analytic families of multilinear operatorsanalytic families of multilinear operators ......

Analytic families of multilinear operators

Mieczysław Mastyło

Adam Mickiewicz University in Poznań

Nonlinar Functional AnalysisValencia 17-20 October 2017

Based on a joint work with Loukas Grafakos

M. Mastyło (UAM) Analytic families of multilinear operators 1 / 39

Outline

1 Introduction

2 Analytic families of multilinear operators

3 Main results

4 Applications to Lorentz and Hardy spaces

5 An application to the bilinear Bochner-Riesz operators


Introduction

IntroductionA bounded measurable function σ on Rn × Rn (called a multiplier) leads to a bilinearoperator Wσ defined by

Wσ(f , g)(x) =∫Rn

∫Rnσ(ξ, η)f (ξ)g(η)e2πi〈x,ξ+η〉dξdη

for every f , g in the Schwartz space S(Rn), where 〈·, ·〉 denotes the inner product in Rn

and f is its Fourier transform of f defined by

f (ξ) =∫Rn

f (x)e−2πi〈x,ξ〉dx , ξ ∈ Rn.

• The study of such bilinear multiplier operators was initiated by Coifman and Meyer(1978). They proved that if 1 < p, q <∞, 1/r = 1/p + 1/q and σ satisfies

|∂αξ ∂βη σ(ξ, η)| ¬ Cα,β(|ξ|+ |η|)−|α|−|β|

for sufficiently large multi-indices α and β, then Wσ extends to a bilinear operatorfrom Lp(Rn)× Lq(Rn) into Lr,∞(Rn) whenever r > 1.


Introduction

• This result was later extended to the range 1 > r > 1/2 by Grafakos and Torres(1996) and Kenig and Stein (1999).

• Multipliers that satisfy the Marcinkiewicz condition were studied by Grafakos andKalton (2001).

• The first significant boundedness results concerning non-smooth symbols wereproved by Lacey and Thiele (1997, 1999) who established that Wσ withσ(ξ, η) = sign(ξ + αη), α ∈ R \ 0, 1 has a bounded extension fromLp(Rn)× Lq(Rn) to Lr (Rn) if 2/3 < r <∞, 1 < p, q ¬ ∞, and 1/r = 1/p + 1/q.

• The bilinear Hilbert transform Hθ is defined for a parameter θ ∈ R by

Hθ(f , g)(x) := limε→0

∫|t|>ε

f (x − t)g(x + θt) 1t dt, x ∈ R

for functions f , g from the Schwartz class S(R). The family Hθ was introducedby Calderón in his study of the first commutator, an operator arising in a seriesdecomposition of the Cauchy integral along Lipschitz curves. In 1977 Calderónposed the question whether Hθ satisfies any Lp estimates.


Introduction

• In their fundamental work (Ann. of Math. 149 (1999), 475-496) Lacey and Thieleproved that if θ 6= −1, then the bilinear Hilbert transform Hθ extends to a bilinearoperator from Lp × Lq into Lr whenever 1 < p, q ¬ ∞ and 1/p + 1/q = 1/r < 3/2.

• The bilinear Hilbert transforms arise in a variety of other related known problems inbilinear Fourier analysis, e.g., in the study of the convergence of the mixed Fourierseries of the form

limN→∞

∑|m−θn|¬N|m−n|¬N

∑f (m)g(n)e2πi(m+n)x

for functions f , g defined on [0, 2π].• Fan and Sato in 2001 were able to show the boundedness of the bilinear Hilbert

transform H on the torus T

H(f , g)(x) :=∫T

f (x − t)g(x + t) ctg(πt) dt, x ∈ T

by transferring the result from R. Their proof relies upon some DeLeeuw (1969)type transference methods for multilinear multipliers.


Introduction

• Stein’s interpolation theorem [Trans. Amer. Math. Soc. (1956)] for analyticfamilies of operators between Lp spaces (p 1) has found several significantapplications in harmonic analysis. This theorem provides a generalization ofthe classical single-operator Riesz -Thorin interpolation theorem to a familyTz of operators that depend analytically on a complex variable z .

• In the framework of Banach spaces, interpolation for analytic families ofmultilinear operators can be obtained via duality in a way similar to that usedin the linear case. For instance, one may adapt the proofs in Zygmund bookand Berg and Lofstrom for a single multilinear operator to a family ofmultilinear operators. However, this duality-based approach is not applicableto quasi-Banach spaces since their topological dual spaces may be trivial.


Introduction

The open strip z ; 0 < Re z < 1 in the complex plane is denoted by S, itsclosure by S and its boundary by ∂S.

Definition Let A(S) be the space of scalar-valued functions, analytic in S andcontinuous and bounded in S. For a given couple (A0,A1) of quasi-Banachspaces and A another quasi-Banach space satisfying A ⊂ A0 ∩ A1, we denoteby F(A) the space of all functions f : S → A that can be written as finitesums of the form

f (z) =N∑

k=1ϕk(z)ak , z ∈ S,

where ak ∈ A and ϕk ∈ A(S). For every f ∈ F(A) we set

‖f ‖F(A) = max

supt∈R‖f (it)‖A0 , sup

t∈R‖f (1 + it)‖A1

.


Introduction

Definition A quasi-Banach couple is said to be admissible whenever ‖ · ‖θis a quasi-norm on A0 ∩ A1, and in this case, the quasi-normed space(A0 ∩ A1, ‖ · ‖θ) is denoted by (A0,A1)θ.

Remark If A is dense in A0 ∩ A1, then for every a ∈ A we have

‖a‖θ = inf‖f ‖F(A); f ∈ F(A), f (θ) = a.

Definition If there is a completion of (A0,A1)θ which is set-theoretically containedin A0 + A1, then it is denoted by [A0,A1]θ.


Introduction

Theorem [A. P. Calderón, Studia Math. (1964)] If (A0,A1) is a Banachcouple, then for every f ∈ F(A0 ∩ A1), and 0 < θ < 1,

log ‖f (θ)‖θ ¬∫ ∞−∞

log ‖f (it)‖A0P0(θ, t) dt +∫ ∞−∞

log ‖f (1 + it)‖A1P1(θ, t) dt,

where P0 and P1 are the Poisson kernels for the strip defined for j ∈ 0, 1 by

Pj(x + iy , t) = e−π(t−y) sinπxsin2 πx + (cos πx − (−1)je−π(t−y))2 , x + iy ∈ S.

Remark The same estimate holds in the case of quasi-Banach spaces; theproof is similar as in the Banach case.


Introduction

Using the fact that the Poisson kernels satisfy∫R

P0(θ, t) dt = 1− θ,∫R

P1(θ, t) dt = θ

together with Calderón’s inequality and Jensen’s inequality, and the concavity ofthe logarithmic function, we obtain the following result:

Lemma Let (A0,A1) be a couple of complex quasi-Banach spaces. For every f inF(A0 ∩ A1), 0 < p0, p1 <∞, and 0 < θ < 1 we have

‖f (θ)‖θ ¬( 1

1− θ

∫ ∞−∞‖f (it)‖p0

A0P0(θ, t) dt

) 1−θp0(1θ

∫ ∞−∞‖f (1 + it)‖p1

A1P1(θ, t) dt

) θp1.




Definition A continuous function F : S → C which is analytic in S is said tobe of admissible growth if there is 0 ¬ α < π such that

supz∈S

log |F (z)|eα|Im z| <∞.

Lemma [I. I. Hirchman, J. Analyse Math. (1953)] If a function F : S → C isanalytic in S, continuous on S, and is of admissible growth, then

log |F (θ)| ¬∫ ∞−∞

log |F (it)|P0(θ, t) dt +∫ ∞−∞

log |F (1 + it)|P1(θ, t) dt.



Definition Let (Ω,Σ, µ) be a measure space and let X1,...,Xm be linearspaces. The family Tzz∈S of multilinear operators

T : X1 × · · · × Xm → L0(µ)

is said to be analytic if for every (x1, ..., xm) ∈ X1 × · · · × Xm and for almostevery ω ∈ Ω the function

z 7→ Tz (x1, ..., xm)(ω), z ∈ S (∗)

is analytic in S and continuous on S. Additionally, if for j = 0 and j = 1 thefunction

(t, ω) 7→ Tj+it(x1, ..., xm)(ω), (t, ω) ∈ R× Ω

is (L × Σ)-measurable for every (x1, ..., xm) ∈ X1 × · · · × Xm, and for almostevery ω ∈ Ω the function given by formula (∗) is of admissible growth, thenthe family Tzz∈S is said to be an admissible analytic family. Here L is theσ-algebra of Lebesgue’s measurable sets in R.



• A quasi-Banach lattice X is said to be maximal whenever 0 ¬ fn ↑ f a.e.,fn ∈ X , and supn1 ‖fn‖X <∞ implies that f ∈ X and ‖fn‖X → ‖f ‖X .

• A quasi-Banach lattice X is said to be p-convex (0 < p <∞) if there existsa constant C > 0 such that for any f1,...,fn ∈ X we have∥∥∥( n∑

k=1|fk |p

)1/p∥∥∥X¬ C

( n∑k=1‖fk‖p

X

)1/p.

The optimal constant C in this inequality is called the p-convexity constantof X , and is denoted, by M(p)(X ).

• A quasi-Banach lattice is said to have nontrivial convexity whenever it isp-convex for some 0 < p <∞.



• If X0 and X1 are quasi-Banach lattices on a given measure space (Ω,Σ, µ)and 0 < θ < 1, we define the quasi-Banach lattice Xθ = X 1−θ

0 X θ1 to be the

space of all f ∈ L0(µ) such that |f | ¬ |f0|1−θ|f1|θ µ-a.e. for some fi ∈ Xi(i = 0, 1) and equipped with the quasi-norm

‖f ‖Xθ = inf‖f0‖1−θ

X0‖f1‖θX1

; |f | ¬ |f0|1−θ|f1|θ µ-a.e..


Main results

Main results

Theorem For each 1 ¬ i ¬ m, let X i = (X0i ,X1i ) be admissible couples ofquasi-Banach spaces, and let (Y0,Y1) be a couple of maximal quasi-Banachlattices on a measure space (Ω,Σ, µ) such that each Yj is pj -convex forj = 0, 1. Assume that Xi is a dense linear subspace of X0i ∩ X1i for each1 ¬ i ¬ m, and that Tzz∈S is an admissible analytic family of multilinearoperators Tz : X1 × · · · × Xm → Y0 ∩ Y1. Suppose that for every(x1, ..., xm) ∈ X1 × · · · × Xm, t ∈ R and j = 0, 1,

‖Tj+it(x1, ..., xm)‖Yj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm

where Kj are Lebesgue measurable functions such that Kj ∈ Lpj (Pj(θ, ·) dt)for all θ ∈ (0, 1). Then for all (x1, ..., xm) ∈ X1 × · · · × Xm, all s ∈ R, and all0 < θ < 1 we have

‖Tθ+is(x1, ..., xm)‖Y 1−θ0 Y θ1

¬ (M(p0)(Y0))1−θ(M(p1)(Y1))θKθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ ,

where

log Kθ(s) =∫R

P0(θ, t) log K0(t + s) dt +∫R

P1(θ, t) log K1(t + s) dt.M. Mastyło (UAM) Analytic families of multilinear operators 18 / 39

Main results

Lemma Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space (Ω,Σ, µ) such that X0 is p0-convex and X1 is p1-convex.Then for every 0 < θ < 1 we have

‖x‖X 1−θ0 Xθ1

¬ (M(p0)(X0))1−θ(M(p1)(X1))θ ‖x‖(X0,X1)θ , x ∈ X0 ∩ X1.

In particular (X0,X1) is an admissible quasi-Banach couple.

Lemma Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure (Ω,Σ, µ). If xj ∈ Xj are such that |xj | (j = 0, 1) are boundedabove and their non-zero values have positive lower bounds, then

|x0|1−θ|x1|θ ∈ (X0,X1)θ

and ∥∥|x0|1−θ|x1|θ∥∥

(X0,X1)θ¬ ‖x0‖1−θ

X0‖x1‖θX1

.


Main results

Corollary Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space (Ω,Σ, µ). If x ∈ X0 ∩ X1 has an order continuous norm inX 1−θ

0 X θ1 , then for every 0 < θ < 1,

‖x‖(X0,X1)θ ¬ ‖x‖X 1−θ0 Xθ1

.

Theorem Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space with nontrivial lattice convexity constants. If the spaceX 1−θ

0 X θ1 has order continuous quasi-norm, then

[X0,X1]θ = X 1−θ0 X θ

1

up equivalences of norms (isometrically, provided that lattice convexityconstants are equal to 1). In particular this holds if at least one of the spacesX0 or X1 is order continuous.


Main results

Theorem For each 1 ¬ i ¬ m, let (X0i ,X1i ) be complex quasi-Banachfunction lattices and let Yj be complex pj -convex maximal quasi-Banachfunction lattices with pj -convexity constants equal 1 for j = 0, 1. Supposethat either X0i or X1i is order continuous for each 1 ¬ i ¬ m. Let T bea multilinear operator defined on (X01 + X11)× · · · × (X0m + X1m) andtaking values in Y0 + Y1 such that

T : Xi1 × · · · × Xim → Yi

is bounded with quasi-norm Mi for i = 0, 1. Then for 0 < θ < 1,

T : (X01)1−θ(X11)θ × · · · × (X0m)1−θ(X1m)θ → Y 1−θ0 Y θ

1

is bounded with the quasi-norm

‖T‖ ¬ M1−θ0 Mθ

1 .


Main results

As an application we obtain the following interpolation theorem for operatorsproved by Kalton (1990), which was applied to study a problem in uniquenessof structure in quasi-Banach lattices (Kalton’s proof uses a deep theorem byNikishin and the theory of Hardy Hp-spaces on the unit disc).

Theorem Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measures spaces. Let Xi ,i = 0, 1, be complex pi -convex quasi-Banach lattices on (Ω1,Σ1, µ1) and letYi be complex pi -convex maximal quasi-Banach lattices on (Ω2,Σ2, µ2) withpi -convexity constants equal 1. Suppose that either X0 or X1 is ordercontinuous. Let T : X0 + X1 → L0(µ2) be a continuous operator such thatT (X0) ⊂ Y0 and T (X1) ⊂ Y1. Then for 0 < θ < 1,

T : X 1−θ0 X θ

1 → Y 1−θ0 Y θ

1

and‖T‖X 1−θ

0 Xθ1→Y 1−θ0 Y θ1

¬ ‖T‖1−θX0→Y0

‖T‖θX1→Y1.


Applications to Lorentz and Hardy spaces


The Lorentz space Lp,q := Lp,q(Ω) on a measure space (Ω,Σ, µ),0 < p <∞, 0 < q ¬ ∞ consists of all f ∈ L0(µ) such that

‖f ‖Lp,q :=

(∫ ∞

0

(t1/pf ∗(t)

)q dtt

)1/qif 0 < q <∞

supt>0 t1/pf ∗(t) if q =∞ .

• The decreasing rearrangement f ∗ of f with respect to µ is defined by

f ∗(t) = infs > 0; µf (s) ¬ t, t 0,

whereµf (s) = µ(ω ∈ Ω; |f (ω)| > s), s > 0.



Lemma Let 0 < pj , qj <∞ and let Lpj ,qj for j = 0, 1 be Lorentz spaces on aninfinite nonatomic measure space (Ω,Σ, µ). Then for 0 < θ < 1 thequasi-norm of

Xθ := (Lp0,q0 )1−θ(Lp1,q1 )θ

is equivalent to that of Lp,q, where 1/p = (1− θ)/p0 + θ/p1 and1/q = (1− θ)/q0 + θ/q1. Moreover for all f ∈ Xθ we have

2−1/p ‖f ‖Lp,q ¬ ‖f ‖Xθ ¬21/p

(log 2)s ss(p1−θ0 pθ1 )s ‖f ‖Lp,q ,

where s = 1 whenever 1 < p0, p1 <∞ and 1 ¬ q0, q1 <∞ ands > max1/p0, 1/q0, 1/p1, 1/q1 otherwise.



Theorem Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure spaces. For1 ¬ i ¬ m, fix 0 < q0, q1, q0i , q1i <∞, 0 < r0, r1, r0i , r1i ¬ ∞ and for0 < θ < 1, define q, r , qi , ri by setting

1qi

= 1− θq0i

+ θ

q1i,

1ri

= 1− θr0i

+ θ

r1i,

1q = 1− θ

q0+ θ

q1,

1r = 1− θ

r0+ θ

r1.

Assume that Xi is a dense linear subspace of Lq0i ,r0i (Ω1) ∩ Lq1i ,r1i (Ω1) andthat Tzz∈S is an admissible analytic family of multilinear operatorsTz : X1 × · · · × Xm → Lq0,r0 (Ω2) ∩ Lq1,r1 (Ω2). Suppose that for every(h1, ..., hm) ∈ X1 × · · · × Xm, t ∈ R and j = 0, 1, we have

‖Tj+it(h1, ..., hm)‖Lqj ,rj (Ω2) ¬ Kj(t)‖h1‖Lqj1,rj1 (Ω1) · · · ‖hm‖Lqjm,rjm (Ω1)

where Kj are Lebesgue measurable functions such that Kj ∈ Lpj (Pj(θ, ·) dt)for all θ ∈ (0, 1), where pj is chosen so that 0 < pj < qj and pj ¬ rj for eachj = 0, 1.



Then for all (f1, ..., fm) ∈ X1 × · · · × Xm, 0 < θ < 1, and s ∈ R we have

‖Tθ+is(f1, ..., fm)‖(Lq0,r0 ,Lq1,r1 )θ ¬ Cθ Kθ(s)m∏

i=1‖fi‖(Lq0,r0 ,Lq1,r1 )θ ,

whereC(θ) :=

( q0q0 − p0

) 1−θp0( q1

q1 − p1

) θp1,

log Kθ(s) =∫R


P1(θ, t) log K1(t + s) dt.



If in addition the measures spaces are infinite and nonatomic, then for all(f1, ..., fm) in X1 × · · · × Xm and s ∈ R, and 0 < θ < 1 we have

‖Tθ+is(f1, ..., fm)‖Lq,r (Ω2) ¬ C( q0

q0 − p0

) 1−θp0( q1

q1 − p1

) θp1 Kθ(s)

m∏i=1‖fi‖Lqi ,ri (Ω1),

whereC = 2

1q +∑m

i=11qi

(u q1−θ

0 qθ1log 2

)u

with u = 1 if 1 < q0, q1 <∞ and 1 ¬ r0, r1 ¬ ∞, whileu > max1/q0, 1/q1, 1/r0, 1/r1 otherwise.



Suppose that there is an operator M defined on a linear subspace of L0(Ω,Σ, µ)taking values in L0(Ω,Σ, µ) such that:

(a) For j = 0 and j = 1 the function (t, x) 7→ M(h(j + it, ·))(x), (t, x) ∈ R× Ωis L × Σ-measurable for any function h : ∂S × Ω→ C such thatω 7→ h(j + it, ω) is Σ-measurable for almost all t ∈ R.

(b) M(λh)(ω) = |λ|M(h)(ω) for all λ ∈ C.(c) For every function h as in above there is an exceptional set Eh ∈ Σ with

µ(Eh) = 0 such that for j ∈ 0, 1

M(∫ ∞−∞

h(j + it, ·)Pj(θ, t) dt)

(ω) ¬∫ ∞−∞M(h(t, ·))(ω)Pj(θ, t) dt

for all θ ∈ (0, 1), and all ω /∈ Eh. Moreover, Eψh = Eh for every analyticfunction ψ on S which is bounded on S.



For each 1 ¬ i ¬ m, let X i = (X0i ,X1i ) be admissible couples ofquasi-Banach spaces, and let (Y0,Y1) be a couple of complex maximalquasi-Banach lattices on a measure space (Ω,Σ, µ) such that each Yj ispj -convex for j = 0, 1. Assume that Xi is a dense linear subspace of X0i ∩ X1ifor each 1 ¬ i ¬ m, and that Tzz∈S is an admissible analytic family ofmultilinear operators Tz : X1 × · · · × Xm → Y0 ∩ Y1. Assume that M isdefined on the range of Tz , takes values in L0(Ω,Σ, µ), and satisfiesconditions (a), (b) and (c).



Theorem Suppose that for every (x1, ..., xm) ∈ X1 × · · · × Xm, t ∈ R and

‖M(Tj+it(x1, ..., xm))‖Yj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm , j = 0, 1,

where Kj are Lebesgue measurable functions such that Kj ∈ Lpj (Pj(θ, ·) dt)for all θ ∈ (0, 1). Then for all (x1, ..., xm) ∈ X1 × · · · × Xm, s ∈ R, and0 < θ < 1,

‖M(Tθ+is(x1, ..., xm))‖Y 1−θ0 Y θ1

¬ Cθ Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ ,

whereCθ = (M(p0)(Y0))1−θ(M(p1)(Y1))θ,

log Kθ(s) =∫R


P1(θ, t) log K1(t + s) dt.



• If φ is a C∞(Rn) function with a compact support, φδ(x) = δ−nϕ(x/δ) forall x ∈ Rn.

Mh(x) := supδ>0|(φδ ∗ h)(x)|, x ∈ Rn.

• Y0 = Lp0 , Y1 = Lp1 , in which case Y 1−θ0 Y θ

1 = Lp, 1/p = (1− θ)/p0 + θ/p1.

Definition The classical Hardy space Hp of Fefferman and Stein is defined by

‖h‖Hp := ‖M(h)‖Lp .



Corollary If Tz is an admissible analytic family is such that

‖Tj+it(x1, ..., xm)‖Hpj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm , j = 0, 1,

then

‖Tθ+s(x1, ..., xm)‖Hp ¬ Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ

for 0 < p0, p1 <∞, s ∈ R, and 0 < θ < 1. Analogous estimates hold for theHardy-Lorentz spaces Hq,r where estimates of the form

‖Tj+it(x1, ..., xm)‖Hqj ,rj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm

for admissible analytic families Tz when j = 0, 1 imply

‖Tθ+is(x1, ..., xm)‖Hq,r ¬ C Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ ,

where 0 < pj < qj <∞, pj ¬ rj ¬ ∞ and 1/q = (1− θ)/q0 + θ/q1,1/r = (1− θ)/r0 + θ/r1 while u = 1 if 1 < q0, q1 <∞ and 1 ¬ r0, r1 ¬ ∞and u > max1/q0, 1/q1, 1/r0, 1/r1 otherwise.


An application to the bilinear Bochner-Riesz operators

An application to the bilinear Bochner-Rieszoperators

Stein’s motivation to study analytic families of operators might have been thestudy of the Bochner-Riesz operators

Bδ(f )(x) :=∫|ξ|¬1

(1− |ξ|2

)δ f (ξ)e2πi〈x ,ξ〉 dξ, f ∈ S(Rn).

in which the “smoothness” variable δ affects the degree p of integrability ofBδ(f ) on Lp(Rn).

Remark Using interpolation for analytic families of operators, Stein showedthat whenever δ > (n − 1)

∣∣ 1p −

12∣∣, then

Bδ : Lp(Rn)→ Lp(Rn)

is bounded for every 1 ¬ p ¬ ∞.



• The bilinear Bochner-Riesz operators are defined on S × S by

Sδ(f , g)(x) :=∫∫|ξ|2+|η|2¬1

(1− |ξ|2 − |η|2

)δ f (ξ)g(η)e2πi〈x ,ξ+η〉 dξdη

for every f , g ∈ S.

• The bilinear Bochner-Riesz means Sz is defined by

Sz (f , g)(x) =∫ ∫

Kz (x − y1, x − y2)f (y1)g(y2)dy1 dy2,

where that the kernel of Kδ+it is given by

Kδ+it(x1, x2) = Γ(δ + 1 + it)πδ+it

Jδ+it+n(2π|x |)|x |δ+it+n , x = (x1, x2).



If δ > n − 1/2, then using known asymptotics for Bessel functions we havethat this kernel satisfies an estimate of the form:

|Kδ+it(x1, x2)| ¬ C(n + δ + it)(1 + |x |)δ+n+1/2 ,

where C(n + δ + it) is a constant that satisfies

C(n + δ + it) ¬ Cn+δeB |t|2

for some B > 0 and so we have

|Kδ+it(x1, x2)| ¬ Cn+δ eB|t|2 1(1 + |x1|)n+ε

1(1 + |x2|)n+ε ,

with ε = 12 (δ − n − 1/2). It follows that the bilinear operator Sδ+it is

bounded by a product of two linear operators, each of which has a goodintegrable kernel. It follows that

Sδ+it : L1 × L1 → L1/2

with constant K1(t) ¬ C ′n+δeB|t|2 whenever δ > n − 1/2.M. Mastyło (UAM) Analytic families of multilinear operators 37 / 39


Theorem Let 1 < p < 2. For any λ > (2n − 1)( 1

p −12)

Sλ : Lp(Rn)× Lp(Rn)→ Lp/2(Rn) is bounded .

Proof. We apply the main Theorem with (X01,X11) = (X02,X12) := (L2, L1),(Y0,Y1) := (L1, L1/2) and X1 = X2 the space of Schwartz functions on Rn,which is dense in L1 and L2. We fix δ > 0 and we consider the bilinearanalytic family Tzz∈S , where Tz := S(n− 1

2 )z+δ for all z ∈ S. We claim thatthis family is admissible. Indeed, for f , g Schwartz functions we have

Tz (f , g)(x) =∫∫|ξ|2+|η|2¬1

(1− |ξ|2 − |η|2

)(n− 12 )z+δ f (ξ)g(η)e2πi〈x ,ξ+η〉 dξdη, x ∈ Rn,

and the map z 7→ Tz (f , g) is analytic in S, continuous and bounded on S,and jointly measurable in (t, x) when z = it or z = 1 + it. Moreover, for allx ∈ Rn we have

supz∈S

log |Tz (f , g)(x)|eα|Im z| <∞, f , g ∈ S

with α = 0 < π; in fact |Tz (f , g)(x)| ¬ ‖f ‖L1‖g ‖L1 .M. Mastyło (UAM) Analytic families of multilinear operators 38 / 39


Based on the preceding discussion, we have that when Re z = 0, Tz mapsL2 × L2 to L1 with constant K0(t) ¬ Cn,δec |t|2 for some Cn,δ, c > 0. We alsohave that when Re z = 1, Tz maps L1 × L1 to L1/2 with constantK1(t) ¬ C ′n,δeB |t|2 for some C ′n,δ, B > 0. We notice that for these functionsKi (t) we have that the constant K (θ, 1, 1/2) <∞; in this case θ = 2( 1

p −12 ).

An application of the main Theorem yields that

Sλ : Lp(Rn)× Lp(Rn)→ Lp/2(Rn)

provided λ = 2(n − 12 ) ( 1

p −12 ) + δ > (2n − 1)( 1

p −12 ).


analytic families of multilinear operatorsanalytic families of multilinear operators ......

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