Existence and non-existence of ergodic Hilbert transform ...
Transcript of Existence and non-existence of ergodic Hilbert transform ...
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Existence and non-existence of ergodic Hilbert transform foradmissible processes
Dogan Comez
Department Colloquium, NDSU
March 29, 2016
![Page 2: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/2.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane:
Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 3: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/3.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 4: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/4.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 5: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/5.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 6: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/6.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 7: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/7.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Set up
Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in theupper half plane and agrees with f (x) on the real line; that is,
∂2u∂x2 + ∂2u
∂x2 = 0, and u(x , 0) = f (x).
Solution: u(x , y) = yπ
∫∞−∞
f (s)
(x−s)2+y2 ds.
Harmonic conjugate of u(x , y) in the upper half plane is
v(x , y) =1
π
∫ ∞−∞
f (x − s)s
s2 + y2ds.
Notice: limy→0 v(x , y) = 1π
∫∞−∞
f (x−s)s
ds.
Hilbert transform of f :
Hf (x) =1
πp.v.
∫ ∞−∞
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (x − s)
sds.
Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 8: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/8.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic setting
Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s
preserves Lebesgue measure, namely, m(U−1s E) = m(E).
Hilbert transform of f can be rewritten as:
Hf (x) = limε→0
∫ε<|s|<1/ε
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (Usx)
sdt.
More generally, if τ = Ttt∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then theergodic Hilbert transform (eHt) of f is defined as
Hτ f (x) = limε→0
∫ε<|s|<1/ε
f (Tsx)
sds.
Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = T nn∈Z), the (discrete)ergodic Hilbert transform of f is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
k.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 9: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/9.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic setting
Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s
preserves Lebesgue measure, namely, m(U−1s E) = m(E).
Hilbert transform of f can be rewritten as:
Hf (x) = limε→0
∫ε<|s|<1/ε
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (Usx)
sdt.
More generally, if τ = Ttt∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then theergodic Hilbert transform (eHt) of f is defined as
Hτ f (x) = limε→0
∫ε<|s|<1/ε
f (Tsx)
sds.
Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = T nn∈Z), the (discrete)ergodic Hilbert transform of f is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
k.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 10: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/10.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic setting
Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s
preserves Lebesgue measure, namely, m(U−1s E) = m(E).
Hilbert transform of f can be rewritten as:
Hf (x) = limε→0
∫ε<|s|<1/ε
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (Usx)
sdt.
More generally, if τ = Ttt∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then theergodic Hilbert transform (eHt) of f is defined as
Hτ f (x) = limε→0
∫ε<|s|<1/ε
f (Tsx)
sds.
Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = T nn∈Z), the (discrete)ergodic Hilbert transform of f is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
k.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 11: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/11.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic setting
Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s
preserves Lebesgue measure, namely, m(U−1s E) = m(E).
Hilbert transform of f can be rewritten as:
Hf (x) = limε→0
∫ε<|s|<1/ε
f (x − s)
sds = lim
ε→0
∫ε<|s|<1/ε
f (Usx)
sdt.
More generally, if τ = Ttt∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then theergodic Hilbert transform (eHt) of f is defined as
Hτ f (x) = limε→0
∫ε<|s|<1/ε
f (Tsx)
sds.
Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = T nn∈Z), the (discrete)ergodic Hilbert transform of f is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
k.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 12: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/12.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic theorems
Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic ifT−1E = E implies µ(E) = 0 or µ(E) = 1.
Crown jewel of ergodic theory:
Birkhoff’s Ergodic Theorem. (1931)
If T : X → X is m.p.t., then, limn→∞1n
∑n−1k=0 f (T kx) = f ∗(x) exists a.e. for all
f ∈ L1. If, furthermore, T is ergodic, then f ∗ =∫f .
Continuous parameter version:
Local Ergodic Theorem. (Wiener, 1939)
If τ is a m.p. flow and f ∈ L1, then limt→0+1t
∫ t0 f (Tsx)ds = f (x) a.e.
Question: Analogous results for the (discrete or local) eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 13: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/13.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic theorems
Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic ifT−1E = E implies µ(E) = 0 or µ(E) = 1.
Crown jewel of ergodic theory:
Birkhoff’s Ergodic Theorem. (1931)
If T : X → X is m.p.t., then, limn→∞1n
∑n−1k=0 f (T kx) = f ∗(x) exists a.e. for all
f ∈ L1. If, furthermore, T is ergodic, then f ∗ =∫f .
Continuous parameter version:
Local Ergodic Theorem. (Wiener, 1939)
If τ is a m.p. flow and f ∈ L1, then limt→0+1t
∫ t0 f (Tsx)ds = f (x) a.e.
Question: Analogous results for the (discrete or local) eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 14: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/14.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic theorems
Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic ifT−1E = E implies µ(E) = 0 or µ(E) = 1.
Crown jewel of ergodic theory:
Birkhoff’s Ergodic Theorem. (1931)
If T : X → X is m.p.t., then, limn→∞1n
∑n−1k=0 f (T kx) = f ∗(x) exists a.e. for all
f ∈ L1. If, furthermore, T is ergodic, then f ∗ =∫f .
Continuous parameter version:
Local Ergodic Theorem. (Wiener, 1939)
If τ is a m.p. flow and f ∈ L1, then limt→0+1t
∫ t0 f (Tsx)ds = f (x) a.e.
Question: Analogous results for the (discrete or local) eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 15: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/15.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Ergodic theorems
Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic ifT−1E = E implies µ(E) = 0 or µ(E) = 1.
Crown jewel of ergodic theory:
Birkhoff’s Ergodic Theorem. (1931)
If T : X → X is m.p.t., then, limn→∞1n
∑n−1k=0 f (T kx) = f ∗(x) exists a.e. for all
f ∈ L1. If, furthermore, T is ergodic, then f ∗ =∫f .
Continuous parameter version:
Local Ergodic Theorem. (Wiener, 1939)
If τ is a m.p. flow and f ∈ L1, then limt→0+1t
∫ t0 f (Tsx)ds = f (x) a.e.
Question: Analogous results for the (discrete or local) eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 16: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/16.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall:
The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 17: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/17.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 18: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/18.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 19: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/19.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 20: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/20.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 21: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/21.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 22: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/22.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Existence of eHt
Recall: The (discrete/local) eHt of f ∈ Lp(X ) is
HT f (x) = limn→∞
∑0<|k|≤n
f (T kx)
kor HT f (x) = lim
ε→0
∫ε<|s|≤1/ε
f (Tsx)
sds.
M. Cotlar (1955)
If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.
K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).
R. Sato (1989) - eHt in the operator setting (discrete).
L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connectionbetween ergodic everages and eHt (discrete).
D. Comez (2006) - eHt for admissible processes (discrete).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 23: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/23.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 24: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/24.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 25: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/25.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 26: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/26.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 27: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/27.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 28: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/28.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 29: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/29.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 30: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/30.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences?
We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 31: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/31.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Results on the modulated eHt
Some other eHt results:
C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for theeHt (discrete).
M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt(local and discrete).
Corollary: limN→∞∑
0<|k|≤Nak f (T k x)
kexists a.e., where ak is a sequence
induced by a trigonometric polynomial.
Comez (2006) - There is a bounded Besicovitch sequence ak such that thelimit fails to exist.
A. Akhmedov & D. Comez, (2014)- Let Mα = a :∑n−n |ak | = O( nα−1
logα n),
1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.
There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it isuniversally good for the eHt in L2.
Question. Does eHt exist along moving averages sequences? We will return!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 32: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/32.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Admissible processes
F = ftt∈R ⊂ Lp is called a τ -admissible process on R ifTs ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.
Discrete version: F = fnn∈Z is T -admissible on Z ifTfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.
An admissible process F = ftt∈R is called
positive if ft ≥ 0 for all t ∈ R,strongly bounded if γF := supt∈R ‖ft‖p <∞,symmetric if T2t f−t = ft for all t ∈ R+.
D. Comez (2006) - If T : X → X is an i.m.p.t and F = fnn∈Z ⊂ L1 is a strongly
bounded symmetric T -admissible process, then limn∑
1≤|k|≤nfk (x)k
exists a.e.
Question. How about extending the local eHt result to the admissible processes?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 33: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/33.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Admissible processes
F = ftt∈R ⊂ Lp is called a τ -admissible process on R ifTs ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.
Discrete version: F = fnn∈Z is T -admissible on Z ifTfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.
An admissible process F = ftt∈R is called
positive if ft ≥ 0 for all t ∈ R,strongly bounded if γF := supt∈R ‖ft‖p <∞,symmetric if T2t f−t = ft for all t ∈ R+.
D. Comez (2006) - If T : X → X is an i.m.p.t and F = fnn∈Z ⊂ L1 is a strongly
bounded symmetric T -admissible process, then limn∑
1≤|k|≤nfk (x)k
exists a.e.
Question. How about extending the local eHt result to the admissible processes?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 34: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/34.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Admissible processes
F = ftt∈R ⊂ Lp is called a τ -admissible process on R ifTs ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.
Discrete version: F = fnn∈Z is T -admissible on Z ifTfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.
An admissible process F = ftt∈R is called
positive if ft ≥ 0 for all t ∈ R,strongly bounded if γF := supt∈R ‖ft‖p <∞,symmetric if T2t f−t = ft for all t ∈ R+.
D. Comez (2006) - If T : X → X is an i.m.p.t and F = fnn∈Z ⊂ L1 is a strongly
bounded symmetric T -admissible process, then limn∑
1≤|k|≤nfk (x)k
exists a.e.
Question. How about extending the local eHt result to the admissible processes?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 35: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/35.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Admissible processes
F = ftt∈R ⊂ Lp is called a τ -admissible process on R ifTs ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.
Discrete version: F = fnn∈Z is T -admissible on Z ifTfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.
An admissible process F = ftt∈R is called
positive if ft ≥ 0 for all t ∈ R,strongly bounded if γF := supt∈R ‖ft‖p <∞,symmetric if T2t f−t = ft for all t ∈ R+.
D. Comez (2006) - If T : X → X is an i.m.p.t and F = fnn∈Z ⊂ L1 is a strongly
bounded symmetric T -admissible process, then limn∑
1≤|k|≤nfk (x)k
exists a.e.
Question. How about extending the local eHt result to the admissible processes?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 36: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/36.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Admissible processes
F = ftt∈R ⊂ Lp is called a τ -admissible process on R ifTs ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.
Discrete version: F = fnn∈Z is T -admissible on Z ifTfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.
An admissible process F = ftt∈R is called
positive if ft ≥ 0 for all t ∈ R,strongly bounded if γF := supt∈R ‖ft‖p <∞,symmetric if T2t f−t = ft for all t ∈ R+.
D. Comez (2006) - If T : X → X is an i.m.p.t and F = fnn∈Z ⊂ L1 is a strongly
bounded symmetric T -admissible process, then limn∑
1≤|k|≤nfk (x)k
exists a.e.
Question. How about extending the local eHt result to the admissible processes?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 37: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/37.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-set up
Recall: Local eHt of f isHf (x) = limε→0
∫ε≤|t|≤1/ε
f (Ts x)s
ds.
Local ergodic Hilbert transform of a symmetric admissible process F = ftt∈R :
HF (x) = limε→0
∫ε≤|t|≤1/ε
fs (x)s
ds.
Let F = ft be a positive, symmetric, strongly bounded τ -admissible process. Then:
M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process g Ttsuch that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a givenadmissible F a positive process.)
∃ a family of non-negative integrable functions ut s.t. ft = Ttu|t| for all t ∈ R∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and ‖δ‖1 = supt∈R ‖ft‖1 (exact dominant of F ).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 38: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/38.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-set up
Recall: Local eHt of f isHf (x) = limε→0
∫ε≤|t|≤1/ε
f (Ts x)s
ds.
Local ergodic Hilbert transform of a symmetric admissible process F = ftt∈R :
HF (x) = limε→0
∫ε≤|t|≤1/ε
fs (x)s
ds.
Let F = ft be a positive, symmetric, strongly bounded τ -admissible process. Then:
M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process g Ttsuch that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a givenadmissible F a positive process.)
∃ a family of non-negative integrable functions ut s.t. ft = Ttu|t| for all t ∈ R∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and ‖δ‖1 = supt∈R ‖ft‖1 (exact dominant of F ).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 39: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/39.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-set up
Recall: Local eHt of f isHf (x) = limε→0
∫ε≤|t|≤1/ε
f (Ts x)s
ds.
Local ergodic Hilbert transform of a symmetric admissible process F = ftt∈R :
HF (x) = limε→0
∫ε≤|t|≤1/ε
fs (x)s
ds.
Let F = ft be a positive, symmetric, strongly bounded τ -admissible process. Then:
M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process g Ttsuch that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a givenadmissible F a positive process.)
∃ a family of non-negative integrable functions ut s.t. ft = Ttu|t| for all t ∈ R∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and ‖δ‖1 = supt∈R ‖ft‖1 (exact dominant of F ).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 40: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/40.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-set up
Recall: Local eHt of f isHf (x) = limε→0
∫ε≤|t|≤1/ε
f (Ts x)s
ds.
Local ergodic Hilbert transform of a symmetric admissible process F = ftt∈R :
HF (x) = limε→0
∫ε≤|t|≤1/ε
fs (x)s
ds.
Let F = ft be a positive, symmetric, strongly bounded τ -admissible process. Then:
M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process g Ttsuch that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a givenadmissible F a positive process.)
∃ a family of non-negative integrable functions ut s.t. ft = Ttu|t| for all t ∈ R
∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and ‖δ‖1 = supt∈R ‖ft‖1 (exact dominant of F ).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 41: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/41.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-set up
Recall: Local eHt of f isHf (x) = limε→0
∫ε≤|t|≤1/ε
f (Ts x)s
ds.
Local ergodic Hilbert transform of a symmetric admissible process F = ftt∈R :
HF (x) = limε→0
∫ε≤|t|≤1/ε
fs (x)s
ds.
Let F = ft be a positive, symmetric, strongly bounded τ -admissible process. Then:
M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process g Ttsuch that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a givenadmissible F a positive process.)
∃ a family of non-negative integrable functions ut s.t. ft = Ttu|t| for all t ∈ R∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and ‖δ‖1 = supt∈R ‖ft‖1 (exact dominant of F ).
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
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Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-results
Let τ = Ttt∈R be an i.m.p. flow and F = ft ⊂ L1 be a symmetric, stronglybounded τ -admissible process with exact dominant δ.
Theorem.1 (Maximal Inequality)
For any λ > 0, there exists a constant C such that,
µx : supq>0|∫q≤|s|≤1/q
fs(x)
sds| > λ ≤
C
λ‖δ‖1.
Theorem.2 (Local eHT for admissible processes)
HF (x) = limq→0
∫q≤|s|≤1/q
fs(x)
sds exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
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Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-results
Let τ = Ttt∈R be an i.m.p. flow and F = ft ⊂ L1 be a symmetric, stronglybounded τ -admissible process with exact dominant δ.
Theorem.1 (Maximal Inequality)
For any λ > 0, there exists a constant C such that,
µx : supq>0|∫q≤|s|≤1/q
fs(x)
sds| > λ ≤
C
λ‖δ‖1.
Theorem.2 (Local eHT for admissible processes)
HF (x) = limq→0
∫q≤|s|≤1/q
fs(x)
sds exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 44: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/44.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
LeHt for admissible processes-results
Let τ = Ttt∈R be an i.m.p. flow and F = ft ⊂ L1 be a symmetric, stronglybounded τ -admissible process with exact dominant δ.
Theorem.1 (Maximal Inequality)
For any λ > 0, there exists a constant C such that,
µx : supq>0|∫q≤|s|≤1/q
fs(x)
sds| > λ ≤
C
λ‖δ‖1.
Theorem.2 (Local eHT for admissible processes)
HF (x) = limq→0
∫q≤|s|≤1/q
fs(x)
sds exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 45: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/45.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
One-sided leHt
One-sided (ergodic) Hilbert transform.
Recall Hilbert transform: limq→0
∫q<|s|<1/q
f (s+t)s
ds exists a.e. for all f ∈ L1.
One-sided variant: limq→0
∫q<|s|≤a
f (s+t)s
ds exists a.e. for all f ∈ L1, a > 0.
Ergodic version:
(∗) Huf (x) := limq→0
∫q<|s|≤a
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Hence: existence of the local eHt and (*) implies that
(∗∗) H l f (x) := limq→0
∫a≤|s|<1/q
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 46: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/46.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
One-sided leHt
One-sided (ergodic) Hilbert transform.
Recall Hilbert transform: limq→0
∫q<|s|<1/q
f (s+t)s
ds exists a.e. for all f ∈ L1.
One-sided variant: limq→0
∫q<|s|≤a
f (s+t)s
ds exists a.e. for all f ∈ L1, a > 0.
Ergodic version:
(∗) Huf (x) := limq→0
∫q<|s|≤a
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Hence: existence of the local eHt and (*) implies that
(∗∗) H l f (x) := limq→0
∫a≤|s|<1/q
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 47: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/47.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
One-sided leHt
One-sided (ergodic) Hilbert transform.
Recall Hilbert transform: limq→0
∫q<|s|<1/q
f (s+t)s
ds exists a.e. for all f ∈ L1.
One-sided variant: limq→0
∫q<|s|≤a
f (s+t)s
ds exists a.e. for all f ∈ L1, a > 0.
Ergodic version:
(∗) Huf (x) := limq→0
∫q<|s|≤a
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Hence: existence of the local eHt and (*) implies that
(∗∗) H l f (x) := limq→0
∫a≤|s|<1/q
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 48: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/48.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
One-sided leHt
One-sided (ergodic) Hilbert transform.
Recall Hilbert transform: limq→0
∫q<|s|<1/q
f (s+t)s
ds exists a.e. for all f ∈ L1.
One-sided variant: limq→0
∫q<|s|≤a
f (s+t)s
ds exists a.e. for all f ∈ L1, a > 0.
Ergodic version:
(∗) Huf (x) := limq→0
∫q<|s|≤a
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Hence: existence of the local eHt and (*) implies that
(∗∗) H l f (x) := limq→0
∫a≤|s|<1/q
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 49: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/49.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
One-sided leHt
One-sided (ergodic) Hilbert transform.
Recall Hilbert transform: limq→0
∫q<|s|<1/q
f (s+t)s
ds exists a.e. for all f ∈ L1.
One-sided variant: limq→0
∫q<|s|≤a
f (s+t)s
ds exists a.e. for all f ∈ L1, a > 0.
Ergodic version:
(∗) Huf (x) := limq→0
∫q<|s|≤a
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Hence: existence of the local eHt and (*) implies that
(∗∗) H l f (x) := limq→0
∫a≤|s|<1/q
f (Tsx)
sds exists a.e. for all f ∈ L1, a > 0.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 50: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/50.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 51: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/51.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 52: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/52.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 53: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/53.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and
γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 54: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/54.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 55: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/55.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 56: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/56.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Sketch of proof of Theorem.2 (assuming Theorem.1):
We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for allt ∈ R.
Let G1 = Ttu0t∈R. Then G1t =
∫ t0 Tsu0ds ≤ Ft for all t ∈ R.
Fix ε > 0, and pick t0 > 0 such that ‖ut0‖1 > ‖δ‖1 − ε, and let G2 = Tt ft0.Then G2
t =∫ t
0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ‖ut0‖1 > γF − ε.
Define the process G = gt by
gt =
Ttu0 when 0 ≤ t < t0
Ttut0 when t0 ≤ t.
Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 57: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/57.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Define a new process Z = zt by zt(x) = ft(x)− Ttg(x) for all t ∈ R.
Then: 0 < zt(x) ≤ Tt(δ − g)(x), and ‖δ − g‖1 < ε.
Furthermore, Z = zt is a strongly bounded, symmetric τ -admissible process withexact dominant δ − g such that γZ = ‖δ − g‖ < ε.
Also, HqF (x) = HqZ(x) +∫q≤|s|≤t0
u0(Ts x)s
ds +∫t0≤|s|≤1/q
ut0 (Ts x)
sds.
Since the last two limits exist a.e. by (*) and (**), it remains to prove thatlimq→0 HqZ(x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 58: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/58.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Define a new process Z = zt by zt(x) = ft(x)− Ttg(x) for all t ∈ R.Then: 0 < zt(x) ≤ Tt(δ − g)(x), and ‖δ − g‖1 < ε.
Furthermore, Z = zt is a strongly bounded, symmetric τ -admissible process withexact dominant δ − g such that γZ = ‖δ − g‖ < ε.
Also, HqF (x) = HqZ(x) +∫q≤|s|≤t0
u0(Ts x)s
ds +∫t0≤|s|≤1/q
ut0 (Ts x)
sds.
Since the last two limits exist a.e. by (*) and (**), it remains to prove thatlimq→0 HqZ(x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 59: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/59.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Define a new process Z = zt by zt(x) = ft(x)− Ttg(x) for all t ∈ R.Then: 0 < zt(x) ≤ Tt(δ − g)(x), and ‖δ − g‖1 < ε.
Furthermore, Z = zt is a strongly bounded, symmetric τ -admissible process withexact dominant δ − g such that γZ = ‖δ − g‖ < ε.
Also, HqF (x) = HqZ(x) +∫q≤|s|≤t0
u0(Ts x)s
ds +∫t0≤|s|≤1/q
ut0 (Ts x)
sds.
Since the last two limits exist a.e. by (*) and (**), it remains to prove thatlimq→0 HqZ(x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 60: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/60.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Define a new process Z = zt by zt(x) = ft(x)− Ttg(x) for all t ∈ R.Then: 0 < zt(x) ≤ Tt(δ − g)(x), and ‖δ − g‖1 < ε.
Furthermore, Z = zt is a strongly bounded, symmetric τ -admissible process withexact dominant δ − g such that γZ = ‖δ − g‖ < ε.
Also, HqF (x) = HqZ(x) +∫q≤|s|≤t0
u0(Ts x)s
ds +∫t0≤|s|≤1/q
ut0 (Ts x)
sds.
Since the last two limits exist a.e. by (*) and (**), it remains to prove thatlimq→0 HqZ(x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 61: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/61.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Define a new process Z = zt by zt(x) = ft(x)− Ttg(x) for all t ∈ R.Then: 0 < zt(x) ≤ Tt(δ − g)(x), and ‖δ − g‖1 < ε.
Furthermore, Z = zt is a strongly bounded, symmetric τ -admissible process withexact dominant δ − g such that γZ = ‖δ − g‖ < ε.
Also, HqF (x) = HqZ(x) +∫q≤|s|≤t0
u0(Ts x)s
ds +∫t0≤|s|≤1/q
ut0 (Ts x)
sds.
Since the last two limits exist a.e. by (*) and (**), it remains to prove thatlimq→0 HqZ(x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 62: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/62.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Let E = x : lim supq HqF (x)− lim infq HqF (x) > λ, then
E ⊂ x : supq|HqZ(x)| >
λ
2.
From the maximal inequality,
µ(E) ≤C
λ‖δ − gε‖1 < ε.
Since ε > 0 is arbitrary, so µ(E) = 0.
Thus HF (x) = limq→0 HqF (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 63: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/63.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Let E = x : lim supq HqF (x)− lim infq HqF (x) > λ, then
E ⊂ x : supq|HqZ(x)| >
λ
2.
From the maximal inequality,
µ(E) ≤C
λ‖δ − gε‖1 < ε.
Since ε > 0 is arbitrary, so µ(E) = 0.
Thus HF (x) = limq→0 HqF (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 64: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/64.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Sketch of proof
Let E = x : lim supq HqF (x)− lim infq HqF (x) > λ, then
E ⊂ x : supq|HqZ(x)| >
λ
2.
From the maximal inequality,
µ(E) ≤C
λ‖δ − gε‖1 < ε.
Since ε > 0 is arbitrary, so µ(E) = 0.
Thus HF (x) = limq→0 HqF (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 65: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/65.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 66: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/66.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.
Examples: (n, n) and (22n ,√
22n ) satisfy CC, but (n,√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 67: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/67.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 68: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/68.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 69: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/69.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 70: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/70.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 71: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/71.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 72: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/72.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving averages
Moving averages sequences.
A sequence w = (vn, rn) in Z× Z+, where rn > 0, satisfies the cone condition (CC)if ∃ C such that ∀ s ∈ R+, |Ωα(s)| ≤ Cs, where
Ωα = (t, s) : ∃n, |t − vn| ≤ α(s − rn), and Ωα(s) = x ∈ R : (x , s) ∈ Ωα.Examples: (n, n) and (22n ,
√22n ) satisfy CC, but (n,
√n) does not.
M.A. Akcoglu and A. delJunco (1975): limn1√n
∑√nk=0 f (T n+kx) fails to exist a.e.
Notice: (n,√n) does not satisfy cone condition.
A. Bellow, R. Jones and J. Rosenblatt (1990): (vn, rn) satisfy CC is necessaryand sufficient for the moving average 1
rn
∑rnk=0 f (T vn+kx) to converge a.e.
S. Ferrando (1995): extended it to superadditive processes setting.
Question. How about the analogous results for the eHt?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 73: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/73.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving eHt
Let T : X → X be an i.m.p.t. and w = (vn, rn). Define the moving eHt of f by
HTw f = limn
∑1≤|k|≤rn
f (T vn+k x)k
, if exists.
As usual, one needs the maximal inequality and the a.e. existence of HTw f for all f in a
dense subset of Lp .
Maximal inequality is OK:For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995).For (discrete) admissible processes (Comez, 2015).
Convergence on a dense subset of Lp?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 74: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/74.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving eHt
Let T : X → X be an i.m.p.t. and w = (vn, rn). Define the moving eHt of f by
HTw f = limn
∑1≤|k|≤rn
f (T vn+k x)k
, if exists.
As usual, one needs the maximal inequality and the a.e. existence of HTw f for all f in a
dense subset of Lp .
Maximal inequality is OK:For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995).For (discrete) admissible processes (Comez, 2015).
Convergence on a dense subset of Lp?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 75: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/75.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving eHt
Let T : X → X be an i.m.p.t. and w = (vn, rn). Define the moving eHt of f by
HTw f = limn
∑1≤|k|≤rn
f (T vn+k x)k
, if exists.
As usual, one needs the maximal inequality and the a.e. existence of HTw f for all f in a
dense subset of Lp .
Maximal inequality is OK:For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995).For (discrete) admissible processes (Comez, 2015).
Convergence on a dense subset of Lp?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 76: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/76.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Moving eHt
Let T : X → X be an i.m.p.t. and w = (vn, rn). Define the moving eHt of f by
HTw f = limn
∑1≤|k|≤rn
f (T vn+k x)k
, if exists.
As usual, one needs the maximal inequality and the a.e. existence of HTw f for all f in a
dense subset of Lp .
Maximal inequality is OK:For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995).For (discrete) admissible processes (Comez, 2015).
Convergence on a dense subset of Lp?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 77: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/77.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2,
andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 78: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/78.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 79: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/79.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n).
w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 80: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/80.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 81: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/81.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences:
w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 82: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/82.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3,
w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 83: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/83.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 84: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/84.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 85: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/85.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Example.Let (X ,F , µ,T ), where X = 0, 1, 2, F = P(X ), µ(i) = 1/3, 0 ≤ i ≤ 2, andT (0) = 1, T (1) = 2, T (2) = 0.
Let w = (n, n). w satisfies the cone condition and vn = n = rn →∞.
Consider two subsequences: w1 consists of those that are multiples of 3, w2 consistsof those that are of the form n = 3k + 1.
Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.
Hf (1) =∑k 6=0
f (T k1)
k= 0,
Hf (0) =∑k 6=0
f (T k0)
k= (1−
1
2) + (
1
4−
1
5) + (
1
7−
1
8) + · · · 6= 0.
.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 86: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/86.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 87: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/87.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k
=∑
0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 88: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/88.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k
→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 89: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/89.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 90: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/90.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k
=∑
0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 91: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/91.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 92: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/92.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k
→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 93: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/93.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 94: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/94.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 95: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/95.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Example
Along w1,
Hw1 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T k0)
k→ Hf (0) 6= 0.
Along w2,
Hw2 f (0) =∑
0<|k|≤rn
f (T vn+k0)
k=
∑0<|k|≤rn
f (T 1+k0)
k
=∑
0<|k|≤rn
f (T k1)
k→ Hf (1) = 0.
Hence limn H(vn,rn)f (0) does not exist!
Question. Is it possible to have an affirmative answer under some additionalconditions?
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 96: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/96.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Some special dynamical systems
Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamicalsystem; positive results are possible if one considers:
restricted classes of functions, or
some special classes of dynamical systems, or
further restrictions of the sequence (vn, rn).
T has Lebesgue spectrum if L02(X ) = ⊕jHj , where Hj = spanT k fj for some fj ∈ L0
2
such that < fj ,Tk fj >= 0 if k 6= 0.
Theorem (Comez, 2016) Let (vn, rn) satisfy CC. If T has Lebesgue spectrum, then,for all f ∈ L0
2, limn H(vn,rn)f (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 97: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/97.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Some special dynamical systems
Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamicalsystem; positive results are possible if one considers:
restricted classes of functions, or
some special classes of dynamical systems, or
further restrictions of the sequence (vn, rn).
T has Lebesgue spectrum if L02(X ) = ⊕jHj , where Hj = spanT k fj for some fj ∈ L0
2
such that < fj ,Tk fj >= 0 if k 6= 0.
Theorem (Comez, 2016) Let (vn, rn) satisfy CC. If T has Lebesgue spectrum, then,for all f ∈ L0
2, limn H(vn,rn)f (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 98: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/98.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Some special dynamical systems
Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamicalsystem; positive results are possible if one considers:
restricted classes of functions, or
some special classes of dynamical systems, or
further restrictions of the sequence (vn, rn).
T has Lebesgue spectrum if L02(X ) = ⊕jHj , where Hj = spanT k fj for some fj ∈ L0
2
such that < fj ,Tk fj >= 0 if k 6= 0.
Theorem (Comez, 2016) Let (vn, rn) satisfy CC. If T has Lebesgue spectrum, then,for all f ∈ L0
2, limn H(vn,rn)f (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 99: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/99.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Some special dynamical systems
Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamicalsystem; positive results are possible if one considers:
restricted classes of functions, or
some special classes of dynamical systems, or
further restrictions of the sequence (vn, rn).
T has Lebesgue spectrum if L02(X ) = ⊕jHj , where Hj = spanT k fj for some fj ∈ L0
2
such that < fj ,Tk fj >= 0 if k 6= 0.
Theorem (Comez, 2016) Let (vn, rn) satisfy CC. If T has Lebesgue spectrum, then,for all f ∈ L0
2, limn H(vn,rn)f (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 100: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/100.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
Some special dynamical systems
Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamicalsystem; positive results are possible if one considers:
restricted classes of functions, or
some special classes of dynamical systems, or
further restrictions of the sequence (vn, rn).
T has Lebesgue spectrum if L02(X ) = ⊕jHj , where Hj = spanT k fj for some fj ∈ L0
2
such that < fj ,Tk fj >= 0 if k 6= 0.
Theorem (Comez, 2016) Let (vn, rn) satisfy CC. If T has Lebesgue spectrum, then,for all f ∈ L0
2, limn H(vn,rn)f (x) exists a.e.
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes
![Page 101: Existence and non-existence of ergodic Hilbert transform ...](https://reader036.fdocument.pub/reader036/viewer/2022070804/62c55491ed82036923649da9/html5/thumbnails/101.jpg)
Set upHistory.
Admissible processes.Local eHt for admissible processes.
Sketch of the proofeHt along moving averages sequences
THANK YOU!
Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes