existence and non-existence of ergodic hilbert transform

Existence and non-existence of ergodic Hilbert transform foradmissible processes

Dogan Comez

Department Colloquium, NDSU

March 29, 2016

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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