A Class of Calderón-Zygmund Operators Arising …CZ Martingales A Class of Calder on-Zygmund Operators Arising from the Projections of Martingale Transforms Michael Perlmutter Department

CZ Martingales

A Class of Calderon-Zygmund Operators Arisingfrom the Projections of Martingale Transforms

Michael Perlmutter

Department of MathematicsPurdue University

1 M. Perlmutter(Purdue) CZ Martingales

CZ Martingales

Calderon-Zygmund Operators

Let T be an operator on S(Rn), which is bounded on L2(Rn) of the form

Tf (x) = p.v .

∫Rn

K (x , x)f (x)dx .

T is called Calderon-Zygmund if K is C 1 on {x 6= x} and

|K (x , x)| ≤ κ

|x − x |n

|∇xK (x , x)| ≤ κ

|x − x |n+1

|∇xK (x , x)| ≤ κ

|x − x |n+1,

for some universal constant κ whenever x 6= x .


CZ Martingales

Examples

Hilbert Transform

Hf (x) =1

π

∫R

1

x − xf (x)dx .

Riesz Transform

Ri f (x) = Cn

∫Rn

xi − xi|x − x |n+1

f (x)dx ,

1 ≤ i ≤ n, Cn =Γ( n+1

2 )π(n+1)/2 .

Beurling-Ahlfors Transform

Bf (z) = − 1

π

∫C

1

(z − w)2f (w)dw .


CZ Martingales

Fourier Multipliers and Pseudo-Differential Operators

Fourier Multipliers

Rj f (ξ) =iξj|ξ|

f (ξ)

Bf (ξ) =ξ2

1 − ξ22 − 2iξ1ξ2

|ξ|2f (ξ)

B = R22 − R2

1 + 2iR1R2.

Pseudo-Differential Operators

Rj f = ∂xj (−4)−1/2f

Bf =∂z∂z

f .


CZ Martingales

Symmetric α-stable Processes

Characteristic Function

E(e iξ·Xt ) = e−t|ξ|α, 0 < α ≤ 2.

Examples:

α = 2: Xt is Brownian Motion with density ht(x) = 1(4πt)n/2 e

−|x |2/4t .

α = 1: Xt is the Cauchy process with density pt(x) = Cnt

(|x |2+t2)(n+1)/2 .

Properties

Semigroup: ψs ∗ ψt = ψs+t .Scaling: ψt(x) = 1

tn/αψ1( x

t1/α ).


CZ Martingales

Setup

I (Xt)t>0, rotationally-invariant α-stable process on Rn, 0 < α ≤ 2.

I For y > 0, ϕy (x) = 1ynϕ( xy ), where ϕ denotes the density of X1.

I Let A(x , y) = (ai ,j(x , y)) be an (n + 1)× (n + 1) matrix-valuedfunction

‖A‖ = ‖ sup|v |≤1

(|A(x , y)v |)‖L∞(Rn×[0,∞)) <∞.

I ai ,j(x , y) = ai ,j(y) whenever i or j = n + 1.


CZ Martingales

Results

Theorem:

Consider the kernel

KA(x , x) =

∫ ∞0

∫Rn

2yA(x , y)∇ϕy (x − x)∇ϕy (x − x)dxdy ,

where ∇ = (∂x1 , . . . , ∂xn , ∂y ). Then the operator

TAf (x) =

∫Rn

K (x , x)f (x)dx

is a CZ operator.


CZ Martingales

Results cont’d

Remark:

If ai ,j(y) = 0 whenever i or j = n + 1, we may also write our kernel interms of the density of Xt .The scaling relation ψt(x) = 1

tn/αψ1( x

t1/α ) implies ϕt1/α = ψt .Therefore,

KA(x , x) =

∫ ∞0

∫Rn

2

αt

2α−1A(x , t1/α)∇ψt(x − x)∇ψt(x − x)dxdt.


CZ Martingales

Boundedness of CZ Operators

Strong type

‖Tf ‖p ≤ Cp,n,κ‖f ‖p, 1 < p <∞ (1)

Weak type

|{x : |Tf (x)| > λ}| ≤ C1,n,κ

λ‖f ‖1. (2)

Questions

For which operators can the constant in (1) and (2) be taken independentof n?What are the best possible constants?


CZ Martingales

Probabilistic Representations of the Riesz Transform

Basic Idea

Lp → Mp → Mp → Lp

Background Radiation

“Time-reversed Brownian motion” in Rn+1+ . B−∞ has Lebesgue

distribution on Rn × {∞}, B0 has Lebesgue distribution on Rn × {0}.

Embedding into Mp

For f ∈ Lp, let uf be it’s Poisson extension to Rn+1+ . Then

(Xt)t≤0 = (uf (Bt))t≤0 is a martingale and

uf (Bt) =

∫ t

−∞∇uf (Bs) · dBs .


CZ Martingales

Probabilistic Representations of the Riesz Transform cont’d

Martingale Transforms

Let A(x , y) an (n + 1)× (n + 1) matrix-valued function, ‖A‖ <∞.

(A ∗ f )t =

∫ t

−∞A(Bs)∇uf (Bs) · dBs .

Theorem: (Banuelos, Wang, Burkholder)

‖(A ∗ f )‖p ≤ (p∗ − 1)‖A‖‖X‖p 1 < p <∞,

p∗ = max{p, pp−1}.


CZ Martingales

Probabilistic Representations of the Riesz Transform cont’d

Projection of the Martingale Transform

TAf (x) = E((A ∗ f )0|B0 = (x , 0))

Theorem: (Banuelos and Wang) ‖TAf ‖p ≤ (p∗ − 1)‖A‖‖f ‖p for1 < p <∞.

Orthogonal Martingale Transforms

Theorem: (Banuelos and Wang) Suppose further that A(x , y)v · v = 0 forall (x , y) ∈ Rn+1

+ , v ∈ Rn+1, then

‖TAf ‖p ≤ cot

(2π

p∗

)‖A‖‖f ‖p for 1 < p <∞.


CZ Martingales

Riesz Transforms

Choose Aj = (ajl ,m),

ajl ,m =

1 l = n + 1, m = j−1 l = j , m = n + 10 otherwise

.

Then TAj f = Rj f and consequently ‖Rj f ‖ ≤ cot(

2πp∗

)‖f ‖p.


Let

B =

2 2i 02i −2 00 0 0

.

Then TB f = Bf and consequently ‖Bf ‖ ≤ 4(p∗ − 1)‖f ‖p.


CZ Martingales

∫Rn

TAf (x)g(x)dx =

∫Rn

E

(∫ 0

−∞A(Bs)∇uf (Bs) · dBs |X0 = x

)g(x)dx

= E

(∫ 0

−∞A(Bs)∇uf (Bs) · dBsug (B0)

)= E

(∫ 0

−∞A(Bs)∇uf (Bs) · dBs

∫ 0

−∞∇ug (Bs) · dBs

)= E

(∫ 0

−∞A(Bs)∇uf (Bs) · ∇ug (Bs)ds

)=

∫ ∞0

∫Rn

2yA(x , y)∇uf (x , y) · ∇ug (x , y)dxdy .

Using the fact that ∇uf (x , y) = ((∇py ) ∗ f )(x) and applying Fubini’stheorem, we see that we have

KA(x , x) =

∫ ∞0

∫Rn

2yA(x , y)∇py (x − x)∇py (x − x)dxdy .


CZ Martingales

Space-time Brownian Motion and Heat Martingales

Space-time Brownian Motion

Fix T > 0 and let (Xt)0≤t≤T be Brownian motion with initial distributiongiven by Lebesgue measure on Rn. Let

(Zt)0≤t≤T = (Xt ,T − t)0≤t≤T .

Heat martingales

For f ∈ Lp, let uf (x , t) = (ht ∗ f )(x). By Ito’s formula, (uf (Zt))0≤t≤T is amartingale and

uf (Zt) =

∫ t

0∇xuf (Zs)dXs .


CZ Martingales

Projections of Heat Martingales

Projection Operator

If A is an n × n matrix valued function, ‖A‖ <∞

STA f (x) = E

(∫ T

0A(Zt)∇xuf (Zs)dXs |XT = x)

),

SAf (x) = limT→∞

STA f (x)

Theorem: (Banuelos and Mendez) ‖SAf (x)‖p ≤ (p∗ − 1)‖A‖‖f ‖p for1 < p <∞.

Analytic Representation

SAf (x) =∫Rn K (x , x)dx where

K (x , x) =

∫ ∞0

∫Rn

A(x , t)∇xht(x − x) · ∇xht(x − x)dxdt.


CZ Martingales

Projections of Heat Martingales cont’d


Let

B =

(1 ii −1

).

Then SB f = Bf , consequently, ‖Bf ‖p ≤ 2(p∗ − 1)‖f ‖p for 1 < p <∞.


CZ Martingales

Proof of Theorem 1

K (i ,j)(x , x) =

∫ ∞0

∫Rn

2yai ,j(x , y)∂xiϕy (x − x)∂xjϕy (x − x)dxdy

Case 1: i or j = n + 1:

K (i ,j)(x , x) =

∫ ∞0

∫Rn

2yai ,j(x , y)∂xiϕy (x − x)∂xjϕy (x − x)dxdy

=

∫ ∞0

2yai ,j(y)∂xi∂xjϕ21/αy (x − x)dy

≤ ‖ai ,j‖∞K (x − x)

K (x) =

∫ ∞0

2y |∂xi∂xjϕ21/αy (x)|dy


CZ Martingales

The Proof of Theorem 1, cont’d

|K (x)| =

∫ ∞0

2y |∂xi∂xjϕ21/αy (x)|dy

=

∫ ∞0

2y |∂xi∂xjϕ21/α|x | y|x|(|x |x ′)|dy

=1

|x |n

∫ ∞0

2y |∂xi∂xjϕ21/αt(x′)|dt.


CZ Martingales

Proof of Theorem 1, cont’d

Case 2: 1 ≤ i , j ,≤ n

|K (i ,j)(x , x)| ≤ ‖ai ,j‖∞∫ ∞

0

∫Rn

2y |∂xiϕy (w)||∂xjϕy (w − (x − x))|dwdy .

K (x) =

∫Rn

∫ ∞0

2y |∂xiϕy (w)||∂xjϕy (w − x)|dwdy

=1

|x |n

∫Rn

∫ ∞0

2y |∂xiϕt(z)||∂xjϕt(z − x ′)|dzdt.


CZ Martingales

Estimates on the Derivatives

Lemma

There exists a constant Cn,α, depending only on n and α, such that for allx ∈ Rn, 1 ≤ i , j ≤ n,

|ϕ(x)| ≤ Cn,α

(1 + |x |2)(n+α)/2

|∂xiϕ(x)| ≤ Cn,α|x |(1 + |x |2)(n+2+α)/2

≤ Cn,α

(1 + |x |2)(n+1+α)/2

and

|∂xi∂xjϕ(x)| ≤ Cn,α

(1 + |x |2)(n+2+α)/2.


CZ Martingales

Subordinated Brownian Motion

Stable Subordinator

Xt = BTt ,

Bt is a standard Brownian motion and Tt is the α/2 stable subordinatorwith density ηα/2(t, ·) .

Estimates

ψt(x) =

∫ ∞0

1

(4πs)n/2e−|x |

2/4sηα/2(t, s)ds,

∂xiϕ(x) =

∫ ∞0

1

(4πs)n/2

xise−|x |

2/4sηα/2 (1, s) ds

ηα/2(t, s) ≤ Cαts−1−α/2,

|∂xiϕ(x)| ≤ Cα|x |n+1+α

∫ ∞0

u(n+α)/2e−udu.


CZ Martingales

Estimate Near 0

Fourier Inversion

ϕ(x) =

∫Rn

e−ix ·ξe−|ξ|α.

|∂xiϕ(x)| =

∣∣∣∣∫Rn

ξie−ix ·ξe−|ξ|

αdξ

∣∣∣∣=

∣∣∣∣∫Rn

ξi (e−ix ·ξ − 1)e−|ξ|

αdξ

∣∣∣∣≤∫Rn

|ξ| |e−ix ·ξ − 1|e−|ξ|αdξ

≤ 2

∫Rn

|ξ|2|x |e−|ξ|αdξ ≤ Cn,α|x |,


CZ Martingales

THANK YOU!


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A Class of Calderón-Zygmund Operators Arising …CZ Martingales A Class of Calder on-Zygmund Operators Arising from the Projections of Martingale Transforms Michael Perlmutter Department