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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