Localization and projections on bi–parameter BMO
R. Lechner
P. F. X. Müller
J. Kepler University, Linz
Warwick, June, 2015
Overview
1 A description of the problem class
2 One dimension
3 Two dimensions
Overview
1 A description of the problem class
2 One dimension
3 Two dimensions
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
A description of the problem class
• Let X be a Banach space
• T : X → X a linear operator
Find conditions on T and X such that the identity on X factors through T
(or Id −T ), i.e.
X
E
Id
P
X
/X
O
T
kEkkP k ≤ C.
/X
• The problem has finite dimensional (quantitative) and infinite
dimensional (qualitative) aspects.
• Classical examples include: `p (Pelczynski), C([0, 1])
(Lindenstrauss-Pelczynski), Lp (Gamlen-Gaudet), L1 (Enflo-Starbird),
`pn (Bourgain-Tzafriri).
• Generically, we expect that T has a large diagonal with respect to an
unconditional basis in X.
• We consider special cases where X = H p (H q ) with the bi-parameter
Haar system as its unconditional basis.
Overview
1 A description of the problem class
2 One dimension
3 Two dimensions
Dyadic H p
k
• D = {[ k−1
2n , 2n [ : k ≥ 0, n ≥ 0} denotes the dyadic intervals on the unit
interval,
• hI the L∞ -normalized Haar function, I ∈ D.
• Let f =
P
I∈D
aI hI be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
a2I h2I
I∈D
1/2
.
• The norm of the one-parameter Hardy space H p , 1 ≤ p < ∞ is defined
by
kf k
Hp
= k S(f )k
Lp
=
Z
0
1
X
I∈D
a2I h2I (x)
p/2
dx
1/p
.
Dyadic H p
k
• D = {[ k−1
2n , 2n [ : k ≥ 0, n ≥ 0} denotes the dyadic intervals on the unit
interval,
• hI the L∞ -normalized Haar function, I ∈ D.
• Let f =
P
I∈D
aI hI be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
a2I h2I
I∈D
1/2
.
• The norm of the one-parameter Hardy space H p , 1 ≤ p < ∞ is defined
by
kf k
Hp
= k S(f )k
Lp
=
Z
0
1
X
I∈D
a2I h2I (x)
p/2
dx
1/p
.
Dyadic H p
k
• D = {[ k−1
2n , 2n [ : k ≥ 0, n ≥ 0} denotes the dyadic intervals on the unit
interval,
• hI the L∞ -normalized Haar function, I ∈ D.
• Let f =
P
I∈D
aI hI be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
a2I h2I
I∈D
1/2
.
• The norm of the one-parameter Hardy space H p , 1 ≤ p < ∞ is defined
by
kf k
Hp
= k S(f )k
Lp
=
Z
0
1
X
I∈D
a2I h2I (x)
p/2
dx
1/p
.
Dyadic H p
k
• D = {[ k−1
2n , 2n [ : k ≥ 0, n ≥ 0} denotes the dyadic intervals on the unit
interval,
• hI the L∞ -normalized Haar function, I ∈ D.
• Let f =
P
I∈D
aI hI be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
a2I h2I
I∈D
1/2
.
• The norm of the one-parameter Hardy space H p , 1 ≤ p < ∞ is defined
by
kf k
Hp
= k S(f )k
Lp
=
Z
0
1
X
I∈D
a2I h2I (x)
p/2
dx
1/p
.
Dyadic H p
k
• D = {[ k−1
2n , 2n [ : k ≥ 0, n ≥ 0} denotes the dyadic intervals on the unit
interval,
• hI the L∞ -normalized Haar function, I ∈ D.
• Let f =
P
I∈D
aI hI be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
a2I h2I
I∈D
1/2
.
• The norm of the one-parameter Hardy space H p , 1 ≤ p < ∞ is defined
by
kf k
Hp
= k S(f )k
Lp
=
Z
0
1
X
I∈D
a2I h2I (x)
p/2
dx
1/p
.
Andrew 1D
Theorem
Let 1 < p < ∞, δ > 0 and T : H p → H p be a linear operator with large
diagonal, i.e. hT hI , hI i ≥ δ|I|. Then we have
Hp
E
Id
kEkkP k ≤ C/δ,
P
Hp
/ Hp
O
T
/ Hp
where the constant C > 0 is universal.
By Gamlen-Gaudet construction (BI ) and a random choice of signs εI there
P
(ε)
exists a block basis bI = K∈BI εK hK of the Haar system such that
(ε)
(ε)
T bI = αI bI + small error,
The orthogonal projection Qf =
(Gamlen-Gaudet).
P
(ε)
hf,bI i
I∈D kb(ε) k2 bI
2
I
αI ≥ δ.
is bounded on H p
Andrew 1D
Theorem
Let 1 < p < ∞, δ > 0 and T : H p → H p be a linear operator with large
diagonal, i.e. hT hI , hI i ≥ δ|I|. Then we have
Hp
E
Id
kEkkP k ≤ C/δ,
P
Hp
/ Hp
O
T
/ Hp
where the constant C > 0 is universal.
By Gamlen-Gaudet construction (BI ) and a random choice of signs εI there
P
(ε)
exists a block basis bI = K∈BI εK hK of the Haar system such that
(ε)
(ε)
T bI = αI bI + small error,
The orthogonal projection Qf =
(Gamlen-Gaudet).
P
(ε)
hf,bI i
I∈D kb(ε) k2 bI
2
I
αI ≥ δ.
is bounded on H p
Andrew 1D
Theorem
Let 1 < p < ∞, δ > 0 and T : H p → H p be a linear operator with large
diagonal, i.e. hT hI , hI i ≥ δ|I|. Then we have
Hp
E
Id
kEkkP k ≤ C/δ,
P
Hp
/ Hp
O
T
/ Hp
where the constant C > 0 is universal.
By Gamlen-Gaudet construction (BI ) and a random choice of signs εI there
P
(ε)
exists a block basis bI = K∈BI εK hK of the Haar system such that
(ε)
(ε)
T bI = αI bI + small error,
The orthogonal projection Qf =
(Gamlen-Gaudet).
P
(ε)
hf,bI i
I∈D kb(ε) k2 bI
2
I
αI ≥ δ.
is bounded on H p
Andrew 1D
Theorem
Let 1 < p < ∞, δ > 0 and T : H p → H p be a linear operator with large
diagonal, i.e. hT hI , hI i ≥ δ|I|. Then we have
Hp
E
Id
kEkkP k ≤ C/δ,
P
Hp
/ Hp
O
T
/ Hp
where the constant C > 0 is universal.
By Gamlen-Gaudet construction (BI ) and a random choice of signs εI there
P
(ε)
exists a block basis bI = K∈BI εK hK of the Haar system such that
(ε)
(ε)
T bI = αI bI + small error,
The orthogonal projection Qf =
(Gamlen-Gaudet).
P
(ε)
hf,bI i
I∈D kb(ε) k2 bI
2
I
αI ≥ δ.
is bounded on H p
Andrew 1D
Theorem
Let 1 < p < ∞, δ > 0 and T : H p → H p be a linear operator with large
diagonal, i.e. hT hI , hI i ≥ δ|I|. Then we have
Hp
E
Id
kEkkP k ≤ C/δ,
P
Hp
/ Hp
O
T
/ Hp
where the constant C > 0 is universal.
By Gamlen-Gaudet construction (BI ) and a random choice of signs εI there
P
(ε)
exists a block basis bI = K∈BI εK hK of the Haar system such that
(ε)
(ε)
T bI = αI bI + small error,
The orthogonal projection Qf =
(Gamlen-Gaudet).
P
(ε)
hf,bI i
I∈D kb(ε) k2 bI
2
I
αI ≥ δ.
is bounded on H p
The Gamlen-Gaudet construction
The Gamlen-Gaudet construction
The Gamlen-Gaudet construction
The Gamlen-Gaudet construction
The Gamlen-Gaudet construction
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Comments on Andrew 1D
• Andrew precedes the study of operators with property A
(Johnson-Maurey-Schechtman-Tzafriri).
• The Rademacher system converges weakly to 0 in H p .
• The operator T is preconditioned by multiplying the Haar system with
highly oscillating Rademacher functions.
(ε)
(ε)
• This gives that hT bI , bJ i ≈ 0, if I 6= J.
• The second part consists of choosing signs εI such that
(ε)
(ε)
(ε)
hT bI , bI i ≥ δkbI k22 .
(ε)
• The block basis {bI
: I ∈ D} is equivalent to the Haar system.
• Andrew’s method of proof used a semi-random choice of signs εI , hence
it is strictly limited to the range 1 < p < ∞.
Overview
1 A description of the problem class
2 One dimension
3 Two dimensions
Mixed-norm Hardy spaces H p (H q )
• R = {I × J : I, J ∈ D} denotes the dyadic rectangles on the unit
square,
• hI×J (x, y) = hI (x)hJ (y) the L∞ -normalized tensor product Haar
function, I × J ∈ R.
P
• Let f = I×J∈R aI×J hI×J be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
I×J∈R
a2I×J h2I×J
1/2
.
• we define the norm of the bi-parameter Hardy spaces H p (H q ),
1 ≤ p, q < ∞ by
kf kH p (H q ) =
Z
0
1Z 1
0
X
I∈D
a2I×J h2I×J (x, y)
q/2
dy
p/q
dx
1/p
.
Mixed-norm Hardy spaces H p (H q )
• R = {I × J : I, J ∈ D} denotes the dyadic rectangles on the unit
square,
• hI×J (x, y) = hI (x)hJ (y) the L∞ -normalized tensor product Haar
function, I × J ∈ R.
P
• Let f = I×J∈R aI×J hI×J be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
I×J∈R
a2I×J h2I×J
1/2
.
• we define the norm of the bi-parameter Hardy spaces H p (H q ),
1 ≤ p, q < ∞ by
kf kH p (H q ) =
Z
0
1Z 1
0
X
I∈D
a2I×J h2I×J (x, y)
q/2
dy
p/q
dx
1/p
.
Mixed-norm Hardy spaces H p (H q )
• R = {I × J : I, J ∈ D} denotes the dyadic rectangles on the unit
square,
• hI×J (x, y) = hI (x)hJ (y) the L∞ -normalized tensor product Haar
function, I × J ∈ R.
P
• Let f = I×J∈R aI×J hI×J be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
I×J∈R
a2I×J h2I×J
1/2
.
• we define the norm of the bi-parameter Hardy spaces H p (H q ),
1 ≤ p, q < ∞ by
kf kH p (H q ) =
Z
0
1Z 1
0
X
I∈D
a2I×J h2I×J (x, y)
q/2
dy
p/q
dx
1/p
.
Mixed-norm Hardy spaces H p (H q )
• R = {I × J : I, J ∈ D} denotes the dyadic rectangles on the unit
square,
• hI×J (x, y) = hI (x)hJ (y) the L∞ -normalized tensor product Haar
function, I × J ∈ R.
P
• Let f = I×J∈R aI×J hI×J be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
I×J∈R
a2I×J h2I×J
1/2
.
• we define the norm of the bi-parameter Hardy spaces H p (H q ),
1 ≤ p, q < ∞ by
kf kH p (H q ) =
Z
0
1Z 1
0
X
I∈D
q/2 p/q 1/p
a2I×J h2I×J (x, y)
dy
dx
.
Mixed-norm Hardy spaces H p (H q )
• R = {I × J : I, J ∈ D} denotes the dyadic rectangles on the unit
square,
• hI×J (x, y) = hI (x)hJ (y) the L∞ -normalized tensor product Haar
function, I × J ∈ R.
P
• Let f = I×J∈R aI×J hI×J be a finite linear combination,
• then the square function S(f ) of f is given by
S(f ) =
X
I×J∈R
a2I×J h2I×J
1/2
.
• we define the norm of the bi-parameter Hardy spaces H p (H q ),
1 ≤ p, q < ∞ by
kf kH p (H q ) =
Z
0
1Z 1
0
X
I∈D
q/2 p/q 1/p
a2I×J h2I×J (x, y)
dy
dx
.
Capon’s theorem: H p (H q ) is primary (augmented)
Theorem
Let 1 < p, q < ∞ or p = q = 1. For any operator T the identity on H p (H q )
factors through H = T or H = Id −T , i.e.
Id
H p (H q )
E
/ H p (H q )
O
P
H p (H q )
H
kEkkP k ≤ C,
/ H p (H q )
where C = C(kT k).
Capon invents a specific bi-parameterPGamlen-Gaudet selection process
which leads to a block basis bI×J = K×L∈BI×J hK×L such that
HbI×J = αI×J bI×J + small error,
and the projection Qf =
P
hf,bI×J i
I×J∈R kbI×J k22 bI×J
αI×J ≥ 1/2.
is bounded on H p (H q ).
Capon’s theorem: H p (H q ) is primary (augmented)
Theorem
Let 1 < p, q < ∞ or p = q = 1. For any operator T the identity on H p (H q )
factors through H = T or H = Id −T , i.e.
Id
H p (H q )
E
/ H p (H q )
O
P
H p (H q )
H
kEkkP k ≤ C,
/ H p (H q )
where C = C(kT k).
Capon invents a specific bi-parameterPGamlen-Gaudet selection process
which leads to a block basis bI×J = K×L∈BI×J hK×L such that
HbI×J = αI×J bI×J + small error,
and the projection Qf =
P
hf,bI×J i
I×J∈R kbI×J k22 bI×J
αI×J ≥ 1/2.
is bounded on H p (H q ).
Capon’s theorem: H p (H q ) is primary (augmented)
Theorem
Let 1 < p, q < ∞ or p = q = 1. For any operator T the identity on H p (H q )
factors through H = T or H = Id −T , i.e.
Id
H p (H q )
E
/ H p (H q )
O
P
H p (H q )
H
kEkkP k ≤ C,
/ H p (H q )
where C = C(kT k).
Capon invents a specific bi-parameterPGamlen-Gaudet selection process
which leads to a block basis bI×J = K×L∈BI×J hK×L such that
HbI×J = αI×J bI×J + small error,
and the projection Qf =
P
hf,bI×J i
I×J∈R kbI×J k22 bI×J
αI×J ≥ 1/2.
is bounded on H p (H q ).
Capon’s theorem: H p (H q ) is primary (augmented)
Theorem
Let 1 < p, q < ∞ or p = q = 1. For any operator T the identity on H p (H q )
factors through H = T or H = Id −T , i.e.
Id
H p (H q )
E
/ H p (H q )
O
P
H p (H q )
H
kEkkP k ≤ C,
/ H p (H q )
where C = C(kT k).
Capon invents a specific bi-parameterPGamlen-Gaudet selection process
which leads to a block basis bI×J = K×L∈BI×J hK×L such that
HbI×J = αI×J bI×J + small error,
and the projection Qf =
P
hf,bI×J i
I×J∈R kbI×J k22 bI×J
αI×J ≥ 1/2.
is bounded on H p (H q ).
Capon’s theorem: H p (H q ) is primary (augmented)
Theorem
Let 1 < p, q < ∞ or p = q = 1. For any operator T the identity on H p (H q )
factors through H = T or H = Id −T , i.e.
Id
H p (H q )
E
/ H p (H q )
O
P
H p (H q )
H
kEkkP k ≤ C,
/ H p (H q )
where C = C(kT k).
Capon invents a specific bi-parameterPGamlen-Gaudet selection process
which leads to a block basis bI×J = K×L∈BI×J hK×L such that
HbI×J = αI×J bI×J + small error,
and the projection Qf =
P
hf,bI×J i
I×J∈R kbI×J k22 bI×J
αI×J ≥ 1/2.
is bounded on H p (H q ).
Localization in H 1 (H 1 )
1 (H 1 ) = span{h
−N } ⊂ H 1 (H 1 ).
HN
I×J : |I|, |J| ≥ 2
N
Theorem (R. L. & P. F. X. M.)
For each n ∈ N there exists N (n) ∈ N, so that for each bounded linear
1 (H 1 ) → H 1 (H 1 ) we have
operator T : HN
N
N
N
Hn1 (Hn1 )
E
Id
P
1 (H 1 )
HN
N
/ H 1 (H 1 )
n O n
H
kEkkP k ≤ C,
/ H 1 (H 1 )
N
N
where H = T or H = Id −T , and C = C(kT k).
• We need quantitative, finite dimensional estimates.
• This leads to a combinatorial problem for coloured dyadic rectangles.
Localization in H 1 (H 1 )
1 (H 1 ) = span{h
−N } ⊂ H 1 (H 1 ).
HN
I×J : |I|, |J| ≥ 2
N
Theorem (R. L. & P. F. X. M.)
For each n ∈ N there exists N (n) ∈ N, so that for each bounded linear
1 (H 1 ) → H 1 (H 1 ) we have
operator T : HN
N
N
N
Hn1 (Hn1 )
E
Id
P
1 (H 1 )
HN
N
/ H 1 (H 1 )
n O n
H
kEkkP k ≤ C,
/ H 1 (H 1 )
N
N
where H = T or H = Id −T , and C = C(kT k).
• We need quantitative, finite dimensional estimates.
• This leads to a combinatorial problem for coloured dyadic rectangles.
Localization in H 1 (H 1 )
1 (H 1 ) = span{h
−N } ⊂ H 1 (H 1 ).
HN
I×J : |I|, |J| ≥ 2
N
Theorem (R. L. & P. F. X. M.)
For each n ∈ N there exists N (n) ∈ N, so that for each bounded linear
1 (H 1 ) → H 1 (H 1 ) we have
operator T : HN
N
N
N
Hn1 (Hn1 )
E
Id
P
1 (H 1 )
HN
N
/ H 1 (H 1 )
n O n
H
kEkkP k ≤ C,
/ H 1 (H 1 )
N
N
where H = T or H = Id −T , and C = C(kT k).
• We need quantitative, finite dimensional estimates.
• This leads to a combinatorial problem for coloured dyadic rectangles.
Localization in H 1 (H 1 )
1 (H 1 ) = span{h
−N } ⊂ H 1 (H 1 ).
HN
I×J : |I|, |J| ≥ 2
N
Theorem (R. L. & P. F. X. M.)
For each n ∈ N there exists N (n) ∈ N, so that for each bounded linear
1 (H 1 ) → H 1 (H 1 ) we have
operator T : HN
N
N
N
Hn1 (Hn1 )
E
Id
P
1 (H 1 )
HN
N
/ H 1 (H 1 )
n O n
H
kEkkP k ≤ C,
/ H 1 (H 1 )
N
N
where H = T or H = Id −T , and C = C(kT k).
• We need quantitative, finite dimensional estimates.
• This leads to a combinatorial problem for coloured dyadic rectangles.
Localization in H 1 (H 1 )
1 (H 1 ) = span{h
−N } ⊂ H 1 (H 1 ).
HN
I×J : |I|, |J| ≥ 2
N
Theorem (R. L. & P. F. X. M.)
For each n ∈ N there exists N (n) ∈ N, so that for each bounded linear
1 (H 1 ) → H 1 (H 1 ) we have
operator T : HN
N
N
N
Hn1 (Hn1 )
E
Id
P
1 (H 1 )
HN
N
/ H 1 (H 1 )
n O n
H
kEkkP k ≤ C,
/ H 1 (H 1 )
N
N
where H = T or H = Id −T , and C = C(kT k).
• We need quantitative, finite dimensional estimates.
• This leads to a combinatorial problem for coloured dyadic rectangles.
We aim for
a block basis bI×J =
P
K×L∈BI×L
1 (H 1 ) such that
hK×L ∈ HN
N
• it well spans a copy of Hn1 (Hn1 ),
• HbI×J = αI×J bI×J + small error,
αI×J ≥ 1/2,
p
q
• and Q is bounded on HN
(HN
) with universal constants, where
Qf =
X hf, bI×J i
bI×J .
kbI×J k22
I×J∈R
To get there we use
a quantitative, finite dimensional replacement for
hx, rm ⊗ h[0,1] i → 0
where {rm } is the Rademacher system.
as m → ∞,
We aim for
a block basis bI×J =
P
K×L∈BI×L
1 (H 1 ) such that
hK×L ∈ HN
N
• it well spans a copy of Hn1 (Hn1 ),
• HbI×J = αI×J bI×J + small error,
αI×J ≥ 1/2,
p
q
• and Q is bounded on HN
(HN
) with universal constants, where
Qf =
X hf, bI×J i
bI×J .
kbI×J k22
I×J∈R
To get there we use
a quantitative, finite dimensional replacement for
hx, rm ⊗ h[0,1] i → 0
where {rm } is the Rademacher system.
as m → ∞,
We aim for
a block basis bI×J =
P
K×L∈BI×L
1 (H 1 ) such that
hK×L ∈ HN
N
• it well spans a copy of Hn1 (Hn1 ),
• HbI×J = αI×J bI×J + small error,
αI×J ≥ 1/2,
p
q
• and Q is bounded on HN
(HN
) with universal constants, where
Qf =
X hf, bI×J i
bI×J .
kbI×J k22
I×J∈R
To get there we use
a quantitative, finite dimensional replacement for
hx, rm ⊗ h[0,1] i → 0
where {rm } is the Rademacher system.
as m → ∞,
We aim for
a block basis bI×J =
P
K×L∈BI×L
1 (H 1 ) such that
hK×L ∈ HN
N
• it well spans a copy of Hn1 (Hn1 ),
• HbI×J = αI×J bI×J + small error,
αI×J ≥ 1/2,
p
q
• and Q is bounded on HN
(HN
) with universal constants, where
Qf =
X hf, bI×J i
bI×J .
kbI×J k22
I×J∈R
To get there we use
a quantitative, finite dimensional replacement for
hx, rm ⊗ h[0,1] i → 0
where {rm } is the Rademacher system.
as m → ∞,
We aim for
a block basis bI×J =
P
K×L∈BI×L
1 (H 1 ) such that
hK×L ∈ HN
N
• it well spans a copy of Hn1 (Hn1 ),
• HbI×J = αI×J bI×J + small error,
αI×J ≥ 1/2,
p
q
• and Q is bounded on HN
(HN
) with universal constants, where
Qf =
X hf, bI×J i
bI×J .
kbI×J k22
I×J∈R
To get there we use
a quantitative, finite dimensional replacement for
hx, rm ⊗ h[0,1] i → 0
where {rm } is the Rademacher system.
as m → ∞,
Combinatorial covering lemma
Let τ, δ > 0,
x ∈ H 1 (H 1 )
and
kxk ≤ 1.
There exists a large collection of pairwise disjoint dyadic intervals K such
that
|hx, hK×[0,1] i| ≤ τ |K| and |K| ≥ δ 2 τ 2
(m ≥ − log2 δ 2 τ 2 ).
Combinatorial covering lemma
Let τ, δ > 0,
x ∈ H 1 (H 1 )
and
kxk ≤ 1.
There exists a large collection of pairwise disjoint dyadic intervals K such
that
|hx, hK×[0,1] i| ≤ τ |K| and |K| ≥ δ 2 τ 2
(m ≥ − log2 δ 2 τ 2 ).
Combinatorial covering lemma
Let τ, δ > 0,
x ∈ H 1 (H 1 )
and
kxk ≤ 1.
There exists a large collection of pairwise disjoint dyadic intervals K such
that
|hx, hK×[0,1] i| ≤ τ |K| and |K| ≥ δ 2 τ 2
[0, 1]
[0, 1]
P
K
|K| ≥ (1 − δ)
(m ≥ − log2 δ 2 τ 2 ).
Combinatorial covering lemma
Let τ, δ > 0,
x ∈ H 1 (H 1 )
and
kxk ≤ 1.
There exists a large collection of pairwise disjoint dyadic intervals K such
that
|hx, hK×[0,1] i| ≤ τ |K| and |K| ≥ δ 2 τ 2
[0, 1]
(m ≥ − log2 δ 2 τ 2 ).
[0, 1]
[0, 1]
[0, 1]
P
K
|K| ≥ (1 − δ)
P
L
|L| ≥ (1 − δ)
Combinatorial covering lemma
Let τ, δ > 0,
x ∈ H 1 (H 1 )∗
and
kxk ≤ 1.
There exists a large collection of pairwise disjoint dyadic intervals K such
that
|hx, hK×[0,1] i| ≤ τ |K| and |K| ≥ δ 2 τ 2
[0, 1]
(m ≥ − log2 δ 2 τ 2 ).
[0, 1]
[0, 1]
[0, 1]
P
K
|K| ≥ (1 − δ)
P
L
|L| ≥ (1 − δ)
Andrew 2D
Theorem (R. L. & P. F. X. M.)
Let 1 ≤ p, q < ∞, δ > 0 and T : H p (H q ) → H p (H q ) be a linear operator
with large diagonal, i.e. hT hI×J , hI×J i ≥ δ|I × J|. Then we have
H p (H q )
E
Id
P
H p (H q )
/ H p (H q )
O
T
kEkkP k ≤ C/δ,
/ H p (H q )
where the constant C > 0 is universal.
By Capon’s bi-parameter construction (BI×J ) and a random choice of signs
P
(ε)
εI×J , there exists a block basis bI×J = K∈BI×J εK×L hK×L of the Haar
system such that
(ε)
(ε)
T bI×J = αI×J bI×J + small error,
αI×J ≥ δ.
P
hf,b
i
The orthogonal projection Qf = I∈D kb I×Jk2 bI×J is bounded on H p (H q ).
I×J 2
Andrew 2D
Theorem (R. L. & P. F. X. M.)
Let 1 ≤ p, q < ∞, δ > 0 and T : H p (H q ) → H p (H q ) be a linear operator
with large diagonal, i.e. hT hI×J , hI×J i ≥ δ|I × J|. Then we have
H p (H q )
E
Id
P
H p (H q )
/ H p (H q )
O
T
kEkkP k ≤ C/δ,
/ H p (H q )
where the constant C > 0 is universal.
By Capon’s bi-parameter construction (BI×J ) and a random choice of signs
P
(ε)
εI×J , there exists a block basis bI×J = K∈BI×J εK×L hK×L of the Haar
system such that
(ε)
(ε)
T bI×J = αI×J bI×J + small error,
αI×J ≥ δ.
P
hf,b
i
The orthogonal projection Qf = I∈D kb I×Jk2 bI×J is bounded on H p (H q ).
I×J 2
Andrew 2D
Theorem (R. L. & P. F. X. M.)
Let 1 ≤ p, q < ∞, δ > 0 and T : H p (H q ) → H p (H q ) be a linear operator
with large diagonal, i.e. hT hI×J , hI×J i ≥ δ|I × J|. Then we have
H p (H q )
E
Id
P
H p (H q )
/ H p (H q )
O
T
kEkkP k ≤ C/δ,
/ H p (H q )
where the constant C > 0 is universal.
By Capon’s bi-parameter construction (BI×J ) and a random choice of signs
P
(ε)
εI×J , there exists a block basis bI×J = K∈BI×J εK×L hK×L of the Haar
system such that
(ε)
(ε)
T bI×J = αI×J bI×J + small error,
αI×J ≥ δ.
P
hf,b
i
The orthogonal projection Qf = I∈D kb I×Jk2 bI×J is bounded on H p (H q ).
I×J 2
Andrew 2D
Theorem (R. L. & P. F. X. M.)
Let 1 ≤ p, q < ∞, δ > 0 and T : H p (H q ) → H p (H q ) be a linear operator
with large diagonal, i.e. hT hI×J , hI×J i ≥ δ|I × J|. Then we have
H p (H q )
E
Id
P
H p (H q )
/ H p (H q )
O
T
kEkkP k ≤ C/δ,
/ H p (H q )
where the constant C > 0 is universal.
By Capon’s bi-parameter construction (BI×J ) and a random choice of signs
P
(ε)
εI×J , there exists a block basis bI×J = K∈BI×J εK×L hK×L of the Haar
system such that
(ε)
(ε)
T bI×J = αI×J bI×J + small error,
αI×J ≥ δ.
P
hf,b
i
The orthogonal projection Qf = I∈D kb I×Jk2 bI×J is bounded on H p (H q ).
I×J 2
Andrew 2D
Theorem (R. L. & P. F. X. M.)
Let 1 ≤ p, q < ∞, δ > 0 and T : H p (H q ) → H p (H q ) be a linear operator
with large diagonal, i.e. hT hI×J , hI×J i ≥ δ|I × J|. Then we have
H p (H q )
E
Id
P
H p (H q )
/ H p (H q )
O
T
kEkkP k ≤ C/δ,
/ H p (H q )
where the constant C > 0 is universal.
By Capon’s bi-parameter construction (BI×J ) and a random choice of signs
P
(ε)
εI×J , there exists a block basis bI×J = K∈BI×J εK×L hK×L of the Haar
system such that
(ε)
(ε)
T bI×J = αI×J bI×J + small error,
αI×J ≥ δ.
P
hf,b
i
The orthogonal projection Qf = I∈D kb I×Jk2 bI×J is bounded on H p (H q ).
I×J 2
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
• Full characterization in 1D for H p by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
I⊂E,|I|=2−i T bI exists almost everywhere,
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
?
• Full characterization in 2D for H p (H q )??
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )?
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).
Summary
Let T : H p (H q ) → H p (H q ) be a linear operator. When does the identity
factor through T (or Id −T )?
• Results for H = T or H = Id −T :
• Capon’s theorem for H p (H q ),
1
1
• local result in HN
(HN
) (R. L. & P. F. X. M.).
• Positive results, if T has a large diagonal:
• Andrew 1D for H p ,
• Andrew 2D for H p (H q ) (R. L. & P. F. X. M.),
p
q
• local version of Andrew 2D for HN
(HN
) (R. L. & P. F. X. M.).
p
• Full characterization in 1D for H by property A (JMST):
• {bI : I ∈ D} equivalent
to the Haar system,
2 P
• Λ(E) = limi→∞ S
T
b
exists almost everywhere,
−i
I
I⊂E,|I|=2
R1
• inf n 0 maxE∈D, |E|=2−n Λp/2 (E) dt > 0.
Outlook
• Full characterization in 2D for H p (H q )?
• 2D version of property A,
• Projection Q onto more general block basis bI×J in H p (H q ).