Probabilistic Approach to Analysis of Lp-operatorsweb.iku.edu.tr/ias/documents/probabilistic...

Probabilistic Approach to Analysis of Lp-operators

Deniz Karlı

Isık Universitywww.isikun.edu.tr/∼deniz.karli

Istanbul Analysis Seminars ’14

Febuary 28, 2014

Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 1 / 45

Figure : John E. Littlewood Raymond E. A. C. Paley


1 Motivation

2 Brownian Motion and Continuous Case

3 Discontinuous Case and Harnack Inequality

4 Regularity

5 Littlewood-Paley Functions


Motivation: Continuous Case

Littlewood-Paley theory was developed to study function spaces inharmonic analysis and partial differential equations (e.g. Sobolev space,Lipschitz space, Hardy space etc.).

The Littlewood-Paley function, which is a non-linear operator, allows oneto give a useful characterization of the Lp norm of a function on Rd .

Littlewood-Paley functions introduce a way to approach singular integrals.

Elias M. Stein,Topics in harmonic analysis, related to the Littlewood-Paley theory,

Princeton University (1970)

Paul-Andre Meyer,Demonstration probabiliste de certaines inegalites de Littlewood-Paley,

Seminaire de probabilites de Strasbourg, 10 (1976),Retour sur la theorie de Littlewood-Paley,

Seminaire de probabilites de Strasbourg, 15 (1981).


Motivation: Continuous Case

Littlewood-Paley theory was developed to study function spaces inharmonic analysis and partial differential equations (e.g. Sobolev space,Lipschitz space, Hardy space etc.).

The Littlewood-Paley function, which is a non-linear operator, allows oneto give a useful characterization of the Lp norm of a function on Rd .

Littlewood-Paley functions introduce a way to approach singular integrals.

Elias M. Stein,Topics in harmonic analysis, related to the Littlewood-Paley theory,

Princeton University (1970)





Brownian motion

(Ω,F ,P) : Probability space,

X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)


Brownian motion


X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)


Brownian motion


X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)


Brownian motion


X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)


Brownian motiontime t-s

time t

time s

Definition

Let (Ω,F ,P) be a probability space. A stochastic process Xt is aone-dimensional Brownian motion started at 0 if

X0 = 0, a.s.;

for all s ≤ t, Xt − Xs is a mean zero Gaussian random variable withvariance t − s ( increments are stationary, i.e. Xt − Xs ∼ Xt−s);

for all s < t, Xt − Xs is independent of σ(Xr ; r ≤ s);

with probability 1 the map t → Xt(w) is continuous.

If X 1t ,X

2t , ...,X

dt are independent one-dimensional Brownian motions, then

Xt = (X 1t ,X

2t , ...,X

dt ) is d-dimensional Brownian motion.



time t

time s

Definition


X0 = 0, a.s.;




If X 1t ,X

2t , ...,X


Xt = (X 1t ,X

2t , ...,X




time t

time s

Definition


X0 = 0, a.s.;




If X 1t ,X

2t , ...,X


Xt = (X 1t ,X

2t , ...,X




time t

time s

Definition


X0 = 0, a.s.;




If X 1t ,X

2t , ...,X


Xt = (X 1t ,X

2t , ...,X



Brownian motion

Definition

Px is the probability measure on (Ω,F) given byPx(Xt ∈ A) = P(Xt + x ∈ A), and we call the pair (Px , Xt), x ∈ Rd ,t ≥ 0, a Brownian motion started at x .

On Rd × R+

Zt

Yt

Xt=(Yt,Zt)

Yt : a d-dimensional Brownian motion,Zt : a 1-dimensional Brownian motion ,Xt = (Yt ,Zt)

T0 = inf t ≥ 0|Zt = 0 : the first time Xt hits the boundaryRd × 0 which is identified with Rd .


Brownian motion

Definition

Px is the probability measure on (Ω,F) given byPx(Xt ∈ A) = P(Xt + x ∈ A), and we call the pair (Px , Xt), x ∈ Rd ,t ≥ 0, a Brownian motion started at x .

On Rd × R+

Zt

Yt

Xt=(Yt,Zt)

Yt : a d-dimensional Brownian motion,Zt : a 1-dimensional Brownian motion ,Xt = (Yt ,Zt)

T0 = inf t ≥ 0|Zt = 0 : the first time Xt hits the boundaryRd × 0 which is identified with Rd .


Martingale

“It’s a fair game”:

Xt : Ω→ Rd for any t ≥ 0.

Define Ftt≥0 as Ft = σ(Xr : r ≤ t). ( Standart filtration.)

Xt is a martingale if E(Xt |Fs) = Xs for any s < t.

Ex: If Xt denotes our stack in a game, then the expected amount ofour stack at a later time t is Xs given that we have Xs at time s. (Noloss no win!)

Brownian motion is martingale.


Martingale








Martingale








Harmonic functions

Definition

A function u on Rd × R+ is harmonic if u(Xt∧T0) is a local martingale.

So u on Rd × R+ is harmonic if E(u(Xt∧T0)|Fs) = u(Xs∧T0) whenevers < t.


Harmonic functions

Definition




Harmonic functions

Definition




Harmonic function

Definition


Equivalent definitions:

Definition

u is harmonic if u is locally integrable and for all x ∈ Rd+ and all

r < dist(x , ∂(Rd+)),

u(x , t) =1

|B(x , r)|

∫B(x ,r)

u(y , s) dy ds.

Definition

u is harmonic if u is in C2 and ∆u = 0 in Rd+. (∆:Laplacian)


Harmonic function

Definition


Equivalent definitions:

Definition

u is harmonic if u is locally integrable and for all x ∈ Rd+ and all

r < dist(x , ∂(Rd+)),

u(x , t) =1

|B(x , r)|

∫B(x ,r)

u(y , s) dy ds.

Definition

u is harmonic if u is in C2 and ∆u = 0 in Rd+. (∆:Laplacian)


Dirichlet problem for the upper half space Rd × R+

Given a “nice” function f (such as f ∈ Lp(Rd)) defined on Rd , consideredas the boundary of Rd ×R+, can we find a “harmonic” function u(x , t) onRd × R+ such that

u(x , 0) = f (x) and limt→∞

u(x , t) = 0?


Dirichlet Problem and Littlewood-Paley functionGiven f : Rd → R, the harmonic extension to Rd × R+ is defined by

u(x , t) = E(x ,t)(f (XT0)) =

∫f (z)P(x ,t)(XT0 ∈ dz),

where P(x ,t)(XT0 ∈ dz) = Pt(x − z)dz and Pt(x − z) = cdt

(|x−z|2+t2)(d+1)/2 .

The Littlewood-Paley function Gf is defined by

Gf (x) =

[∫ ∞0

t |∇u(x , t)|2 dt

]1/2

Theorem

Suppose that f ∈ Lp(Rd). There are constants c1 and c2 such that

c1||f ||p ≤ ||Gf ||p ≤ c2||f ||p

if p ∈ (1,∞).



u(x , t) = E(x ,t)(f (XT0)) =

∫f (z)P(x ,t)(XT0 ∈ dz),


(|x−z|2+t2)(d+1)/2 .


Gf (x) =

[∫ ∞0

t |∇u(x , t)|2 dt

]1/2

Theorem


c1||f ||p ≤ ||Gf ||p ≤ c2||f ||p

if p ∈ (1,∞).



u(x , t) = E(x ,t)(f (XT0)) =

∫f (z)P(x ,t)(XT0 ∈ dz),


(|x−z|2+t2)(d+1)/2 .


Gf (x) =

[∫ ∞0

t |∇u(x , t)|2 dt

]1/2

Theorem


c1||f ||p ≤ ||Gf ||p ≤ c2||f ||p

if p ∈ (1,∞).



u(x , t) = E(x ,t)(f (XT0)) =

∫f (z)P(x ,t)(XT0 ∈ dz),


(|x−z|2+t2)(d+1)/2 .


Gf (x) =

[∫ ∞0

t |∇u(x , t)|2 dt

]1/2

Theorem


c1||f ||p ≤ ||Gf ||p ≤ c2||f ||p

if p ∈ (1,∞).


Discontinuous Case

Nicolas Bouleau and Damien LambertonTheorie de Littlewood-Paley-Stein et processus stables,

Seminaire de probabilites de Strasbourg, 20 (1986)

N. Varopoulos,Aspects of probabilistic Littlewood-Paley theory,

J. of Functional Analysis 38 (1980),





Symmetric α-Stable Process

The characteristic function of Brownian Motion Xt is

E(e iuXt ) = e−t|u|2

Consider the process (preserving “some” properties as before) whosecharacteristic equation is

E(e iuXt ) = e−t|u|α

(a) Brownian Mo-tion

(b) α = 1.9 (c) α = 1.5 (d) α = 1








(b) α = 1.9 (c) α = 1.5 (d) α = 1








(b) α = 1.9 (c) α = 1.5 (d) α = 1








(b) α = 1.9 (c) α = 1.5 (d) α = 1








(b) α = 1.9 (c) α = 1.5 (d) α = 1


Setup

Definition

Let (ΩY ,FY ,P) be probability space. A stochastic process Yt is asymmetric α-stable process if it is a Markov process with independent andstationary increments and if its characteristic function is

E(e iξYt ) = e−t|ξ|α.

Xt=(Yt,Zt)

Yt

Zt

On the upper half-space Rd × R+:

Yt : a d-dim’l symmetric α-stable process,Zt : a 1-dim’l Brownian motion.

T0 = inf t ≥ 0|Zt = 0 : the first time Xt hits theboundary Rd × 0 which is identified with Rd .

P(x ,t): probability measure corresponding Xt startingat (x , t).


Setup

Definition

Let (ΩY ,FY ,P) be probability space. A stochastic process Yt is asymmetric α-stable process if it is a Markov process with independent andstationary increments and if its characteristic function is

E(e iξYt ) = e−t|ξ|α.

Xt=(Yt,Zt)

Yt

Zt

On the upper half-space Rd × R+:

Yt : a d-dim’l symmetric α-stable process,Zt : a 1-dim’l Brownian motion.

T0 = inf t ≥ 0|Zt = 0 : the first time Xt hits theboundary Rd × 0 which is identified with Rd .

P(x ,t): probability measure corresponding Xt startingat (x , t).


Harmonic Functions & Harnack Inequality

Scaling: (Yt − Y0) ∼ 1c (Ytcα − Y0) and (Zt − Z0) ∼ 1

c (Ztc2 − Z0)

B(0,r)

Zr2tZt

B(0,1)

Rectangular box :Dr (x , t) = (y , s) ∈ Rd × R+ : |yi − xi | < rα/2/2, |s − t| < r/2.

Dr

t 1/2

Zt

Yt

1/t



Scaling: (Yt − Y0) ∼ 1c (Ytcα − Y0) and (Zt − Z0) ∼ 1

c (Ztc2 − Z0)

B(0,r)

Zr2tZt

B(0,1)

Rectangular box :Dr (x , t) = (y , s) ∈ Rd × R+ : |yi − xi | < rα/2/2, |s − t| < r/2.

Dr

t 1/2

Zt

Yt

1/t



Definition

A continuous function u is harmonic (with respect to Xt) in Rd × R+ ifu(Xt∧T0) is a local martingale with respect to P(x ,t) for any starting point(x , t) ∈ Rd × R+.

Theorem (Harnack Inequality)

There exists c > 0 such that if h is non-negative and bounded onRd × R+, harmonic in D16(x , t) and D32(x , t) ⊂ Rd × R+ then

h(y , t) ≤ c h(y ′, t ′) , (y , t), (y ′, t ′) ∈ D1(x , t).



Definition

A continuous function u is harmonic (with respect to Xt) in Rd × R+ ifu(Xt∧T0) is a local martingale with respect to P(x ,t) for any starting point(x , t) ∈ Rd × R+.



h(y , t) ≤ c h(y ′, t ′) , (y , t), (y ′, t ′) ∈ D1(x , t).


Main steps of the proof : Step I

Exit time from Dr (x , t) is comparable to r 2, that is, there exists c > 0

c−1r 2 ≤ E(y ,s)(τDr (x ,t)) ≤ cr 2

for (y , s) ∈ Dεr (x , t) and D2r (x , t) ⊂ Rd × R+.

Dr

(y,s)


Step II

Suppose (y , s) ∈ D2(x , t) and K = E × [a, b] is a rectangular box inD1(x , t) such that E ⊂ Rd . Then

P(y ,s)(TK < τD3(x ,t)) ≥ c ·m(E ) · (b − a)

for some positive constant c and D6(x , t) ⊂ Rd × R+.

D1

D2

D3

(y,s)

Ex[a,b]


D1

D2

D3

(y,s)

Ex[a,b]

P(y ,s)(TK < τD3(x ,t)) ≥ E(y ,s)

∑v≤τD3(x,t)

1IZv∈[a,b]1IYv 6=Yv−,Yv∈E

= E(y ,s)

(∫ τD3(x,t)

01IZv∈[a,b]

∫E

dz

|Yv − z |d+αdv

)

≥ c m(E )E(y ,s)

(∫ τD3(x,t)

01IZv∈[a,b]dv

).


E(y ,s)

(∫ τD3(x,t)

01IZv∈[a,b]dv

)is the average time spent by Brownian

motion in the interval [a, b] before leaving the interval [t − 3, t + 3].

E(y ,s)

(∫ τD3(x,t)

01IZv∈[a,b]dv

)≥ c(b − a)

P(y ,s)(TK < τD3(x ,t)) ≥ c m(E )E(y ,s)

(∫ τD3(x,t)

01IZv∈[a,b]dv

)≥ c m(E ) · (b − a)


Step III

Corollary

There exists a non-decreasing function ϕ : (0, 1)→ (0, 1) such that if A isa compact set inside D1 such that |A| > 0 and (y , t) ∈ D2 then

P(y ,t)(TA < τD3) ≥ ϕ(|A|).

(y,s)

AD1

D2

D3

(y,s)

We use Krylov-Safanov’s method to prove the above corollary (covering acompact set with union of cubes).




h(y , t) ≤ c h(y ′, t ′) , (y , t), (y ′, t ′) ∈ D1(x , t).

Assume infD1

h = 1/2.

Aim: The function h is bounded by a constant not depending on h.

If h(x0, t0) = K for K large enough, then define the setA′ = h > K, and take a compact subset A with positive measure.

Hitting probability of A is positive. Using an argument based onhitting to A, we find β > 1 so that h(x1, t1) > βK .

By induction and hitting A repeatedly, this results in a sequence(xn, tn) on which h is unbounded. (Contradiction)

We conclude that supD1(x ,t)

h < c for some c > 0.




h(y , t) ≤ c h(y ′, t ′) , (y , t), (y ′, t ′) ∈ D1(x , t).

Assume infD1

h = 1/2.










h(y , t) ≤ c h(y ′, t ′) , (y , t), (y ′, t ′) ∈ D1(x , t).

Assume infD1

h = 1/2.








Regularity

Theorem

If f is a bounded function on Rd × R+ and it is harmonic onD4(x , t) ⊂ Rd ×R+, then f is Holder continuous in D1(x , t), that is, thereexists γ > 0 such that

|f (y , s)− f (y ′, s ′)| ≤ c‖f ‖∞|(y , s)− (y ′, s ′)|γ , (y , s), (y ′, s ′) ∈ D1(x , t).

For r ′ > r , P(y ,s)(XτDr (y,s)∧u 6∈ Dr ′(y , s)) ≤ c1

(rr ′

)2.

Dr

Dr


Regularity

Theorem

If f is a bounded function on Rd × R+ and it is harmonic onD4(x , t) ⊂ Rd ×R+, then f is Holder continuous in D1(x , t), that is, thereexists γ > 0 such that

|f (y , s)− f (y ′, s ′)| ≤ c‖f ‖∞|(y , s)− (y ′, s ′)|γ , (y , s), (y ′, s ′) ∈ D1(x , t).

For r ′ > r , P(y ,s)(XτDr (y,s)∧u 6∈ Dr ′(y , s)) ≤ c1

(rr ′

)2.

Dr

Dr


Since f is harmonic in D4(y , s),

f (y , s)− f (y ′, s ′) = E(y ,s)[f (XτD

θk(y,s)

)− f (y ′, s ′)]

Partition the expectation into sets

XτDθk

(y,s)∈ Dθk−1−i (y , s)− Dθk−i (y , s)

|f (y , s)− f (y ′, s ′)| ≤ βk for some β. Also |(y , s)− (y ′, s ′)| ≥ (θk)2/α


Since f is harmonic in D4(y , s),

f (y , s)− f (y ′, s ′) = E(y ,s)[f (XτD

θk(y,s)

)− f (y ′, s ′)]

Partition the expectation into sets

XτDθk

(y,s)∈ Dθk−1−i (y , s)− Dθk−i (y , s)

|f (y , s)− f (y ′, s ′)| ≤ βk for some β. Also |(y , s)− (y ′, s ′)| ≥ (θk)2/α


Resolvent

Define Uλ by

Uλf (x , t) = E(x ,t)

∫ ∞0

e−λs f (Xs)ds.

Theorem

Suppose f is bounded and λ > 0. Then we have

|Uλf (y , s)− Uλf (y ′, s ′)| ≤ c ||f ||∞(|(y , s)− (y ′, s ′)| ∧ 1)γ

The argument of the proof uses Holder continiuty of the harmonicfunction

(y , s)→ E(z,u)[U0f (XτDr (y,s)

)].


Littlewood-Paley FunctionsHarmonic Extension

Let f ∈ Lp(Rd). Find a harmonic function (with respect to Xt) u(x , t)on Rd × R+ such that u(x , 0) = f (x) and u(x , t)→ 0 as t →∞.

Pt denotes the semi-group corresponding to the symmetric stableprocess Yt . (That is, Pt f (x) = Ex(f (Yt)))

Ps f (x) =

∫Rd

f (y)p(s, x , y)dy

c−1(

s−d/α ∧ s|x−y |d+α

)≤ p(s, x , y) ≤ c

(s−d/α ∧ s

|x−y |d+α

)





Ps f (x) =

∫Rd

f (y)p(s, x , y)dy

c−1(


)≤ p(s, x , y) ≤ c

(s−d/α ∧ s

|x−y |d+α

)





Ps f (x) =

∫Rd

f (y)p(s, x , y)dy

c−1(


)≤ p(s, x , y) ≤ c

(s−d/α ∧ s

|x−y |d+α

)





Ps f (x) =

∫Rd

f (y)p(s, x , y)dy

c−1(


)≤ p(s, x , y) ≤ c

(s−d/α ∧ s

|x−y |d+α

)


We define Qt =

∫ ∞0

Ps µt(ds) where µt(ds) = Pt(T0 ∈ ds).

Qt is a semi-group. [Q0 = id and QtQs = Qt+s ].

One can extend f to Rd+ by

u(x , t) = E(x ,t)(f (XT0)) =

∫ ∞0

Ps f (x)µt(ds) = Qt f (x).

(We will denote u(x , t) by f (x , t).)


We define Qt =

∫ ∞0

Ps µt(ds) where µt(ds) = Pt(T0 ∈ ds).

Qt is a semi-group. [Q0 = id and QtQs = Qt+s ].

One can extend f to Rd+ by

u(x , t) = E(x ,t)(f (XT0)) =

∫ ∞0

Ps f (x)µt(ds) = Qt f (x).

(We will denote u(x , t) by f (x , t).)


f (x , t) =

∫ ∞0

Ps f (x)µt(ds) =

∫ ∞0

∫Rd

f (y)p(s, x , y)dy µt(ds).

µt(ds) =t

2√π

e−t2/4ss−3/2ds

f (x , t) is harmonic (with respect to Xt) in Rd × R+.


Littlewood-Paley FunctionsCarre du Champ

Continuous case: Gf (x) =

[∫ ∞0

t |∇f (x , t)|2 dt

]1/2

. Here |∇f (x , t)|2

is the Carre du Champ corresponding to the Brownian motion.

Question: What is the Carre du Champ for a symmetric α-stableprocess?

In our setup: Carre du Champ Γ for the “horizontal” component is

Γ(f (t, ·), f (t, ·))(x) = c

∫[f (x + h, t)− f (x , t)]2

dh

|h|d+α.




[∫ ∞0

t |∇f (x , t)|2 dt

]1/2

. Here |∇f (x , t)|2




Γ(f (t, ·), f (t, ·))(x) = c

∫[f (x + h, t)− f (x , t)]2

dh

|h|d+α.




[∫ ∞0

t |∇f (x , t)|2 dt

]1/2

. Here |∇f (x , t)|2




Γ(f (t, ·), f (t, ·))(x) = c

∫[f (x + h, t)− f (x , t)]2

dh

|h|d+α.


Littlewood-Paley FunctionsThe General Function

Denote f (x , t) by ft(x).

Define Littlewood-Paley function Gf as G 2f = (~Gf )2 + (G ↑f )2 where

~Gf (x) =

[∫ ∞0

t

∫[ft(x + h)− ft(x)]2

dh

|h|d+αdt

]1/2

G ↑f (x) =

[∫ ∞0

t

[∂

∂tft(x)

]2

dt

]1/2


Lp − inequalities

(p=2) : ‖~Gf ‖2 = c1 ‖G ↑f ‖2 = c2 ‖f ‖2.

By P.A. Meyer , ‖Gf ‖p ≤ c ‖f ‖p for p > 2.

By N. Varopoulos, ‖G ↑f ‖p ≤ c ‖f ‖p for p > 1.

But ‖~Gf ‖p ≤ c ‖f ‖p fails for p ∈ (1, 2)!


Lp − inequalities

(p=2) : ‖~Gf ‖2 = c1 ‖G ↑f ‖2 = c2 ‖f ‖2.





Lp − inequalities

(p=2) : ‖~Gf ‖2 = c1 ‖G ↑f ‖2 = c2 ‖f ‖2.





We restrict the horizontal component to a parabolic-like domain :

C(x)

x

I C (x) is the parabolic-like domain with vertex atthe point x .

I C (x) = (h, t) ∈ Rd × R+ : |x − h| < t2/α

I−→G f ,α(x) =∫ ∞

0

t

∫|h|<t2/α

(ft(x + h)− ft(x))2 dh

|h|d/αdt.

Theorem

If p ∈ (1, 2) and f ∈ Lp(Rd) then

‖−→G f ,α‖p ≤ c ‖f ‖p.


We restrict the horizontal component to a parabolic-like domain :

C(x)

x

I C (x) is the parabolic-like domain with vertex atthe point x .

I C (x) = (h, t) ∈ Rd × R+ : |x − h| < t2/α

I−→G f ,α(x) =∫ ∞

0

t

∫|h|<t2/α


|h|d/αdt.

Theorem

If p ∈ (1, 2) and f ∈ Lp(Rd) then

‖−→G f ,α‖p ≤ c ‖f ‖p.


Connection Between Harmonic Analysis and Probability

Let Mt = f (Xt∧T0), m: Lebesque measure on Rd , εa:point mass onR+ and ma = m ⊗ εa.

Xt=(Yt,Zt)

Yt

Zt

Define the measure Pma =

∫P(x ,a) m(dx) and the expectation Ema

with respect to Pma (that is, Ema =∫Pma). Note that Pma is not a

probability measure. Then

Ema(MpT0

) = Ema(f p(XT0)) = ||f ||pp

.


Ema(f p(XT0)) =

∫E(x ,a)(f p(XT0)) dx =

∫Qaf p(x) dx

=

∫ ∫ ∞0

∫f p(y)p(s, x , y) dy µa(ds) dx

=

∫ ∞0

∫f p(y)

∫p(s, x , y) dx dy µa(ds)

=

∫ ∞0

∫f p(y)dy µa(ds)

= ||f ||pp∫ ∞

0µa(ds).


Idea of the proof of ‖−→G f ,α‖p ≤ c ‖f ‖p.:

C(x)

x

Dr

t 1/2

Zt

Yt

1/t

For any t > 0 and |h| < t2/α, ft(x) ∼ ft(x + h) by the Harnackinequality. The constant of comparison does not depend on t.

ft(x) = f ∗ qt(x) ≤ cM(f )(x) where M(f ) is the Hardy-Littlewoodmaximal function.

Hence ft(x + h) ≤ cM(f )(x) for any t > 0 and |h| < t2/α.



C(x)

x

Dr

t 1/2

Zt

Yt

1/t






C(x)

x

Dr

t 1/2

Zt

Yt

1/t





Lemma (Ito’s Lemma)

For g ∈ C2

g(Mt) = g(M0) +

∫g ′(Ms−)dMs +

1

2

∫g ′′(Ms−)d〈Mc〉s

+∑

[g(Ms)− g(Ms−)− g ′(Ms−)∆Ms ].

Take g(x) = xp and f > 0.Recall MT0 = f (XT0), and by Ito’s formula

‖f ‖pp = Ema(MpT0

)

≥ Ema

∑s≤T0

[(Ms)p − (Ms−)p − p(Ms−)p−1(Ms −Ms−)

] .


Theorem (Levy System Formula)

If g is a positive measurable function on Rd × Rd and zero on thediagonal then

Ex

∑s≤t

f (Ys−,Ys)

= Ex

[∫ t

0

∫f (Ys ,Ys + u)

du

|u|d+αds

]

for any x ∈ Rd .

By Levy system formula,

‖f ‖pp ≥ Ema

∑s≤T0

[(Ms)p − (Ms−)p − p(Ms−)p−1(Ms −Ms−)

] .

= Ema

(∫ T0

0

∫ [(f pZs

(Ys + h)− f pZs

(Ys)

−pf p−1Zs

(Ys)(fZs (Ys + h)− fZs (Ys))] dh

|h|d+αds

),


By invariance of Pt under m, and the Green Kernel of Browniancomponent∫

E(x ,a)

(∫ T0

0

∫ [(f pZs

(Ys + h)− f pZs

(Ys)

−pf p−1Zs

(Ys)(fZs (Ys + h)− fZs (Ys))] dh

|h|d+αds

)dx

≥∫

Ea

(∫ T0

0

∫ [(f pZs

(x + h)− f pZs

(x)

−pf p−1Zs

(x)(fZs (x + h)− fZs (x))] dh

|h|d+αds

)dx ,

= c

∫ ∫ ∞0

(t ∧ a)

∫ξp−2(ft(x + h)− ft(x))2 dh

|h|d+αdt dx ,

for some ξ between ft(x) and ft(x + h).


Hence after taking the limit a→∞

‖f ‖pp ≥ c

∫ ∫ ∞0

t

∫|h|<t2/α

ξp−2(ft(x + h)− ft(x))2 dh

|h|d+αdt dx ,

(say) = c

∫If (x) dx ,


Recall that

−→G 2

f ,α(x) =

∫ ∞0

t

∫|h|<t2/α


|h|d/αdt

−→G 2

f ,α(x) =

∫ ∞0

t

∫|h|<t2/α

ξ2−pξp−2(ft(x + h)− ft(x))2 dh

|h|d/αdt

≤ c M(f )2−p(x) ·∫ ∞

0t

∫|h|<t2/α

ξp−2(ft(x + h)− ft(x))2 dh

|h|d/αdt

≤ c M(f )2−p(x) · If (x)

Then the Holder inequality with the exponents 2/p and 2/(2− p) gives

‖−→G f ,α‖p ≤ c‖f ‖p


Some Futher ResultsFor f ∈ Lp(Rd), ft(·) = Qt f (·) and

Γα(ft , ft)(x) =

∫|h|<t2/α

[ft(x + h)− ft(x)]2dh

|h|d+α

Then the following operators are bounded on Lp for p > 2:

i. L∗f (x) =

[∫ ∞0

t · QtΓα(ft , ft)(x) dt

]1/2

,

ii. Af (x) =

[∫ ∞0

∫|y |<t2/α

t1−2d/αΓα(ft , ft)(x − y)dy dt

]1/2

,

iii. For Kλt (x) = t−2d/α

[t2/α

t2/α+|x |

]λd, t > 0

−→G ∗λ,f (x) =

[∫ ∞0

t · Kλt ∗ Γα(ft , ft)(x) dt

]1/2

.



Γα(ft , ft)(x) =

∫|h|<t2/α


|h|d+α


i. L∗f (x) =

[∫ ∞0


]1/2

,

ii. Af (x) =

[∫ ∞0

∫|y |<t2/α


]1/2

,


[t2/α

t2/α+|x |

]λd, t > 0

−→G ∗λ,f (x) =

[∫ ∞0


]1/2

.



Γα(ft , ft)(x) =

∫|h|<t2/α


|h|d+α


i. L∗f (x) =

[∫ ∞0


]1/2

,

ii. Af (x) =

[∫ ∞0

∫|y |<t2/α


]1/2

,


[t2/α

t2/α+|x |

]λd, t > 0

−→G ∗λ,f (x) =

[∫ ∞0


]1/2

.


Theorem (Classical Result)

Suppose κ is the kernel of a convolution operator T . If κ satisfies

|κ(x)| ≤ c |x |−d and |∇κ(x)| ≤ c |x |−d−1, x 6= 0

and the cancelation condition∫r<|x |<R κ(x)dx = 0 for all 0 < r < R then

T is bounded on Lp.

Theorem

Suppose α ∈ (1, 2), κ : Rd → R is a function with the cancelation propertysuch that

i. |κ(x)| ≤ c

|x |d1I|x |≤1 +

c

|x |d−1+α/21I|x |>1

ii. |∇κ(x)| ≤ c

|x |d+11I|x |≤1 +

c

|x |d+α/21I|x |>1.

Suppose T is a convolution operator with kernel κ. Then for f ∈ C1K

‖Tf ‖p ≤ c ‖f ‖p, p ∈ (1,∞).


Theorem (Classical Result)

Suppose κ is the kernel of a convolution operator T . If κ satisfies

|κ(x)| ≤ c |x |−d and |∇κ(x)| ≤ c |x |−d−1, x 6= 0

and the cancelation condition∫r<|x |<R κ(x)dx = 0 for all 0 < r < R then

T is bounded on Lp.

Theorem

Suppose α ∈ (1, 2), κ : Rd → R is a function with the cancelation propertysuch that

i. |κ(x)| ≤ c

|x |d1I|x |≤1 +

c

|x |d−1+α/21I|x |>1

ii. |∇κ(x)| ≤ c

|x |d+11I|x |≤1 +

c

|x |d+α/21I|x |>1.

Suppose T is a convolution operator with kernel κ. Then for f ∈ C1K

‖Tf ‖p ≤ c ‖f ‖p, p ∈ (1,∞).


Thank You

Some of these results may reached atHarnack Inequality and Regularity for a Product of SymmetricStable Process and Brownian Motion, Potential Analysis, 38,95-117 (2013),and others will be available in two papers in preparation.

References

R. F. Bass: Probabilistic Techniques in Analysis. Springer, New York (1995)

R. F. Bass and D. A. Levin: Harnack inequalities for jump processes.Potential Anal. 17, 375-388 (2002)

N. Bouleau, D. Lamberton: Theorie de Littlewood-Paley-Stein et processusstables. Seminaire de probabilites (Strasbourg) 20, 162-185 (1986)

L. Grafakos: Classical and Modern Foruier Analysis. Prentice Hall, NewJersey(2004)

P. A. Meyer: Demonstration probabiliste de certaines inegalites deLittlewood-Paley. Seminaire de probabilites (Strasbourg) 10, 164-174 (1976)


P. A. Meyer: Retour sur la theorie de Littlewood-Paley. Seminaire deprobabilites (Strasbourg) 15, 151-166 (1981)

P. A. Meyer: Un cours sur les integrales stochastiques. Seminaire deprobabilites (Strasbourg) 10, 245-400 (1976)

E. M. Stein: Singular Integrals and Differentiability Properties ofFunctions. Princeton University Press, Princeton, New Jersey (1970)

E. M. Stein: Beijing Lectures in Harmonic Analysis. PrincetonUniversity Press, Princeton, New Jersey (1986)


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