Multipliers: Sharp estimates - Purdue Universitybanuelos/Lectures/Banff2011.pdf · These operators...

Sharp Estimates

Multipliers: Sharp estimates

Rodrigo Banuelos

Department of Mathematics

9/22/11

R. Banuelos (Purdue) Sharp Estimates 9/22/11

Sharp Estimates

Definition

m : Rd → C in L∞ produces a Fourier multiplier operator

Mm : L2(Rd)→ L2(Rd)

via Plancherel’s theorem and

Mmf (ξ) = m(ξ) f (ξ)

Goals

1 Study Fourier multipliers which arise from some basic transformations ofstochastic integrals (or other such objects) and that extend to boundedoperators on Lp(Rd), 1 < p <∞. (Fourier Lp–multipliers).

2 Identify, precisely, their operator norms: That is, determine

||Mm : Lp(Rd)→ Lp(Rd)|| = ‖Mm‖p.

3 Applications to some problems in analysis.


Sharp Estimates

Theorem (Hormander multipliers 1960)

k an integer greater than d/2 and m : Rd → C a C k function with the propertythat there is a constant c such that for all multi-indices α = (α1, · · · , αd)satisfying |α| = |α1|+ · · ·+ |αd | ≤ k we have

supx∈Rd

|x ||α|

∣∣∣∂αm(x)

∂xα

∣∣∣ = C <∞

Then‖Mmf ‖p ≤ Cp‖f ‖p, 1 < p <∞

with Cp depending on C , d and p.

Theorem (C. Fefferman 1971–“The Multiplier Problem for the Ball”)

If m = χB where B is the unit ball in Rd , d > 1, then Mm is an Lp–multiplier ifand only if p = 2.


Sharp Estimates

Levy measure ν ≥ 0 on Rd . So, ν(0) = 0 and∫Rd

min(|z |2, 1)dν(z) <∞

Let µ ≥ 0 be a finite Borel measure on the unit sphere S ⊂ Rd , and

ϕ : Rd → C, ψ : S→ C, ϕ, ψ ∈ L∞(C)

Consider the “Levy multiplier”

m (ξ) =

∫Rd

(1− cos ξ ·z

)ϕ (z) dν(z) + 1

2

∫S |ξ ·θ|

2ψ (θ) dµ(θ)∫Rd

(1− cos ξ ·z

)dν(z) + 1

2

∫S |ξ ·θ|2dµ(θ)

,

Note that ‖m‖∞ <∞.

m (ξ) =

∫Rd

(1− cos ξ ·z

)ϕ (z) dν(z) + 1

2Aξ · ξ∫Rd

(1− cos ξ ·z

)dν(z) + 1

2Bξ · ξ,


Sharp Estimates

A =

[∫Sϕ (θ) θiθj dµ(θ)

]i,j=1...d

and B =

[∫Sθiθj dµ(θ)

]i,j=1...d

with both A and B symmetric and B non–negative definite.

Theorem (R.B & P. Hernandez-Mendez (2003), R.B & K. Bogdan (2007), R.B.,A. Bielaszewski & K. Bogdan (2010))

‖φ‖∞ & ‖ψ‖∞ ≤ 1⇒ ‖Mmf ‖p ≤ (p∗ − 1)‖f ‖p, 1 < p <∞,

p∗ − 1 =

1

p−1 , 1 < p ≤ 2,

p − 1, 2 ≤ p <∞.

Remarks

The constant is best possible: Due to a result of Geiss, Montgomery-Smith andSaksman 2008 on Riesz transforms.

Remarks

New result in this talk: A sharper theorem with more information on the norm.


Sharp Estimates

Theorem (Burkholder 1966 & 1984–“symmetric multipliers”)

f = fn, n ≥ 0 a martingale with difference sequence dk = fk − fk−1, k ≥ 0,g = v ∗ f its martingale transforms with difference sequence vk dk , k ≥ 0,v = vk , k ≥ 0 predictable with values in [−1, 1]. For 1 < p <∞

‖g‖p ≤ Cp‖f ‖p, (1966)

18 Years Later:‖g‖p ≤ (p∗ − 1)‖f ‖p, (1984)

and the bound p∗ − 1 is best possible.

Theorem (Choi 1992– Sharp “one-sided multipliers”)

If v = vk , k ≥ 0 takes values in [0, 1].

‖g‖p ≤ cp‖f ‖p, 1 < p <∞.

cp =p

2+

1

2log

(1 + e−2

2

)+α2

p+ · · ·


Sharp Estimates

Definition

Let −∞ < b < B <∞ and 1 < p <∞ be given and fixed. Define Cp,b,B as theleast positive number C such that for any real-valued martingale f and for anytransform g = v ∗ f of f by a predictable sequence v = vk , k ≥ 0 with values in[b,B], we have

||g ||p ≤ C ||f ||p.

Example

I Cp,−a,a = a(p∗ − 1), by Burkholder (symmetric)

I Cp,0,a = a cp, by Choi’s Theorem (one sided)

Theorem (R.B. & A. Osekowski–2011)

max

(B − b

2

)(p∗ − 1), max|B|, |b|

≤ Cp,b,B ≤ maxB, |b|(p∗ − 1)


Sharp Estimates


Suppose ϕ, ψ take values in [b,B] for some −∞ < b < B <∞. Then Mm withLevy multipler

m (ξ) =

∫Rd (1− cos ξ ·z)ϕ (z) dν(z) + 1

2

∫S |ξ ·θ|

2ψ (θ) dµ(θ)∫Rd (1− cos ξ ·z) dν(z) + 1

2

∫S |ξ ·θ|2dµ(θ)

⇒ ‖Mmf ‖p ≤ Cp,b,B‖f ‖p, 1 < p <∞.

The constant is best possible.


Let m be a real and even multiplier which is homogeneous of order 0 on Rd . Letb and B be the minimal and the maximal term of the sequence(

m(1, 0, 0, . . . , 0),m(0, 1, 0, . . . , 0), . . . ,m(0, 0, . . . , 0, 1)).

Then‖Mm‖p ≥ Cp,b,B .


Sharp Estimates

Example (From stable processes we get: Marcinkiewicz multipliers (Stein’s bookon Singular integrals) with correct norms)

1 0 < α ≤ 2, d ≥ 2. For J ( 1, 2, . . . , d, set

m(ξ) =

∑j∈J |ξj |α∑dj=1 |ξj |α

.

2 0 < α ≤ 2, d = 2n, even. Set

m(ξ) =|ξ2

1 + ξ22 + . . .+ ξ2

n |α/2

|ξ21 + ξ2

2 + . . .+ ξ2n |α/2 + |ξ2

n+1 + ξ2n+2 + . . .+ ξ2

2n|α/2.

In both cases,||Mm||p = Cp,0,1 = cp, 1 < p <∞,

cp is the Choi constant.


Sharp Estimates

These operators are motivated by

(i) A 1982 conjecture of Tadeusz Iwaniec on the Lp–norm of Calderon-Zygmundsingular integral operator called ”Beurling–Ahlfors transform” which plays afundamental role in many areas of analysis.

(ii) A conjecture of Morrey (1952): Ψ :Mn×m → R rank-one convex ⇒quasi-convex with implications to the calculus of variations

Some References:

I K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equationsand Quasiconformal Mappings in the Plane, Princeton University Press,2009.

I V. Vasylunin and A. Volberg, 2011– “Burkholder’s function viaMonge-Ampere equation”

I R.B. 2011–“The foundational inequalities of D.L. Burkholder and some oftheir ramifications”


Sharp Estimates

Conjecture: Tadeusz Iwaniec, 1982

Bf (z) = − 1

π

∫∫C

f (w)

(z − w)2dA(w) ⇒ ‖Bf ‖p ≤ (p∗ − 1)‖f ‖p.

As a Calderon–Zygmund singular integral, ‖Bf ‖p ≤ Cp‖f ‖p, 1 < p <∞. Longknown (Lehto (1965)) that (p∗ − 1) cannot be improved.

1 R.B. & Wang 1995 (via Burkholder’s 1984 result) :

‖B‖p ≤ 4(p∗ − 1)

2 Nazarov–Volberg (2003):

‖B‖p ≤ 2(p∗ − 1)

Via a ”Littlewood-Paley inequality” proved with Bellman functions.Burkholder’s result is used for the construction of the function–hence notmartingale independent either. NO non-martingale proofs exists!!


Sharp Estimates

3 R.B. & Mendez (2003):‖B‖p ≤ 2(p∗ − 1)

Exact same proof as the ”4” result with Wang except applied to ”heatmartingales”.

4 R.B. & Janakiraman (2007): Taking into account some additional conformalstructure of martingales that arise from the Beurling-Ahlfors operator,

‖B‖p ≤ 1.575(p∗ − 1), 1 < p <∞

Basic Building blocks: Riesz transforms

Except for the 1.575 result, the other estimates come from decomposing the B-Aoperator into its basic components of Riesz transforms.


Sharp Estimates

Riesz Transforms in Rd , d ≥ 2

Rj(f )(x) = − ∂

∂xj(−∆)−1/2f , j = 1, . . . , d , (first order)

RjRk f =∂2

∂xj∂xk(−∆)−1f , (second order)

Rj f (ξ) =iξj|ξ|

f (ξ), RjRk f (ξ) =−ξjξk|ξ|2

f (ξ)

Basic properties of Beurling-Ahlfors

Bf =∂2

∂z2(−∆)−1f , B∂ = ∂

Bf (ξ) =ξ

2

|ξ|2f (ξ) ⇒ B = R2

2 − R21 + 2iR2R1


Sharp Estimates

R.B. & G. Wang 1995

⇒∥∥∥R2

1 − R22

∥∥∥p≤ 2(p∗ − 1),

∥∥∥R1R2f∥∥∥p≤ (p∗ − 1)

⇒∥∥∥Bf

∥∥∥p≤ 4(p∗ − 1)

∥∥∥f∥∥∥p

Nazarov–Volberg 2003 & (separately) R.B. P. Mendez 2003

⇒∥∥∥R2

1 − R22

∥∥∥p≤ (p∗ − 1),

∥∥∥2R1R2f∥∥∥p≤ (p∗ − 1)

⇒∥∥∥Bf

∥∥∥p≤ 2(p∗ − 1)

∥∥∥f∥∥∥p

Geiss, Montgomery-Smith and Saksman 2008

||R21 − R2

2 ||p = ||2R1R2||p = p∗ − 1.

Question (Geiss, Montgomery-Smith and Saksman 2008)

What is the norm of R2j in Lp? That is, what is ‖R2

j ‖p, 1 < p <∞?


Sharp Estimates

Theorem (R.B. & A. Osekowski 2011)

Let d ≥ 2 and assume that A = (aij)di ,j=1 is a d × d symmetric matrix with

real entries and eigenvalues λ1 ≤ λ2 ≤ . . . ≤ λd . Consider the operator

Mm =d∑

i ,j=1

aijRiRj with multiplier m(ξ) =(Aξ, ξ)

|ξ|2.

⇒ ‖Mm‖p = Cp,λ1,λd , 1 < p <∞

Corollary

If d ≥ 2 and J ( 1, 2, . . . , d, then

||∑j∈J

R2j ||p = Cp,0,1 = cp, 1 < p <∞,

where cp is the Choi constant.


Sharp Estimates

Proof ideas, the Riesz transform case

Upper Bound for ‖Mm‖p: A two step procedure:

1 A continuous–time martingale inequality involving Cp,λ1,λd

2 A stochastic integral representation of Mm. A “lifting” of functions inLp(Rd) “up to” Lp(Ω) and a “projection” from Lp(Ω) “down to” Lp(Rd).

For (1) we use


Suppose −∞ < b < B <∞ and Xt , Yt two real valued martingales withright-continuous & left-limits paths satisfying[

B − b

2X ,

B − b

2X

]t

−[

Y − b + B

2X ,Y − b + B

2X

]t

≥ 0

and nondecreasing for all t ≥ 0 (differential subordination). Then

||Y ||p ≤ Cp,b,B ||X ||p, 1 < p <∞,

and the inequality is sharp.


Sharp Estimates

Proof ideas, the Riesz transform case

Upper Bound for ‖Mm‖p: A two step procedure:

1 A continuous–time martingale inequality involving Cp,λ1,λd

2 A stochastic integral representation of Mm. A “lifting” of functions inLp(Rd) “up to” Lp(Ω) and a “projection” from Lp(Ω) “down to” Lp(Rd).

For (1) we use


Suppose −∞ < b < B <∞ and Xt , Yt two real valued martingales withright-continuous & left-limits paths satisfying[

B − b

2X ,

B − b

2X

]t

−[

Y − b + B

2X ,Y − b + B

2X

]t

≥ 0

and nondecreasing for all t ≥ 0 (differential subordination). Then

||Y ||p ≤ Cp,b,B ||X ||p, 1 < p <∞,

and the inequality is sharp.R. Banuelos (Purdue) Sharp Estimates 9/22/11

Sharp Estimates

“Proof” : For (1), follow Burkholder’s method

Define V (x , y) = |y |p − C pp,b,B |x |p. Want EV (Xt ,Yt) ≤ 0. Ito’s formula,

V (Xt ,Yt) =

∫ t

0

dVs +1

2

∫ t

0

d [V ]s

Enter Burkholder: Find ”the” smallest function U(x , y) satisfying U(0, 0) = 0,

V (x , y) ≤ U(x , y),

d [U(X ,Y )]s ≤ 0,

for all s, for all (Xt ,Yt) satisfying our condition. As it turns out, such functionmust satisfy

Uxx ± 2Uxy + Uyy ≤ 0 and Uyy ≥ 0 on R2.

Symmetric case B = 1, b = −1 and Cp,1,−1 = p∗ − 1, Burkholder 1988:

U(x , y) = αp (|y | − (p∗ − 1)|x |) (|x |+ |y |)p−1, αp = p

(1− 1

p∗

)p−1


Sharp Estimates

For (2). Follow a “lifting” and “projection” strategy.

The semigroup and the space-time martingales–Brownian motion case

Let Pt be the semigroup of Brownian motion acting on “NICE” f . Consider the”space-time” martingale

P(T−t)(Wt)− PT f (W0) =

∫ t

0

∇xP(T−s)f (Ws) · dWs , 0 < t ≤ T .

f (WT ) ∼ X =

∫ T

0

∇xUf (Ws ,T − s) · dWs , for very large T

Definition (Martingale transform)

For any d × d matrix A (could even be a predictable function!) set

Y =

∫ T

0

A∇xP(T−s)f (Ws) · dWs .


Sharp Estimates

Lemma

A symmetric, real entries, eigenvalues λ1 ≤ λ2 ≤ . . . ≤ λd .The martingales X andY satisfy the differential subordination condition with b = λ1 and B = λd .

Proof.

Set ξ = ∇P(T−t)f (Wt).

d

[Y − b + B

2X ,Y − b + B

2X

]t

− d

[B − b

2X ,

B − b

2X

]t

=

(∣∣∣∣Aξ − b + B

2ξ

∣∣∣∣2 − ∣∣∣∣B − b

2ξ

∣∣∣∣2)

dt =(|Aξ|2 − (b + B)(Aξ, ξ) + bB|ξ|2

)dt

=⟨(A− BI)(A− bI)ξ, ξ

⟩dt,

A− BI is nonpositive-definite, A− bI is nonnegative-definite and theycommute.

Corollary

‖Y ‖Lp(Ω) ≤ Cp,b,B‖X‖Lp(Ω)


Sharp Estimates

Lemma

STA f (x) = E[

Y∣∣WT = (x , 0)

].

Then STA : Lp(Rd)→ Lp(Rd) and in fact

‖STA f (x)‖p ≤ Cp,b,B‖f ‖Lp(Rd ), 1 < p <∞.

Lemma

limT→∞

STA =Mm =d∑

i,j=1

aijRiRj with multiplier m(ξ) =(Aξ, ξ)

|ξ|2.


Sharp Estimates

Beurling-Ahlfors from Gaussian: ν = 0

Take µ point mass at 1, i , e−iπ/4, e iπ/4

ψ(1) = 1, ψ(e−iπ/4) = i , ψ(i) = −1, ψ(e iπ/4) = −i .

Compute:∫S|ξ · θ|2 ψ(θ)dµ(θ) = ξ2

1 − ξ22 + i(ξ1

1√2− ξ2

1√2

)2 − i(ξ11√2

+ ξ21√2

)2 =

= ξ21 − ξ2

2 − 2i ξ1ξ2 = ξ2,

∫S|ξ · θ|2 µ(dθ) = ξ2

1 + ξ22 + (ξ1

1√2− ξ2

1√2

)2 + (ξ11√2

+ ξ21√2

)2 = 2 |ξ|2 ,

⇒ m(ξ) =

∫S |ξ · θ|

2ψ(θ)dµ(θ)∫

S |ξ · θ|2 dµ(θ)

=ξ2

2 |ξ|2,

⇒ ‖B‖p ≤ 2(p∗ − 1). (Disappointing!)


Sharp Estimates

—————————


Sharp Estimates

Proposition (A disappointing fact)

If µ is a measure on the circle S in R2 and ψ : S→ C with ‖ψ‖∞ ≤ 1 and∫S |ξ · θ|

2 ψ(θ)dµ(θ)∫S |ξ · θ|

2 dµ(θ)=

ξ2

c |ξ|2, ξ ∈ R2 \ 0,

then |c | ≥ 2.


Sharp Estimates

Beurling-Ahlfors from stable processes: µ = 0

For 0 < α < 2, consider a stable Levy measure in R2 (in polar coordinates)

dνα(r , θ) = r−1−αdr dσ(θ) and ϕ(z) = ϕ

(z

|z |

),

∫R2

(1− cos(ξ · z)

)ϕ(z)dνα(z)

=

∫S

∫ ∞0

(1− cos(rθ · ξ)

)ϕ(rθ)r−1−α dr dσ(θ)

=

∫S|ξ · θ|α ϕ(θ)

∫ ∞0

1− cos(s)

s1+αdsdσ(θ)

= cα

∫S|ξ · θ|α ϕ(θ)dσ(θ),

cα =

∫ ∞0

1− cos(s)

s1+αds


Sharp Estimates

The Levy “Stable Multiplier” is:

m(ξ) =

∫S |ξ · θ|

αϕ(θ) dσ(θ)∫

S |ξ · θ|α dσ(θ)

.

Setθ = (cos(t), sin(t)),

ϕ(cos(t), sin(t)) = e−i2t

ξ = |ξ| e iu = |ξ| (cos(u), sin(u))

Compute:

∫S|ξ · θ|α ϕ(θ) dσ(θ) = 2 |ξ|α ξ

2

|ξ|2α

α + 2B(α + 1

2,

1

2

),

For ϕ = 1 then we have∫S|ξ · θ|α dσ(θ) = 2 |ξ|α B

(α + 1

2,

1

2

).


Sharp Estimates

m(ξ) =ξ

2

|ξ|2α

α + 2(Disappointing Again!)

and

‖Bf ‖p ≤α + 2

α(p∗ − 1)‖f ‖p,

and letting α→ 2 gives

‖B‖ ≤ 2(p∗ − 1)

Conjecture

ϕ : S→ C, ‖ϕ‖L∞ ≤ 1. For any 0 < r <∞, and any d ≥ 2, set

m(ξ) =

∫S |ξ · θ|

rϕ(θ) dσ(θ)∫

S |ξ · θ|r dσ(θ)

, (1)

where σ denotes the surface measure on the unite sphere S of Rd . Then

‖Mm‖p ≤ p∗ − 1. (2)


Sharp Estimates

Morrey (1952), Iwaniec (1982), Burkholder (1984)

I (f ) =

∫Ω

F

(∂fi∂xj

(x)

)dx , f : Ω ⊂ Rd → Rd , f ∈W 1,p(Ω,Rd).

Morrey (1952): “Quasi–convexity and lower semicontinuity of multipleintegrals.”

I I is weakly lower semicontinuous ⇐⇒ F quasi-convex

I The Euler equations I ′(f ) = 0 are elliptic ⇐⇒ F is rank–one convex

I Quasi-convexity: F : Rd×d → R for each A ∈ Rd×d , each boundedD ⊂ Rd , each compactly supported Lipschitz function f : D → Rd ,

F (A) ≤ 1

|D|

∫D

F (A +∂fi∂xj

)

I Rank-one convexity: F : Rd×d → R, A, B ∈ Rd×d , rank B = 1,

h(t) = F (A + tB) is convex


Sharp Estimates

I d = 1, quasi-convex or rank–one convex ⇐⇒ convex.

I If d ≥ 2, convexity =⇒ quasi–convexity =⇒ rank–one convexity.

Conjecture

Morrey 1952: Rank–one convexity does not imply quasi–convexity.

Sverak 1992: True if d ≥ 3. Case d = 2, open.

Enter the Burkholder function U: ∀z , w , h, k ∈ C, |k | ≤ |h|,

h(t) = −U(z + th, w + tk) is convex–(Burkholder 1984)

Define Γ: R2×2 → C× C by Γ

(a bc d

)= (z ,w),

z = (a + d) + i(c − b), w = (a− d) + i(c + b)

FU = −U Γ, is rank–one convex–(R.B–Lindeman 1997).


Sharp Estimates

f : C→ C,(∂fi∂xj

)=

[ux uy

vx vy

], f = u + iv ∈ C∞0 (C)

FU

(∂fi∂xj

)= −U

(∂f , ∂f

).

Quasiconvexity of FU at 0 ∈ R2×2 ⇐⇒

−∫

supp fU(∂f , ∂f

)≥0.

Question (The “Win-Win Question”–R.B. Wang 1995, R.B. Lindeman1997)

Is FU quasiconvex?

1 If true: Iwaniec’s conjecture true. (Using B∂ = ∂, and the fact thatV (x , y) = |y |p − (p∗ − 1)p|x |p ≤ U(x , y).)

2 If false: Morrey’s conjecture true for d = 2.


Sharp Estimates

THANK YOU


Multipliers: Sharp estimates - Purdue Universitybanuelos/Lectures/Banff2011.pdf · These operators...

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