Pointwise multipliers and di eomorphisms in function spaces fileIntroduction Pointwise multipliers...
Transcript of Pointwise multipliers and di eomorphisms in function spaces fileIntroduction Pointwise multipliers...
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Pointwise multipliers and diffeomorphisms in functionspaces
Benjamin Scharf
Friedrich Schiller University Jena
May 27, 2011
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 1 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Table of contents
1 IntroductionProblem settingSome classical examples for pointwise multipliers
2 Pointwise multipliers in function spacesThe definition of function spaces on Rn
Known results for multipliers in function spacesAtomic characterizations of function spacesA simple approach to pointwise multipliers in function spaces
3 Diffeomorphisms in function spacesA theorem on diffeomorphisms in function spaces
4 Non-smooth atomic representation theoremsNon-smooth atomic representation theorems
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 2 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The problem setting
We want to observe the behaviour of the linear mappings
Pϕ : f 7→ ϕ · f
and
Dϕ : f 7→ f ◦ ϕ,
where f is an element of a function space (Besov, Triebel-Lizorkin type)and ϕ is a suitably smooth function.
The aim:
If ϕ fulfils . . ., then Pϕ resp. Dϕ maps the function space A into A.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 3 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The problem setting
We want to observe the behaviour of the linear mappings
Pϕ : f 7→ ϕ · f
and
Dϕ : f 7→ f ◦ ϕ,
where f is an element of a function space (Besov, Triebel-Lizorkin type)and ϕ is a suitably smooth function.
The aim:
If ϕ fulfils . . ., then Pϕ resp. Dϕ maps the function space A into A.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 3 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The problem setting
We want to observe the behaviour of the linear mappings
Pϕ : f 7→ ϕ · f
and
Dϕ : f 7→ f ◦ ϕ,
where f is an element of a function space (Besov, Triebel-Lizorkin type)and ϕ is a suitably smooth function.
The aim:
If ϕ fulfils . . ., then Pϕ resp. Dϕ maps the function space A into A.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 3 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The spaces C k
Let C k be the space of all k-times differentiable functions f : Rn → Rsuch that
‖f |C k‖ :=∑|α|≤k
sup |Dαf (x)| <∞.
Then
f , g ∈ C k ⇒ f · g ∈ C k and ‖f · g |C k‖ ≤ ck‖f |C k‖ · ‖g |C k‖
and
(∀f ∈ C k : f · g ∈ C k)⇒ g ∈ C k and ‖Pg : C k → C k‖ ≥ ‖g |C k‖.
Proof: Leibniz rule and 1 ∈ C k .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 4 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The spaces C k
Let C k be the space of all k-times differentiable functions f : Rn → Rsuch that
‖f |C k‖ :=∑|α|≤k
sup |Dαf (x)| <∞.
Then
f , g ∈ C k ⇒ f · g ∈ C k and ‖f · g |C k‖ ≤ ck‖f |C k‖ · ‖g |C k‖
and
(∀f ∈ C k : f · g ∈ C k)⇒ g ∈ C k and ‖Pg : C k → C k‖ ≥ ‖g |C k‖.
Proof: Leibniz rule and 1 ∈ C k .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 4 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Holder spaces Ck
Let 0 < σ ≤ 1 and f : Rn → R be continuous. We define
‖f |lipσ‖ := supx ,y∈Rn,x 6=y
|f (x)− f (y)||x − y |σ
,
Let s > 0 and s = bsc+ {s} with bsc ∈ Z and {s} ∈ (0, 1]. Then theHolder space with index s is given by
Cs ={
f ∈ C bsc : ‖f |Cs‖ := ‖f |C bsc−‖+∑|α|=bsc
‖Dαf |lip{s}‖ <∞}.
It holds
f , g ∈ Cs ⇒ f · g ∈ Cs and ‖f · g |Cs‖ ≤ cs‖f |Cs‖ · ‖g |Cs‖.and
(∀f ∈ Cs : f · g ∈ Cs)⇒ g ∈ Cs and ‖Pg : Cs → Cs‖ ≥ ‖g |Cs‖Proof: Leibniz rule for Holder spaces and 1 ∈ Cs .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 5 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Holder spaces Ck
Let 0 < σ ≤ 1 and f : Rn → R be continuous. We define
‖f |lipσ‖ := supx ,y∈Rn,x 6=y
|f (x)− f (y)||x − y |σ
,
Let s > 0 and s = bsc+ {s} with bsc ∈ Z and {s} ∈ (0, 1]. Then theHolder space with index s is given by
Cs ={
f ∈ C bsc : ‖f |Cs‖ := ‖f |C bsc−‖+∑|α|=bsc
‖Dαf |lip{s}‖ <∞}.
It holds
f , g ∈ Cs ⇒ f · g ∈ Cs and ‖f · g |Cs‖ ≤ cs‖f |Cs‖ · ‖g |Cs‖.and
(∀f ∈ Cs : f · g ∈ Cs)⇒ g ∈ Cs and ‖Pg : Cs → Cs‖ ≥ ‖g |Cs‖Proof: Leibniz rule for Holder spaces and 1 ∈ Cs .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 5 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Lebesgue spaces Lp(Rn)
Let 0 < p ≤ ∞ and Lp(Rn) the usual set of equivalence classes ofmeasurable functions f with finite
‖f |Lp(Rn)‖ :=
{(∫Rn |f (x)|p dx
) 1p , 0 < p <∞
ess sup |f (x)| , p =∞
Then
f ∈ Lp(Rn), g ∈ L∞(Rn)⇒ f · g ∈ Lp(Rn) and
‖f · g |Lp(Rn)‖ ≤ ‖f |Lp‖ · ‖f |L∞(Rn)‖
and
(∀f ∈ Lp(Rn) : f · g ∈ Lp(Rn))⇒ g ∈ L∞(Rn) and
‖Pg : Lp(Rn)→ Lp(Rn)‖ ≥ ‖g |L∞(Rn)‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 6 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Lebesgue spaces Lp(Rn)
Let 0 < p ≤ ∞ and Lp(Rn) the usual set of equivalence classes ofmeasurable functions f with finite
‖f |Lp(Rn)‖ :=
{(∫Rn |f (x)|p dx
) 1p , 0 < p <∞
ess sup |f (x)| , p =∞
Then
f ∈ Lp(Rn), g ∈ L∞(Rn)⇒ f · g ∈ Lp(Rn) and
‖f · g |Lp(Rn)‖ ≤ ‖f |Lp‖ · ‖f |L∞(Rn)‖
and
(∀f ∈ Lp(Rn) : f · g ∈ Lp(Rn))⇒ g ∈ L∞(Rn) and
‖Pg : Lp(Rn)→ Lp(Rn)‖ ≥ ‖g |L∞(Rn)‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 6 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (i)
Let 1 < p <∞, k ∈ N0 and W kp (Rn) the set of equivalence classes of
measurable functions f with finite
‖f |W kp (Rn)‖ :=
∑|α|≤k
‖Dαf (x)|Lp(Rn)‖.
Then
f ∈W kp (Rn), g ∈ C k ⇒ f · g ∈W k
p (Rn) and ‖f · g |W kp (Rn)‖ ≤ ‖f |W k
p ‖ · ‖f |C k‖.
The converse is not true!
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 7 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (i)
Let 1 < p <∞, k ∈ N0 and W kp (Rn) the set of equivalence classes of
measurable functions f with finite
‖f |W kp (Rn)‖ :=
∑|α|≤k
‖Dαf (x)|Lp(Rn)‖.
Then
f ∈W kp (Rn), g ∈ C k ⇒ f · g ∈W k
p (Rn) and ‖f · g |W kp (Rn)‖ ≤ ‖f |W k
p ‖ · ‖f |C k‖.
The converse is not true!
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 7 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (ii)
Theorem (Sobolev embedding)
Let k1 < k2 and k1 − np1≤ k2 − n
p2. Then
W k2p2
(Rn) ↪→W k1p1
(Rn).
Theorem (Multiplier algebra)
If k > np , then
‖f · g |W kp (Rn)‖ ≤ ‖f |W k
p (Rn)‖ · ‖g |W kp (Rn)‖.
Proof: We start with
‖Dα(f · g)|Lp(Rn)‖ ≤ c∑‖(Dβf ) · (Dα−βg)‖Lp(Rn)‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 8 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (ii)
Theorem (Sobolev embedding)
Let k1 < k2 and k1 − np1≤ k2 − n
p2. Then
W k2p2
(Rn) ↪→W k1p1
(Rn).
Theorem (Multiplier algebra)
If k > np , then
‖f · g |W kp (Rn)‖ ≤ ‖f |W k
p (Rn)‖ · ‖g |W kp (Rn)‖.
Proof: We start with
‖Dα(f · g)|Lp(Rn)‖ ≤ c∑‖(Dβf ) · (Dα−βg)‖Lp(Rn)‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 8 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (ii)
Theorem (Sobolev embedding)
Let k1 < k2 and k1 − np1≤ k2 − n
p2. Then
W k2p2
(Rn) ↪→W k1p1
(Rn).
Theorem (Multiplier algebra)
If k > np , then
‖f · g |W kp (Rn)‖ ≤ ‖f |W k
p (Rn)‖ · ‖g |W kp (Rn)‖.
Proof: We start with
‖Dα(f · g)|Lp(Rn)‖ ≤ c∑‖(Dβf ) · (Dα−βg)‖Lp(Rn)‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 8 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (iii)
‖Dα(f · g)|Lp(Rn)‖ ≤ c∑‖(Dβf ) · (Dα−βg)‖Lp(Rn)‖
≤ c∑‖(Dβf )|Lp1‖ · ‖(Dα−βg)|Lp2‖
≤ c∑‖f |W |β|
p1 (Rn)‖ · ‖g |W |α|−|β|p2 (Rn)‖
≤ c ′‖f |W kp (Rn)‖ · ‖g |W k
p (Rn)‖.
Here (|α| ≤ k)
1
p1+
1
p2=
1
p
|β| − n
p1≤ k − n
p
|α| − |β| − n
p2≤ k − n
p
This is possible, if k > np .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 9 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The Sobolev spaces W kp (Rn) (iii)
‖Dα(f · g)|Lp(Rn)‖ ≤ c∑‖(Dβf ) · (Dα−βg)‖Lp(Rn)‖
≤ c∑‖(Dβf )|Lp1‖ · ‖(Dα−βg)|Lp2‖
≤ c∑‖f |W |β|
p1 (Rn)‖ · ‖g |W |α|−|β|p2 (Rn)‖
≤ c ′‖f |W kp (Rn)‖ · ‖g |W k
p (Rn)‖.
Here (|α| ≤ k)
1
p1+
1
p2=
1
p
|β| − n
p1≤ k − n
p
|α| − |β| − n
p2≤ k − n
p
This is possible, if k > np .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 9 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
1 IntroductionProblem settingSome classical examples for pointwise multipliers
2 Pointwise multipliers in function spacesThe definition of function spaces on Rn
Known results for multipliers in function spacesAtomic characterizations of function spacesA simple approach to pointwise multipliers in function spaces
3 Diffeomorphisms in function spacesA theorem on diffeomorphisms in function spaces
4 Non-smooth atomic representation theoremsNon-smooth atomic representation theorems
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 10 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Resolution of unity
Let ϕ0 ∈ S(Rn) such that supp ϕ0 ⊂{|x | ≤ 3
2
}and ϕ0(x) = 1 for
|x | ≤ 1. We define
ϕ(x) := ϕ0(x)− ϕ0(2x) and ϕj(x) := ϕ(2−jx) for j ∈ N.
Then we have
∞∑j=0
ϕj(x) = 1.
|Dαϕj(x)| ≤ cα2−j |α|,
supp ϕj ⊂{
2j−1 ≤ |x | ≤ 2j+1},
(1)
A sequence of functions {ϕj}∞j=0 with (1), ϕj ∈ S(Rn) and ϕ0 as abovewill be called resolution of unity.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 11 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Resolution of unity
Let ϕ0 ∈ S(Rn) such that supp ϕ0 ⊂{|x | ≤ 3
2
}and ϕ0(x) = 1 for
|x | ≤ 1. We define
ϕ(x) := ϕ0(x)− ϕ0(2x) and ϕj(x) := ϕ(2−jx) for j ∈ N.
Then we have
∞∑j=0
ϕj(x) = 1.
|Dαϕj(x)| ≤ cα2−j |α|,
supp ϕj ⊂{
2j−1 ≤ |x | ≤ 2j+1},
(1)
A sequence of functions {ϕj}∞j=0 with (1), ϕj ∈ S(Rn) and ϕ0 as abovewill be called resolution of unity.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 11 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The definition of B sp,q(Rn)
Let {ϕj}∞j=0 be a resolution of unity. Let 0 < p ≤ ∞, 0 < q ≤ ∞ ands ∈ R. For f ∈ S ′(Rn) we define
‖f |Bsp,q(Rn)‖ϕ :=
∞∑j=0
2jsq‖(ϕj f ) |Lp(Rn)‖q 1
q
(modified in case q =∞) and
Bs,ϕp,q (Rn) :=
{f ∈ S ′(Rn) : ‖f |Bs
p,q(Rn)‖ϕ <∞}.
Then (Bs,ϕp,q (Rn), ‖ · |Bs
p,q(Rn)‖ϕ) is a quasi-Banach space. It does notdepend on the choice of the resolution of unity {ϕj}∞j=0 in the sense ofequivalent norms. So we denote it shortly by Bs
p,q(Rn).
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 12 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The definition of F sp,q(Rn)
Let {ϕj}∞j=0 be a resolution of unity. Let 0 < p <∞, 0 < q ≤ ∞ ands ∈ R. For f ∈ S ′(Rn) we define
‖f |F sp,q(Rn)‖ϕ :=
∥∥∥∥∥∥∥ ∞∑
j=0
2jsq|(ϕj f ) |q 1
q ∣∣∣Lp(Rn)
∥∥∥∥∥∥∥(modified in case q =∞) and
F s,ϕp,q (Rn) :=
{f ∈ S ′(Rn) : ‖f |F s
p,q(Rn)‖ϕ <∞}.
Then (F s,ϕp,q (Rn), ‖ · |F s
p,q(Rn)‖ϕ) is a quasi-Banach space. It does notdepend on the choice of the resolution of unity {ϕj}∞j=0 in the sense ofequivalent norms. So we denote it shortly by F s
p,q(Rn).
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 13 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Function spaces as multiplier algebras
Definition
A function space Asp,q(Rn) is said to be a multiplier algebra iff there is a
bounded bilinear symmetric mapping
P : Asp,q(Rn) · As
p,q(Rn)→ Asp,q(Rn)
such that
P(f , g) = f · g
for f ∈ S(Rn), g ∈ Asp,q(Rn).
If such a mapping exists, then it is uniquely determined. This follows bycompletion resp. local arguments.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 14 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Function spaces as multiplier algebras
Definition
A function space Asp,q(Rn) is said to be a multiplier algebra iff there is a
bounded bilinear symmetric mapping
P : Asp,q(Rn) · As
p,q(Rn)→ Asp,q(Rn)
such that
P(f , g) = f · g
for f ∈ S(Rn), g ∈ Asp,q(Rn).
If such a mapping exists, then it is uniquely determined. This follows bycompletion resp. local arguments.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 14 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Known necessary results for function spaces (i)Theorem (see Runst and Sickel 1996, 4.3.2)
Let ϕ ∈ S(Rn). Then
‖Pϕ|Asp,q(Rn)→ As
p,q(Rn)‖ := supf ∈As
p,q(Rn),f 6=0
‖ϕ · f |Asp,q(Rn)‖
‖f |Asp,q(Rn)‖
≥ c · ‖ϕ|L∞(Rn)‖,
where c > 0 does not depend on f .
Hence, if Asp,q(Rn) is a multiplier algebra, then
‖ϕ|L∞(Rn)‖ ≤ c ′‖Pϕ|Asp,q(Rn)→ As
p,q(Rn)‖ ≤ c ′′‖ϕ|Asp,q(Rn)‖
for ϕ ∈ S(Rn). So
Asp,q(Rn) ↪→ L∞(Rn).
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 15 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Known necessary results for function spaces (i)Theorem (see Runst and Sickel 1996, 4.3.2)
Let ϕ ∈ S(Rn). Then
‖Pϕ|Asp,q(Rn)→ As
p,q(Rn)‖ := supf ∈As
p,q(Rn),f 6=0
‖ϕ · f |Asp,q(Rn)‖
‖f |Asp,q(Rn)‖
≥ c · ‖ϕ|L∞(Rn)‖,
where c > 0 does not depend on f .
Hence, if Asp,q(Rn) is a multiplier algebra, then
‖ϕ|L∞(Rn)‖ ≤ c ′‖Pϕ|Asp,q(Rn)→ As
p,q(Rn)‖ ≤ c ′′‖ϕ|Asp,q(Rn)‖
for ϕ ∈ S(Rn). So
Asp,q(Rn) ↪→ L∞(Rn).
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 15 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Known necessary results for function spaces (ii)
More general:
Theorem
If there is a (Besov or Triebel-Lizorkin) function space A such that
‖ϕ · f |Asp,q(Rn)‖ ≤ c‖ϕ|A‖ · ‖f |As
p,q(Rn)‖,
then
A ↪→ L∞.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 16 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Known sufficient results for function spaces
Theorem (see e.g Runst and Sickel 1996)
For s > σp the spaces Asp,q(Rn) ∩ L∞(Rn) are multiplier algebras, even
‖f · g |Asp,q(Rn)‖
≤ c(‖f |As
p,q(Rn)‖ · ‖g |L∞‖+ ‖g |Asp,q(Rn)‖ · ‖f |L∞‖
).
Theorem (see e.g. Triebel 2008)
If F sp,q(Rn) is a multiplier algebra, then ϕ is a pointwise multiplier for
F sp,q(Rn) iff
supm∈Z‖ψ(· −m) · ϕ|F s
p,q(Rn)‖ <∞,
where ψ is a nonnegative C∞0 -function with∑m
ψ(x −m) = 1 for x ∈ Rn.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 17 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Known sufficient results for function spaces
Theorem (see e.g Runst and Sickel 1996)
For s > σp the spaces Asp,q(Rn) ∩ L∞(Rn) are multiplier algebras, even
‖f · g |Asp,q(Rn)‖
≤ c(‖f |As
p,q(Rn)‖ · ‖g |L∞‖+ ‖g |Asp,q(Rn)‖ · ‖f |L∞‖
).
Theorem (see e.g. Triebel 2008)
If F sp,q(Rn) is a multiplier algebra, then ϕ is a pointwise multiplier for
F sp,q(Rn) iff
supm∈Z‖ψ(· −m) · ϕ|F s
p,q(Rn)‖ <∞,
where ψ is a nonnegative C∞0 -function with∑m
ψ(x −m) = 1 for x ∈ Rn.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 17 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Atomic characterization of B sp,q(Rn)
Theorem
Let 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ R. Let K , L ≥ 0, K > s andL > σp − s. Then f ∈ S ′(Rn) belongs to Bs
p,q(Rn) if and only if it can berepresented as
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m with convergence in S ′(Rn).
Here aν,m are (s, p)K ,L-atoms located at Qν,m and ‖λ|bp,q‖ <∞ .Furthermore, we have in the sense of equivalence of norms
‖f |Bsp,q(Rn)‖ ∼ inf ‖λ|bp,q‖,
where the infimum on the right-hand side is taken over all admissiblerepresentations of f .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 18 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Atomic characterization of F sp,q(Rn)
Theorem
Let 0 < p <∞, 0 < q ≤ ∞ and s ∈ R. Let K , L ≥ 0, K > s andL > σp,q − s. Then f ∈ S ′(Rn) belongs to F s
p,q(Rn) if and only if it can berepresented as
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m with convergence in S ′(Rn).
Here aν,m are (s, p)K ,L-atoms located at Qν,m and ‖λ|fp,q‖ <∞.Furthermore, we have in the sense of equivalence of norms
‖f |F sp,q(Rn)‖ ∼ inf ‖λ|fp,q‖,
where the infimum on the right-hand side is taken over all admissiblerepresentations of f .
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 19 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Treatment of products using atomic decompositions
f ∈ Asp,q(Rn)
↓
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m
↓
ϕ · f =∞∑ν=0
∑m∈Zn
λν,m · ϕ · aν,m
↓
If ϕ · aν,m are atoms: ϕ · f ∈ Asp,q(Rn)
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 20 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Treatment of products using atomic decompositions
f ∈ Asp,q(Rn)
↓
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m
↓
ϕ · f =∞∑ν=0
∑m∈Zn
λν,m · ϕ · aν,m
↓
If ϕ · aν,m are atoms: ϕ · f ∈ Asp,q(Rn)
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 20 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Treatment of products using atomic decompositions
f ∈ Asp,q(Rn)
↓
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m
↓
ϕ · f =∞∑ν=0
∑m∈Zn
λν,m · ϕ · aν,m
↓
If ϕ · aν,m are atoms: ϕ · f ∈ Asp,q(Rn)
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 20 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Treatment of products using atomic decompositions
f ∈ Asp,q(Rn)
↓
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m
↓
ϕ · f =∞∑ν=0
∑m∈Zn
λν,m · ϕ · aν,m
↓
If ϕ · aν,m are atoms: ϕ · f ∈ Asp,q(Rn)
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 20 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The definition of atoms
A function a : Rn → R is called classical (s, p)K ,L-atom located at Qν,m ifsupp a ⊂ d · Qν,m
|Dαa(x)| ≤ C · 2−ν(s− n
p
)+|α|ν
for all |α| < K + 1, (2)∫Rn
xβa(x) dx = 0 for all |β| < L. (3)
A function a : Rn → R is called (s, p)K ,L-atom located at Qν,m if insteadof (2) and (3) it holds (for all ψ ∈ CL)
‖a(2−ν ·)|CK‖ ≤ C · 2−ν(s− np
)∣∣∣∣∣∫d ·Qν,m
ψ(x)a(x) dx
∣∣∣∣∣ ≤ C · 2−ν(s+L+n
(1− 1
p
))‖ψ|CL‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 21 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The definition of atoms
A function a : Rn → R is called classical (s, p)K ,L-atom located at Qν,m ifsupp a ⊂ d · Qν,m
|Dαa(x)| ≤ C · 2−ν(s− n
p
)+|α|ν
for all |α| < K + 1, (2)∫Rn
xβa(x) dx = 0 for all |β| < L. (3)
A function a : Rn → R is called (s, p)K ,L-atom located at Qν,m if insteadof (2) and (3) it holds (for all ψ ∈ CL)
‖a(2−ν ·)|CK‖ ≤ C · 2−ν(s− np
)∣∣∣∣∣∫d ·Qν,m
ψ(x)a(x) dx
∣∣∣∣∣ ≤ C · 2−ν(s+L+n
(1− 1
p
))‖ψ|CL‖
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 21 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Atomic representations revisitedEvery classical (s, p)K ,L-atom is an (s, p)K ,L-atom.
Theorem
The atomic representation theorem for Bsp,q(Rn) and F s
p,q(Rn) is valid withboth forms of atoms. Hence every f which can be represented as a linearcombination of classical (s, p)K ,L-atom resp. (s, p)K ,L-atom belongs toBsp,q(Rn) resp. F s
p,q(Rn). Hereby
K > s and
L > σp − s = σp = n
(1
p− 1
)+
− s resp.
L > σp,q − s = n
(1
min(p, q)− 1
)+
− s
The proof for classical atoms goes back to Triebel ’97. The modifi- cationswere suggested by Skrzypczak ’98, Triebel/Winkelvoss ’96.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 22 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
Atomic representations revisitedEvery classical (s, p)K ,L-atom is an (s, p)K ,L-atom.
Theorem
The atomic representation theorem for Bsp,q(Rn) and F s
p,q(Rn) is valid withboth forms of atoms. Hence every f which can be represented as a linearcombination of classical (s, p)K ,L-atom resp. (s, p)K ,L-atom belongs toBsp,q(Rn) resp. F s
p,q(Rn). Hereby
K > s and
L > σp − s = σp = n
(1
p− 1
)+
− s resp.
L > σp,q − s = n
(1
min(p, q)− 1
)+
− s
The proof for classical atoms goes back to Triebel ’97. The modifi- cationswere suggested by Skrzypczak ’98, Triebel/Winkelvoss ’96.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 22 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The pointwise multiplier theorem (i)
Now we get
Lemma
There exists a constant c with the following property: For all ν ∈ N0,m ∈ Z, all (s, p)K ,L-atoms aν,m with support in d · Qν,m and all ϕ ∈ C ρ
with ρ ≥ max(K , L) the product
c · ‖ϕ|Cρ‖−1 · ϕ · aν,m
is an (s, p)K ,L-atom with support in d · Qν,m.
Proof: Use that Cρ is a multiplication algebra.
This does not work for classical atoms (s, p)K ,L-atoms with L ≥ 1, since ingeneral moment conditions are destroyed when multiplying by ϕ!
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 23 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The pointwise multiplier theorem (i)
Now we get
Lemma
There exists a constant c with the following property: For all ν ∈ N0,m ∈ Z, all (s, p)K ,L-atoms aν,m with support in d · Qν,m and all ϕ ∈ C ρ
with ρ ≥ max(K , L) the product
c · ‖ϕ|Cρ‖−1 · ϕ · aν,m
is an (s, p)K ,L-atom with support in d · Qν,m.
Proof: Use that Cρ is a multiplication algebra.
This does not work for classical atoms (s, p)K ,L-atoms with L ≥ 1, since ingeneral moment conditions are destroyed when multiplying by ϕ!
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 23 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The pointwise multiplier theorem (ii)
We get as a corollary
Theorem
Let s ∈ R and 0 < q ≤ ∞.(i) Let 0 < p ≤ ∞ and ρ > max(s, σp − s). Then there exists a positivenumber c such that
‖ϕ · f |Bsp,q(Rn)‖ ≤ c‖ϕ|Cρ‖ · ‖f |Bs
p,q(Rn)‖
for all ϕ ∈ Cρ and all f ∈ Bsp,q(Rn).
(ii) Let 0 < p <∞ and ρ > max(s, σp,q − s). Then there exists a positivenumber c such that
‖ϕ · f |F sp,q(Rn)‖ ≤ c‖ϕ|Cρ‖ · ‖f |F s
p,q(Rn)‖
for all ϕ ∈ Cρ and all f ∈ F sp,q(Rn).
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 24 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
1 IntroductionProblem settingSome classical examples for pointwise multipliers
2 Pointwise multipliers in function spacesThe definition of function spaces on Rn
Known results for multipliers in function spacesAtomic characterizations of function spacesA simple approach to pointwise multipliers in function spaces
3 Diffeomorphisms in function spacesA theorem on diffeomorphisms in function spaces
4 Non-smooth atomic representation theoremsNon-smooth atomic representation theorems
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 25 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The diffeomorphism theorem (i)
In the same way we can treat the mapping Dϕ:
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m ⇒ f ◦ ϕ =∞∑ν=0
∑m∈Zn
λν,m · (aν,m ◦ ϕ).
Hence we have to investigate if aν,m ◦ ϕ is an (s, p)K ,L-atom when aν,m isan (s, p)K ,L-atom.
Definition
Let ρ ≥ 1. We say that the one-to-one mapping ϕ : Rn → Rn is aρ-diffeomorphism if the components of ϕ(x) = (ϕ1(x), . . . , ϕn(x)) haveclassical derivatives up to order bρc with ∂ϕ
∂xj∈ Cρ−1 and if
| detϕ∗| ≥ c > 0 for some c and all x ∈ Rn. Here ϕ∗ stands for theJacobian matrix.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 26 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The diffeomorphism theorem (i)
In the same way we can treat the mapping Dϕ:
f =∞∑ν=0
∑m∈Zn
λν,m · aν,m ⇒ f ◦ ϕ =∞∑ν=0
∑m∈Zn
λν,m · (aν,m ◦ ϕ).
Hence we have to investigate if aν,m ◦ ϕ is an (s, p)K ,L-atom when aν,m isan (s, p)K ,L-atom.
Definition
Let ρ ≥ 1. We say that the one-to-one mapping ϕ : Rn → Rn is aρ-diffeomorphism if the components of ϕ(x) = (ϕ1(x), . . . , ϕn(x)) haveclassical derivatives up to order bρc with ∂ϕ
∂xj∈ Cρ−1 and if
| detϕ∗| ≥ c > 0 for some c and all x ∈ Rn. Here ϕ∗ stands for theJacobian matrix.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 26 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The diffeomorphism theorem (ii)
Theorem
(i) Let 0 < p ≤ ∞, ρ ≥ 1 and ρ > max(s, σp − s). If ϕ is aρ-diffeomorphism, then there exists a constant c such that
‖f (ϕ(·))|Bsp,q(Rn)‖ ≤ c‖f |Bs
p,q(Rn)‖.
for all f ∈ Bsp,q(Rn). Hence Dϕ maps Bs
p,q(Rn) onto Bsp,q(Rn).
(ii) Let 0 < p <∞, ρ ≥ 1 and ρ > max(s, σp,q − s). If ϕ is aρ-diffeomorphism, then there exists a constant c such that
‖f (ϕ(·))|F sp,q(Rn)‖ ≤ c‖f |F s
p,q(Rn)‖.
for all f ∈ F sp,q(Rn). Hence Dϕ maps F s
p,q(Rn) onto F sp,q(Rn).
Proof: Show that aν,m is an (s, p)K ,L-atom and control the support of theatoms.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 27 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The diffeomorphism theorem (ii)
Theorem
(i) Let 0 < p ≤ ∞, ρ ≥ 1 and ρ > max(s, σp − s). If ϕ is aρ-diffeomorphism, then there exists a constant c such that
‖f (ϕ(·))|Bsp,q(Rn)‖ ≤ c‖f |Bs
p,q(Rn)‖.
for all f ∈ Bsp,q(Rn). Hence Dϕ maps Bs
p,q(Rn) onto Bsp,q(Rn).
(ii) Let 0 < p <∞, ρ ≥ 1 and ρ > max(s, σp,q − s). If ϕ is aρ-diffeomorphism, then there exists a constant c such that
‖f (ϕ(·))|F sp,q(Rn)‖ ≤ c‖f |F s
p,q(Rn)‖.
for all f ∈ F sp,q(Rn). Hence Dϕ maps F s
p,q(Rn) onto F sp,q(Rn).
Proof: Show that aν,m is an (s, p)K ,L-atom and control the support of theatoms.
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 27 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
1 IntroductionProblem settingSome classical examples for pointwise multipliers
2 Pointwise multipliers in function spacesThe definition of function spaces on Rn
Known results for multipliers in function spacesAtomic characterizations of function spacesA simple approach to pointwise multipliers in function spaces
3 Diffeomorphisms in function spacesA theorem on diffeomorphisms in function spaces
4 Non-smooth atomic representation theoremsNon-smooth atomic representation theorems
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 28 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The characteristic function and the Haar wavelet
It is known that
χ[0,1]n ∈ Asp,q(Rn)
for 0 < s < 1p and p ≥ 1. But it can’t be understood as a (classical) atom
for these spaces! This would be of interest in connection with Haarwavelets.
The question: Is there a more general “non-smooth” atomic representationtheorem?
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 29 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The characteristic function and the Haar wavelet
It is known that
χ[0,1]n ∈ Asp,q(Rn)
for 0 < s < 1p and p ≥ 1. But it can’t be understood as a (classical) atom
for these spaces! This would be of interest in connection with Haarwavelets.
The question: Is there a more general “non-smooth” atomic representationtheorem?
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 29 / 30
Introduction Pointwise multipliers in function spaces Diffeomorphisms Non-smooth atomic representation theorems
The end
Thank you for your attention
Questions?
Benjamin Scharf (Uni Jena) Pointwise multipliers and diffeomorphisms May 27, 2011 30 / 30