, f, g) for M = K. 1 q1 sition 3.11 to conclude that T is bounded in L (R) × L (R) × Lq2 (R) → Ls1 (R) where q11 + q12 = s11 ≤ 1 and it has norm bounded by 1. Assume first that p11 + p12 ≤ 1. Hence T is bounded in L1 (R)×Lp1 (R)×Lp2 (R) → Lp (R) for p11 + p12 = p1 . On the other hand, using H¨ older’s inequality Z sup | f (x − t)g(x + t)K(t)dt| ≤ kf kp1 kgkp2 kKkp′ . x

R

′

This shows that T is also bounded in Lp (R) × Lp1 (R) × Lp2 (R) → L∞ (R). Therefore, by interpolation, selecting 0 < θ < 1 such that p13 = 1−θ p , one obtains that T 1 p3 p2 q p1 is bounded in L (R) × L (R) × L (R) → L (R) for p1 + p12 − p13 = 1 − q1 .


Assume now that 1 < p11 + p12 ≤ 2. R Using that R f (x − t)g(x + t)dt = f ∗ g(2x), Young’s inequality implies that Z ∞ (|f | ∗ |g|)kr3 ≤ Ckf kr1 kgkr2 kKk∞ k f (x − t)g(x + t)K(t)dtkr3 ≤ kKk∞ kD1/2 R

whenever r11 + r12 ≥ 1 and r11 + r12 − 1 = r13 . Hence T is bounded in L∞ (R) × Lp1 (R) × Lp2 (R) → Lp (R) where p11 + p12 − 1 = 1 p ≤ 1. Using duality, hT (K, f, g), hi = hT (h, f¯, g), Ki, where f¯(x) = f (−x, that is Z Z Z f (x − t)g(x + t)K(t)h(x)dtdx = ( f¯(t − x)g(x + t)h(x)dx)K(t)dt. R2

R

R

′

Therefore T is also bounded in Lp (R) × Lp1 (R) × Lp2 (R) → L1 (R). Select 0 ≤ θ ≤ 1 such that p13 = p1 + pθ′ . Now using interpolation T will be bounded in Lq (R)×Lp1 (R)×Lp2 (R) → Lp3 (R) for q1 = pθ′ = p13 − p1 = p13 − p11 − p12 +1. References [1] Bergh, J., L¨ ofstr¨ om, J., Interpolation spaces. Springer Verlag [1970]. [2] Blasco,O. Bilinear multipliers and transference. Int. J. Math. Math. Sci. 2005(4) [2005], 545554. [3] Berkson, E., Blasco, O., Carro, M. and Gillespie, A. Discretization versus transference for bilinear operators. Banach Spaces and their Applications in Analysis, 11-30 , De Gruyter [2007]. [4] Blasco, O., Carro, M., and Gillespie, A. Bilinear Hilbert transform on measure spaces. J. Fourier Anal. and Appl.11 [2005], 459-470. [5] Blasco, O., Villarroya, F., Transference of bilinear multipliers on Lorentz spaces. Illinois J. Math. 47(4),[2005], 1327-1343 [6] Coifman R.R., Meyer, Y. Fourier Analysis of multilinear convolution, Calder´ on theorem and analysis of Lipschitz curves. Euclidean Harmonic Analysis (Proc. Sem. Univ. Maryland, College Univ., Md) Lecture Notes in Maths. 779, [1979], 104-122. [7] Fan, D., Sato, S., Transference of certain multilinear multipliers operators. J. Austral. Math. Soc. 70, [2001], pp. 37-55. [8] Gilbert, J, Nahmod, A.,Bilinear operators with non-smooth symbols, J. Fourier Anal. Appl., 7[2001], pp. 435-467. [9] Gilbert, J, Nahmod, A.,Boundedness Bilinear operators with non-smooth symbols, Mat. Res. Letters, 7[2000], pp. 767-778. [10] Grafakos, L., Kalton, N., Some remarks on multilinear maps and interpolation. Math. Annalen 319 [2001], pp. 151-180 [11] Kenig, C. E., Stein, E.M., Multilinear estimates and fractional integration. Math. Res. Lett. 6 [1999], pp. 1-15. [12] Lacey M., Thiele C., Lp estimates on the bilinear Hilbert transform for 2 < p < ∞ Annals Math. 146, [1997], pp. 693-724. [13] Lacey, M., Thiele, C., On Calder´ on’s conjecture. Ann. Math. 149 2 [1999] pp. 475-496. [14] Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean spaces. Princeton Univ. Press [1971]. [15] Villarroya, P., La transformada de Hilbert bilineal. Doctoral Dissertation. Univ. of Valencia [2002]. [16] Villarroya, P., Bilinear multipliers on Lorentz spaces. Czechoslovac Math. J. (to appear) [2007]. Department of Mathematics, Universitat de Valencia, Burjassot 46100 (Valencia) Spain E-mail address: [email protected]

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