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


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



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.



The problem setting


Pϕ : f 7→ ϕ · f

and

Dϕ : f 7→ f ◦ ϕ,


The aim:




The problem setting


Pϕ : f 7→ ϕ · f

and

Dϕ : f 7→ f ◦ ϕ,


The aim:




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 .



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 .



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 .



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 .



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)‖



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)‖



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!



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!



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)‖






p2. Then

W k2p2

(Rn) ↪→W k1p1

(Rn).


If k > np , then

‖f · g |W kp (Rn)‖ ≤ ‖f |W k

p (Rn)‖ · ‖g |W kp (Rn)‖.








p2. Then

W k2p2

(Rn) ↪→W k1p1

(Rn).


If k > np , then

‖f · g |W kp (Rn)‖ ≤ ‖f |W k

p (Rn)‖ · ‖g |W kp (Rn)‖.





The Sobolev spaces W kp (Rn) (iii)


≤ 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 .



The Sobolev spaces W kp (Rn) (iii)


≤ 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 .










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.



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.



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).



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).



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.



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.



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).



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).



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∞.



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.



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.



Atomic characterization of B sp,q(Rn)

Theorem

Let 0 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 .



Atomic characterization of F sp,q(Rn)

Theorem

Let 0 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 .



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)




f ∈ Asp,q(Rn)

↓

f =∞∑ν=0

∑m∈Zn

λν,m · aν,m

↓

ϕ · f =∞∑ν=0

∑m∈Zn


↓





f ∈ Asp,q(Rn)

↓

f =∞∑ν=0

∑m∈Zn

λν,m · aν,m

↓

ϕ · f =∞∑ν=0

∑m∈Zn


↓





f ∈ Asp,q(Rn)

↓

f =∞∑ν=0

∑m∈Zn

λν,m · aν,m

↓

ϕ · f =∞∑ν=0

∑m∈Zn


↓




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‖



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‖



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.



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.



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!



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!



The pointwise multiplier theorem (ii)

We get as a corollary

Theorem

Let s ∈ R and 0 < q ≤ ∞.(i) Let 0 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 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).










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.



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.



The diffeomorphism theorem (ii)

Theorem

(i) Let 0 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 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.



The diffeomorphism theorem (ii)

Theorem

(i) Let 0 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 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.










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?



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?



The end

Thank you for your attention

Questions?

Date post:	08-Oct-2019
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