probabilistic approach to analysis of lp-operatorsweb.iku.edu.tr/ias/documents/probabilistic...
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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
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Figure : John E. Littlewood Raymond E. A. C. Paley
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1 Motivation
2 Brownian Motion and Continuous Case
3 Discontinuous Case and Harnack Inequality
4 Regularity
5 Littlewood-Paley Functions
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 3 / 45
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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).
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 4 / 45
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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).
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 4 / 45
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Brownian motion
(Ω,F ,P) : Probability space,
X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)
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Brownian motion
(Ω,F ,P) : Probability space,
X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)
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Brownian motion
(Ω,F ,P) : Probability space,
X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)
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Brownian motion
(Ω,F ,P) : Probability space,
X :[0,∞)× Ω→ R(t, ω) → X (t, ω) = Xt(ω)
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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.
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 6 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 6 / 45
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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.
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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 .
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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 .
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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.
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 8 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 8 / 45
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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.
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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.
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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.
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Harmonic function
Definition
A function u on Rd × R+ is harmonic if u(Xt∧T0) is a local martingale.
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)
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Harmonic function
Definition
A function u on Rd × R+ is harmonic if u(Xt∧T0) is a local martingale.
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)
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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?
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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,∞).
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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,∞).
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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,∞).
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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,∞).
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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),
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).
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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
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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
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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
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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
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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
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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).
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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).
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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
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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
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Harmonic Functions & Harnack Inequality
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).
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Harmonic Functions & Harnack Inequality
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).
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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)
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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]
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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
).
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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)
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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).
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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).
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.
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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).
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.
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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).
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.
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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
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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
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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
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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
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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)
)].
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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
)
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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
)
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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
)
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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
)
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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).)
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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).)
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 28 / 45
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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+.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 29 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 30 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 30 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 30 / 45
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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
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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)!
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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)!
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 32 / 45
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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)!
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 32 / 45
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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.
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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.
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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
.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 34 / 45
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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).
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 35 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 36 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 36 / 45
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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.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 36 / 45
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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−)
] .
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 37 / 45
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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
),
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 38 / 45
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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).
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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 ,
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 40 / 45
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Recall that
−→G 2
f ,α(x) =
∫ ∞0
t
∫|h|<t2/α
(ft(x + h)− ft(x))2 dh
|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
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 41 / 45
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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
.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 42 / 45
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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
.
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 42 / 45
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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
.
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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,∞).
Deniz Karlı (Isık University) Probabilistic Approach to Analysis of Lp -operators Febuary 28, 2014 43 / 45
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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,∞).
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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
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P. A. Meyer: Retour sur la theorie de Littlewood-Paley. Seminaire deprobabilites (Strasbourg) 15, 151-166 (1981)
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