![Page 1: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/1.jpg)
Atomic norms and finite sums of atoms
Fulvio Ricci
Scuola Normale Superiore di Pisa
Bardonecchia, June 16, 2009
Convegnetto in honour of Poor Little Guido
![Page 2: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/2.jpg)
This is joint work with J. Verdera, to appear in TAMS
Everything is done in good old Euclidean Rn.
![Page 3: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/3.jpg)
(p, q)-atoms
Let 0 < p ≤ 1 ≤ q ≤ ∞ and p < q.
A (p,q)-atom is an Lq-function supported on a ball B and suchthat• ‖a‖q ≤ m(B)
1q−
1p ;
•∫
xαa(x) dx = 0 for |α| ≤ n(1
p − 1).
![Page 4: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/4.jpg)
(p, q)-atoms
Let 0 < p ≤ 1 ≤ q ≤ ∞ and p < q.
A (p,q)-atom is an Lq-function supported on a ball B and suchthat• ‖a‖q ≤ m(B)
1q−
1p ;
•∫
xαa(x) dx = 0 for |α| ≤ n(1
p − 1).
![Page 5: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/5.jpg)
Atomic norms
With p,q as above, Hp,q is the space of distributions f(functions if p = 1) that can be represented by atomic series ofthe form
f =∞∑
j=0
λjaj , (in the sense of distributions)
where the aj are (p,q)-atoms, and the sequence λ = (λj) isin `p.
The (p,q)-atomic norm of f is
‖f‖Hp,q = inf{‖λ‖`p : f =
∞∑j=0
λjaj , aj (p,q)-atoms}.
![Page 6: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/6.jpg)
Atomic norms
With p,q as above, Hp,q is the space of distributions f(functions if p = 1) that can be represented by atomic series ofthe form
f =∞∑
j=0
λjaj , (in the sense of distributions)
where the aj are (p,q)-atoms, and the sequence λ = (λj) isin `p.The (p,q)-atomic norm of f is
‖f‖Hp,q = inf{‖λ‖`p : f =
∞∑j=0
λjaj , aj (p,q)-atoms}.
![Page 7: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/7.jpg)
Equivalence of atomic norms
For fixed p, all Hp,q spaces coincide with the Hardy space Hp
defined in terms of maximal operators, Riesz transforms, etc.(Coifman&Weiss, Latter).
![Page 8: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/8.jpg)
Finite atomic norms
In 1985, Meyer, Taibleson and Weiss observed a specialfeature of (p,∞)-atomic norms:a single (p,∞)- atom a can have arbitrary small Hp,∞-norm.
In fact they showed that the ratio between the atomic norm of a(p,∞)-atom a and its finite (p,∞)-atomic norm,
‖a‖F p,∞ = inf{‖λ‖`p : a =
∑finite
λjaj , aj (p,∞)-atoms},
can be arbitrary small.
This shows that the simple fact that a linear operator isuniformly bounded on atoms may not be sufficient to imply thatthe operator is bounded on Hp.
![Page 9: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/9.jpg)
Finite atomic norms
In 1985, Meyer, Taibleson and Weiss observed a specialfeature of (p,∞)-atomic norms:a single (p,∞)- atom a can have arbitrary small Hp,∞-norm.
In fact they showed that the ratio between the atomic norm of a(p,∞)-atom a and its finite (p,∞)-atomic norm,
‖a‖F p,∞ = inf{‖λ‖`p : a =
∑finite
λjaj , aj (p,∞)-atoms},
can be arbitrary small.
This shows that the simple fact that a linear operator isuniformly bounded on atoms may not be sufficient to imply thatthe operator is bounded on Hp.
![Page 10: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/10.jpg)
Finite atomic norms
In 1985, Meyer, Taibleson and Weiss observed a specialfeature of (p,∞)-atomic norms:a single (p,∞)- atom a can have arbitrary small Hp,∞-norm.
In fact they showed that the ratio between the atomic norm of a(p,∞)-atom a and its finite (p,∞)-atomic norm,
‖a‖F p,∞ = inf{‖λ‖`p : a =
∑finite
λjaj , aj (p,∞)-atoms},
can be arbitrary small.
This shows that the simple fact that a linear operator isuniformly bounded on atoms may not be sufficient to imply thatthe operator is bounded on Hp.
![Page 11: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/11.jpg)
Recent developments
In 2005 Bownik exhibited a linear functional, defined on(1,∞)-atoms and uniformly bounded on them, which is notcontrolled by the H1,∞-norms of the same atoms, and thereforedoes not admit a continuous extension to H1.
It was recently shown by Yang and Zhou for q = 2, and byMeda, Sjögren and Vallarino (MSV) in general, that thisphenomenon is typical of (p,∞)-atoms, and it does not occurfor any p if q <∞.
![Page 12: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/12.jpg)
Recent developments
In 2005 Bownik exhibited a linear functional, defined on(1,∞)-atoms and uniformly bounded on them, which is notcontrolled by the H1,∞-norms of the same atoms, and thereforedoes not admit a continuous extension to H1.
It was recently shown by Yang and Zhou for q = 2, and byMeda, Sjögren and Vallarino (MSV) in general, that thisphenomenon is typical of (p,∞)-atoms, and it does not occurfor any p if q <∞.
![Page 13: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/13.jpg)
F p,q versus Hp,q norms
It is natural to introduce at this point the spaces F p,q of finitelinear combinations of (p,q)-atoms, with the norm
‖f‖F p,q = inf{‖λ‖`p : f =
∑finite
λjaj , aj (p,q)-atoms}.
A linear functional that is uniformly bounded on single(p,q)-atoms automatically extends boundedly to F p,q.
MSV proved that the F p,q-norm and the Hp,q-norm areequivalent on F p,q when q is finite, in contrast with the q =∞case.Therefore, for q <∞, the completion of F p,q is Hp and thisexplains the absence of Bownik-type examples for q <∞.
![Page 14: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/14.jpg)
F p,q versus Hp,q norms
It is natural to introduce at this point the spaces F p,q of finitelinear combinations of (p,q)-atoms, with the norm
‖f‖F p,q = inf{‖λ‖`p : f =
∑finite
λjaj , aj (p,q)-atoms}.
A linear functional that is uniformly bounded on single(p,q)-atoms automatically extends boundedly to F p,q.
MSV proved that the F p,q-norm and the Hp,q-norm areequivalent on F p,q when q is finite, in contrast with the q =∞case.Therefore, for q <∞, the completion of F p,q is Hp and thisexplains the absence of Bownik-type examples for q <∞.
![Page 15: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/15.jpg)
Sums of continuous atoms
Let Hp,c (resp. F p,c) be the space of finite linear combinations(resp. finite l.c.) of continuous (p,∞)-atoms, with the norm
‖f‖Hp,c = inf{‖λ‖`p : f =
∞∑j=0
λjaj , aj continuous (p,q)-atoms},
resp.
‖f‖F p,c = inf{‖λ‖`p : f =
∑finite
λjaj , aj continuous (p,q)-atoms}.
![Page 16: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/16.jpg)
The completion F p,∞ of F p,∞
Lemma (MSV). The Hp-, F p,∞-, F p,c-norms are all equivalenton F p,c , and F p,c is dense in Hp. Hence the completion of F p,c
is Hp.
Consider the inclusion maps
F p,c T−→ F p,∞ U−→ Hp ,
and their continuous extension to the completions,
Hp eT−→ F p,∞ eU−→ Hp .
Since U ◦ T = I, we have the following
Lemma. T (Hp) is closed in F p,∞, Np = ker U is nontrivial, and
F p,∞ = T (Hp)⊕ Np .
![Page 17: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/17.jpg)
The completion F p,∞ of F p,∞
Lemma (MSV). The Hp-, F p,∞-, F p,c-norms are all equivalenton F p,c , and F p,c is dense in Hp. Hence the completion of F p,c
is Hp.
Consider the inclusion maps
F p,c T−→ F p,∞ U−→ Hp ,
and their continuous extension to the completions,
Hp eT−→ F p,∞ eU−→ Hp .
Since U ◦ T = I, we have the following
Lemma. T (Hp) is closed in F p,∞, Np = ker U is nontrivial, and
F p,∞ = T (Hp)⊕ Np .
![Page 18: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/18.jpg)
The completion F p,∞ of F p,∞
Lemma (MSV). The Hp-, F p,∞-, F p,c-norms are all equivalenton F p,c , and F p,c is dense in Hp. Hence the completion of F p,c
is Hp.
Consider the inclusion maps
F p,c T−→ F p,∞ U−→ Hp ,
and their continuous extension to the completions,
Hp eT−→ F p,∞ eU−→ Hp .
Since U ◦ T = I, we have the following
Lemma. T (Hp) is closed in F p,∞, Np = ker U is nontrivial, and
F p,∞ = T (Hp)⊕ Np .
![Page 19: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/19.jpg)
The completion F p,∞ of F p,∞
Lemma (MSV). The Hp-, F p,∞-, F p,c-norms are all equivalenton F p,c , and F p,c is dense in Hp. Hence the completion of F p,c
is Hp.
Consider the inclusion maps
F p,c T−→ F p,∞ U−→ Hp ,
and their continuous extension to the completions,
Hp eT−→ F p,∞ eU−→ Hp .
Since U ◦ T = I, we have the following
Lemma. T (Hp) is closed in F p,∞, Np = ker U is nontrivial, and
F p,∞ = T (Hp)⊕ Np .
![Page 20: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/20.jpg)
The atomic structure of F p,∞
Theorem. Given any sequence of (p,∞) atoms aj and any`p-sequence of scalars λj , the series
∑∞j=0 λj aj converges in
F p,∞ to an element ξ such that ‖ξ‖peF p,∞ ≤∑∞
j=0 |λj |p.
Conversely, each ξ ∈ F p,∞ can be written as
ξ =∞∑
j=0
λj aj , (∗)
with each aj a (p,∞) atom and the sum converging in F p,∞.Moreover,
‖ξ‖fF p = inf( ∞∑
j=0
|λj |p) 1
p,
where the infimum is taken over all decompositions (*) of ξ.
![Page 21: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/21.jpg)
The atomic structure of F p,∞
Theorem. Given any sequence of (p,∞) atoms aj and any`p-sequence of scalars λj , the series
∑∞j=0 λj aj converges in
F p,∞ to an element ξ such that ‖ξ‖peF p,∞ ≤∑∞
j=0 |λj |p.
Conversely, each ξ ∈ F p,∞ can be written as
ξ =∞∑
j=0
λj aj , (∗)
with each aj a (p,∞) atom and the sum converging in F p,∞.
Moreover,
‖ξ‖fF p = inf( ∞∑
j=0
|λj |p) 1
p,
where the infimum is taken over all decompositions (*) of ξ.
![Page 22: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/22.jpg)
The atomic structure of F p,∞
Theorem. Given any sequence of (p,∞) atoms aj and any`p-sequence of scalars λj , the series
∑∞j=0 λj aj converges in
F p,∞ to an element ξ such that ‖ξ‖peF p,∞ ≤∑∞
j=0 |λj |p.
Conversely, each ξ ∈ F p,∞ can be written as
ξ =∞∑
j=0
λj aj , (∗)
with each aj a (p,∞) atom and the sum converging in F p,∞.Moreover,
‖ξ‖fF p = inf( ∞∑
j=0
|λj |p) 1
p,
where the infimum is taken over all decompositions (*) of ξ.
![Page 23: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/23.jpg)
Atomic series in F p,∞ and in Hp
Corollary. If a series∑∞
j=0 λj aj of (p,∞)-atoms converges inthe F p,∞-norm to ξ, then it converges to Uξ in the Hp-norm.
The non-trivial elements of Np are those ξ =∑∞
j=0 λj aj suchthat the series converges to 0 in Hp.
Formally, consider the “free atomic space” `p(Ap), where Apthe set of (p,∞)-atoms.Both Hp and F p are quotient spaces of `p(Ap), and theprojection onto Hp factors through F p.
![Page 24: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/24.jpg)
Atomic series in F p,∞ and in Hp
Corollary. If a series∑∞
j=0 λj aj of (p,∞)-atoms converges inthe F p,∞-norm to ξ, then it converges to Uξ in the Hp-norm.
The non-trivial elements of Np are those ξ =∑∞
j=0 λj aj suchthat the series converges to 0 in Hp.
Formally, consider the “free atomic space” `p(Ap), where Apthe set of (p,∞)-atoms.Both Hp and F p are quotient spaces of `p(Ap), and theprojection onto Hp factors through F p.
![Page 25: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/25.jpg)
The Stone space Rn
Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.
The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras
L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .
Rn is defined as the set of non-zero multiplicative linearfunctionals ϕ on L∞0 (Rn) with the weak-* topology.
![Page 26: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/26.jpg)
The Stone space Rn
Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras
L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .
Rn is defined as the set of non-zero multiplicative linearfunctionals ϕ on L∞0 (Rn) with the weak-* topology.
![Page 27: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/27.jpg)
The Stone space Rn
Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras
L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .
Rn is defined as the set of non-zero multiplicative linearfunctionals ϕ on L∞0 (Rn) with the weak-* topology.
![Page 28: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/28.jpg)
The projection of Rn onto Rn
By the Riesz representation theorem, ϕ|C0(Rn)is given by
evaluation at a point x ∈ Rn.
The map π : ϕ 7−→ x is a continuous projection of Rn onto Rn.
For f ∈ C0(Rn), f = f ◦ π.
![Page 29: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/29.jpg)
The projection of Rn onto Rn
By the Riesz representation theorem, ϕ|C0(Rn)is given by
evaluation at a point x ∈ Rn.
The map π : ϕ 7−→ x is a continuous projection of Rn onto Rn.
For f ∈ C0(Rn), f = f ◦ π.
![Page 30: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/30.jpg)
A description of Rn - I
A continuous linear functional ϕ on L∞0 (Rn) is uniquelydetermined by the values it takes on characteristic functions ofbounded sets.
If ϕ is multiplicative, ϕ(χE) is either 0 or 1.
Then ϕ is uniquely determined by the family of boundedmeasurable sets (modulo sets of measure 0)Eϕ = {E : ϕ(χE) = 1}.
![Page 31: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/31.jpg)
A description of Rn - I
A continuous linear functional ϕ on L∞0 (Rn) is uniquelydetermined by the values it takes on characteristic functions ofbounded sets.
If ϕ is multiplicative, ϕ(χE) is either 0 or 1.
Then ϕ is uniquely determined by the family of boundedmeasurable sets (modulo sets of measure 0)Eϕ = {E : ϕ(χE) = 1}.
![Page 32: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/32.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:
• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 33: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/33.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;
• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 34: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/34.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;
• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 35: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/35.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;
• if E ∈ E is the disjoint union of E1 and E2, then one andonly one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 36: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/36.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 37: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/37.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 38: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/38.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 39: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/39.jpg)
A description of Rn - II
The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and
only one between E1 and E2 is in E .
Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.
(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)
![Page 40: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/40.jpg)
The Stone space of a ball
Consider now a ball B ⊂ Rn. By the Gelfand-Mazur theoremagain,
L∞(B) ∼= C(B) ,
where now B is a compact space. In fact, there is a naturalinclusion of B into Rn that makes the following diagramcommutative:
B −→ Rn
−→πB π
−→
B −→ Rn
![Page 41: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/41.jpg)
About B
B consists of those multiplicative functionals ϕ such thatϕ(χB) = 1.
B is closed and open in Rn, π(B) = B, and
π−1(oB) ⊂ B ⊂ π−1(B) ,
with proper inclusions.
![Page 42: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/42.jpg)
The measure on Rn
For f ∈ L∞0 (Rn), supp f ⊂ B if and only if supp f ⊂ B. Inparticular, L∞c (Rn) = Cc(Rn).
There is a unique positive Borel measure m on Rn such that∫Rn
f dm =
∫cRn
f dm
for every f ∈ L∞c (Rn).
Lemma. The map f 7−→ f extends to an isomorphism fromL1
loc(Rn) onto L1
loc(Rn). In particular, every locally boundedm-measurable function on Rn coincides m-a.e. with acontinuous function.
![Page 43: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/43.jpg)
The measure on Rn
For f ∈ L∞0 (Rn), supp f ⊂ B if and only if supp f ⊂ B. Inparticular, L∞c (Rn) = Cc(Rn).
There is a unique positive Borel measure m on Rn such that∫Rn
f dm =
∫cRn
f dm
for every f ∈ L∞c (Rn).
Lemma. The map f 7−→ f extends to an isomorphism fromL1
loc(Rn) onto L1
loc(Rn). In particular, every locally boundedm-measurable function on Rn coincides m-a.e. with acontinuous function.
![Page 44: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/44.jpg)
The measure on Rn
For f ∈ L∞0 (Rn), supp f ⊂ B if and only if supp f ⊂ B. Inparticular, L∞c (Rn) = Cc(Rn).
There is a unique positive Borel measure m on Rn such that∫Rn
f dm =
∫cRn
f dm
for every f ∈ L∞c (Rn).
Lemma. The map f 7−→ f extends to an isomorphism fromL1
loc(Rn) onto L1
loc(Rn). In particular, every locally boundedm-measurable function on Rn coincides m-a.e. with acontinuous function.
![Page 45: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/45.jpg)
The dual space of F 1,∞
Theorem.• Let ` be a bounded linear functional on F 1,∞.
There exist a function b ∈ BMO(Rn) and a Radon measureµ on Rn, singular with respect to m, satisfying
|µ|(B) ≤ C m(B), for each ball B, (1)
such that, if f =∑
j λjaj ,
`(f ) =∑
j
λj
∫aj(b − bBj ) dm +
∫f dµ, f ∈ F 1,∞. (2)
• Conversely, if b and µ are as above, then the identity (??)defines a bounded linear functional on F 1,∞ and
‖`‖(eF 1,∞)∗
∼= ‖b‖BMO + supB
|µ|(B)
m(B).
![Page 46: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/46.jpg)
The dual space of F 1,∞
Theorem.• Let ` be a bounded linear functional on F 1,∞.
There exist a function b ∈ BMO(Rn) and a Radon measureµ on Rn, singular with respect to m, satisfying
|µ|(B) ≤ C m(B), for each ball B, (1)
such that, if f =∑
j λjaj ,
`(f ) =∑
j
λj
∫aj(b − bBj ) dm +
∫f dµ, f ∈ F 1,∞. (2)
• Conversely, if b and µ are as above, then the identity (??)defines a bounded linear functional on F 1,∞ and
‖`‖(eF 1,∞)∗
∼= ‖b‖BMO + supB
|µ|(B)
m(B).
![Page 47: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/47.jpg)
Proof
Let B be a ball in Rn. Any function f ∈ L∞(B) with mean valuezero on B, is a scalar multiple of a (1,∞)-atom. More precisely,
‖f‖F 1,∞ ≤ ‖f‖∞m(B) .
Hence ` induces a linear functional on the space ofL∞-functions on B with mean value zero, with norm not greaterthan m(B)‖`‖.
By one-dimensional extension, this is the same as saying aone-parameter family of linear functionals on L∞(B), alldiffering one from the other by a scalar multiple of f 7−→
∫B f .
![Page 48: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/48.jpg)
Proof
Let B be a ball in Rn. Any function f ∈ L∞(B) with mean valuezero on B, is a scalar multiple of a (1,∞)-atom. More precisely,
‖f‖F 1,∞ ≤ ‖f‖∞m(B) .
Hence ` induces a linear functional on the space ofL∞-functions on B with mean value zero, with norm not greaterthan m(B)‖`‖.
By one-dimensional extension, this is the same as saying aone-parameter family of linear functionals on L∞(B), alldiffering one from the other by a scalar multiple of f 7−→
∫B f .
![Page 49: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/49.jpg)
Proof
Let B be a ball in Rn. Any function f ∈ L∞(B) with mean valuezero on B, is a scalar multiple of a (1,∞)-atom. More precisely,
‖f‖F 1,∞ ≤ ‖f‖∞m(B) .
Hence ` induces a linear functional on the space ofL∞-functions on B with mean value zero, with norm not greaterthan m(B)‖`‖.
By one-dimensional extension, this is the same as saying aone-parameter family of linear functionals on L∞(B), alldiffering one from the other by a scalar multiple of f 7−→
∫B f .
![Page 50: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/50.jpg)
Proof, cont.dIdentify now L∞(B) with C(B).
By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral
`(a) =
∫bB a dνB ,
with νB a Borel measure on B, and as well by any othermeasure νB + cχbBm with c ∈ C.
By a standard construction, we obtain a measure ν on Rn,unique up to contant multiples of m, such that
`(a) =
∫cRn
a dν ,
for every (1,∞)-atom a, and
‖`‖ ∼= supB
1m(B)
∣∣∣ν − ν(B)
m(B)m∣∣∣(B) .
![Page 51: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/51.jpg)
Proof, cont.dIdentify now L∞(B) with C(B).By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral
`(a) =
∫bB a dνB ,
with νB a Borel measure on B, and as well by any othermeasure νB + cχbBm with c ∈ C.
By a standard construction, we obtain a measure ν on Rn,unique up to contant multiples of m, such that
`(a) =
∫cRn
a dν ,
for every (1,∞)-atom a, and
‖`‖ ∼= supB
1m(B)
∣∣∣ν − ν(B)
m(B)m∣∣∣(B) .
![Page 52: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/52.jpg)
Proof, cont.dIdentify now L∞(B) with C(B).By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral
`(a) =
∫bB a dνB ,
with νB a Borel measure on B, and as well by any othermeasure νB + cχbBm with c ∈ C.
By a standard construction, we obtain a measure ν on Rn,unique up to contant multiples of m, such that
`(a) =
∫cRn
a dν ,
for every (1,∞)-atom a, and
‖`‖ ∼= supB
1m(B)
∣∣∣ν − ν(B)
m(B)m∣∣∣(B) .
![Page 53: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/53.jpg)
Proof, cont.d
Let ν = G m + µ be the Lebesgue decomposition of ν w.r. to m(with G unique up to constants). Then
‖`‖ ∼= supB
1m(B)
∫bB∣∣∣G − ν(B)
m(B)
∣∣∣dm + supB
|µ|(B)
m(B).
By the Lemma, G = g for some g ∈ L1loc(R
n).
![Page 54: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/54.jpg)
Proof, cont.d
Let ν = G m + µ be the Lebesgue decomposition of ν w.r. to m(with G unique up to constants). Then
‖`‖ ∼= supB
1m(B)
∫bB∣∣∣G − ν(B)
m(B)
∣∣∣dm + supB
|µ|(B)
m(B).
By the Lemma, G = g for some g ∈ L1loc(R
n).
![Page 55: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/55.jpg)
The dual space of F p,∞, p < 1
Theorem. The bounded linear functionals on F p,∞, 0 < p < 1,extend uniquely to Hp(Rn). Thus (F p,∞)∗ = Hp(Rn)∗,0 < p < 1.
Proof. A repetition of the above argument leads to theexistence of a measure ν = G m + µ on Rn, where now
‖`‖ ∼= supB
1
m(B)1p
∫bB |G − P
ν,bB|dm + supB
|µ|(B)
m(B)1p
.
The condition |µ|(B) ≤ Cm(B)1p implies that µ = 0.
![Page 56: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/56.jpg)
The dual space of F p,∞, p < 1
Theorem. The bounded linear functionals on F p,∞, 0 < p < 1,extend uniquely to Hp(Rn). Thus (F p,∞)∗ = Hp(Rn)∗,0 < p < 1.
Proof. A repetition of the above argument leads to theexistence of a measure ν = G m + µ on Rn, where now
‖`‖ ∼= supB
1
m(B)1p
∫bB |G − P
ν,bB|dm + supB
|µ|(B)
m(B)1p
.
The condition |µ|(B) ≤ Cm(B)1p implies that µ = 0.
![Page 57: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/57.jpg)
Concluding remarks
In particular, the dual space of F p,∞ does not separate points.
Corollary. For p < 1, let T be a linear operator with values in aBanach space, defined on (p,∞)-atoms, and uniformlybounded on them. Then T admits a continuous extension toHp.
![Page 58: Atomic norms and finite sums of atoms - polito.itcalvino.polito.it/~anfunz/Bardonecchia/slides-bardo/... · 2009-06-16 · Atomic norms and finite sums of atoms Fulvio Ricci Scuola](https://reader033.vdocuments.mx/reader033/viewer/2022052722/5f0cb7907e708231d436c8e1/html5/thumbnails/58.jpg)
Concluding remarks
In particular, the dual space of F p,∞ does not separate points.
Corollary. For p < 1, let T be a linear operator with values in aBanach space, defined on (p,∞)-atoms, and uniformlybounded on them. Then T admits a continuous extension toHp.