# Anisotropic, Mixed-Norm Lizorkin-Triebel Spaces and Diffeomorphic Maps

Authors: J. Johnsen; S. Munch Hansen; W. Sickel
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

This paper gives general results on invariance of anisotropic Lizorkin-Triebel spaces with mixed norms under coordinate transformations on Euclidean space, open sets, and cylindrical domains.


---

## Body

## 1. Introduction

This paper continues a study of anisotropic Lizorkin-Triebel spacesF
p
→
,
q
s
,
a
→
(
ℝ
n
) with mixed norms, which was begun in [1, 2] and followed up in our joint work [3].First Sobolev embeddings and completeness of the scaleF
p
→
,
q
s
,
a
→
(
ℝ
n
) were established in [1], using the Nikol’skiĭ-Plancherel-Polya inequality for sequences of functions in the mixed-norm space L
p
→
(
ℝ
n
), which was obtained straightforwardly in [1]. Then a detailed trace theory for hyperplanes in ℝ
n was worked out in [2], for example, with the novelty that the well-known borderline s
=
1
/
p has to be shifted upwards in some cases, because of the mixed norms.Secondly, our joint paper [3] presented some general characterisations of F
p
→
,
q
s
,
a
→
(
ℝ
n
), which may be specialised to kernels of local means, in Triebel’s sense [4]. One interest of this is that local means have recently been useful for obtaining wavelet bases of Sobolev spaces and especially of their generalisations to the Besov and Lizorkin-Triebel scales (cf. works of Vybiral [5, Theorem  2.12], Triebel [6, Theorem  1.20], Hansen [7, Theorem  4.3.1]).In the present paper, we treat the invariance ofF
p
→
,
q
s
,
a
→ under coordinate changes. During the discussions below, the results in [3] are crucial for the entire strategy.Indeed, we address the main technical challenge to obtain invariance ofF
p
→
,
q
s
,
a
→
(
ℝ
n
) under the map
(1)
f
⟼
f
∘
σ
,
when σ is a bounded diffeomorphism on ℝ
n. (Cf. Theorems 20 and 21) Not surprisingly, this will require the condition on σ that only affects blocks of variables x
j in which the corresponding integral exponents p
j are equal, and similarly for the anisotropic weights a
j. Moreover, when estimating the operator norm of f
↦
f
∘
σ, that is, obtaining the inequality
(2)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
,
the Fourier analytic definition of the spaces seems difficult to manage directly, so as done by Triebel [4] we have chosen to characterise F
p
→
,
q
s
,
a
→
(
ℝ
n
) in terms of local means as developed in [3].However, the diffeomorphism invariance relies not just on the local means, but first of all also on techniques underlying them. In particular, we use the following inequality for the maximal functionψ
j
*
f
(
x
) of Peetre-Fefferman-Stein type, which was established in [3, Theorem  2] for mixed norms and with uniformity with respect to a general parameter θ:
(3)
∥
{
2
s
j
sup
⁡
θ
∈
Θ
ψ
θ
,
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
{
2
s
j
φ
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
.
Hereby the “cut-off” functions ψ
j, φ
j should fulfill a set of Tauberian and moment conditions; cf. Theorem 14 for the full statement. In the isotropic case this inequality originated in a well-known article of Rychkov [8], which contains a serious flaw (as pointed out in [7]); this and other inaccuracies were corrected in [3].A second adaptation of Triebel’s approach is caused by the anisotropya
→ we treat here. In fact, our proof only extends to, for example, s
<
0 by means of the unconventional lift operator
(4)
Λ
r
=
OP
(
λ
r
)
,
λ
r
(
ξ
)
=
∑
j
=
1
n
‍
(
1
+
ξ
j
2
)
r
/
2
a
j
.
Moreover, to cover all a
→
=
(
a
1
,
…
,
a
n
), especially to allow irrational ratios a
j
/
a
k, we found it useful to invoke the corresponding pseudodifferential operators (
1
-
∂
j
2
)
μ
=
OP
(
(
1
+
ξ
j
2
)
μ
) that for μ
∈
ℝ are shown here to be bounded F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
2
a
j
μ
,
a
→
(
ℝ
n
) for all s.Local versions of our result, in whichσ is only defined on subsets of ℝ
n, are also treated below. In short form we have, for example, the following result (cf. Theorem 22).Theorem 1.
LetU
,
V
⊂
ℝ
n be open and let σ
:
U
→
V be a C
∞-bijection on the form σ
(
x
)
=
(
σ
′
(
x
1
,
…
,
x
n
-
1
)
,
x
n
). When f
∈
F
p
→
,
q
s
,
a
→
(
V
) has compact support and all p
j are equal for j
<
n, and similarly for the a
j, then f
∘
σ
∈
F
p
→
,
q
s
,
a
→
(
U
) and
(5)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
≤
c
(
supp
⁡
f
,
σ
)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
.This is useful for introduction of Lizorkin-Triebel spaces on cylindrical manifolds. However, this subject is postponed to our forthcoming paper [9]. (Already this part of the mixed-norm theory has seemingly not been elucidated before.) Moreover, in [9] we also carry over trace results from [2] to spaces over a smooth cylindrical domain in Euclidean space, for example, by analysing boundedness and ranges for traces on the flat and curved parts of its boundary.To elucidate the importance of the results here and in [9], we recall that the F
p
→
,
q
s
,
a
→ are relevant for parabolic differential equations with initial and boundary value conditions: when solutions are sought in a mixed-norm Lebesgue space L
p
→ (in order to allow different properties in the space and time directions), then F
p
→
,
q
s
,
a
→-spaces are in general inevitable for a correct description of nontrivial data on the curved boundary.This conclusion was obtained in works of Weidemaier [10–12], who treated several special cases; one may also consult the introduction of [2] for details.Contents. Section 2 contains a review of our notation, and the definition of anisotropic Lizorkin-Triebel spaces with mixed norms is recalled, together with some needed properties, a discussion of different lift operators and a pointwise multiplier assertion.In Section3 results from [3] on characterisation of F
p
→
,
q
s
,
a
→-spaces by local means are recalled and used to prove an important lemma for compactly supported elements in F
p
→
,
q
s
,
a
→. Sufficient conditions for f
↦
f
∘
σ to leave the spaces F
p
→
,
q
s
,
a
→
(
ℝ
n
) invariant for all s
∈
ℝ are deduced in Section 4, when σ is a bounded diffeomorphism. Local versions for spaces on domains are derived in Section 5 together with isotropic results.


## 2. Preliminaries

### 2.1. Notation

The Schwartz space𝒮
(
ℝ
n
) contains all rapidly decreasing C
∞-functions. It is equipped with the family of seminorms, using D
α
:
=
(
-
i
∂
x
1
)
α
1
⋯
(
-
i
∂
x
n
)
α
n for each multi-index α
=
(
α
1
,
…
,
α
n
) with α
j
∈
ℕ
0
:
=
ℕ
∪
{
0
}, and 〈
x
〉
2
:
=
1
+
|
x
|
2,
(6)
p
M
(
φ
)
≔
sup
⁡
{
〈
x
〉
M
|
D
α
φ
(
x
)
|
∣
x
∈
ℝ
n
,
|
α
|
≤
M
}
,
≔
s
{
〈
x
〉
M
|
D
α
φ
(
x
)
|
∣
x
∈
ℝ
n
,
|
α
|
≤
M
M
}
,
M
∈
ℕ
0
or with
(7)
q
N
,
α
(
ψ
)
:
=
∫
ℝ
n
‍
〈
x
〉
N
|
D
α
ψ
(
x
)
|
d
x
,
N
∈
ℕ
0
,
α
∈
ℕ
0
n
.
The Fourier transformation ℱ
g
(
ξ
)
=
g
^
(
ξ
)
=
∫
ℝ
n
‍
e
-
i
x
·
ξ
g
(
x
)
d
x for g
∈
𝒮
(
ℝ
n
) extends by duality to the dual space 𝒮
′
(
ℝ
n
) of temperate distributions.Inequalities for vectorsp
→
=
(
p
1
,
…
,
p
n
) are understood componentwise; as are functions, for example, p
→
!
=
p
1
!
⋯
p
n
!. Moreover, t
+
:
=
max
⁡
(
0
,
t
) for t
∈
ℝ.For0
<
p
→
≤
∞ the space L
p
→
(
ℝ
n
) consists of all Lebesgue measurable functions such that
(8)
∥
u
∣
L
p
→
(
ℝ
n
)
∥
≔
(
∫
-
∞
∞
‍
(
⋯
(
∫
-
∞
∞
‍
|
u
(
x
1
,
…
,
x
n
)
|
p
1
d
x
1
)
p
2
/
p
1
∫
-
∞
∞
‍
⋯
)
p
n
/
p
n
-
1
d
x
n
)
1
/
p
n
<
∞
,
with the modification of using the essential supremum over x
j in case p
j
=
∞. Equipped with this quasinorm, L
p
→
(
ℝ
n
) is a quasi-Banach space (normed if p
j
≥
1 for all j).Furthermore, for0
<
q
≤
∞ we will use the notation L
p
→
(
ℓ
q
)
(
ℝ
n
) for the space of all sequences {
u
k
}
k
=
0
∞ of Lebesgue measurable functions u
k
:
ℝ
n
→
ℂ such that
(9)
∥
{
u
k
}
k
=
0
∞
∣
L
p
→
(
ℓ
q
)
(
ℝ
n
)
∥
:
=
∥
(
∑
k
=
0
∞
‍
|
u
k
(
·
)
|
q
)
1
/
q
∣
L
p
→
(
ℝ
n
)
∥
<
∞
,
with supremum over k in case q
=
∞. This quasi-norm is often abbreviated to ∥
u
k
∣
L
p
→
(
ℓ
q
)
∥ and when p
→
=
(
p
,
…
,
p
) we simplify L
p
→ to L
p. If max
⁡
(
p
1
,
…
,
p
n
,
q
)
<
∞ sequences of C
0
∞-functions are dense in L
p
→
(
ℓ
q
).Generic constants will primarily be denoted byc or C and when relevant, their dependence on certain parameters will be explicitly stated. B
(
0
,
r
) stands for the ball in ℝ
n centered at 0 with radius r
>
0, and U
¯ denotes the closure of a set U
⊂
ℝ
n.


### 2.2. Anisotropic Lizorkin-Triebel Spaces with Mixed Norms

The scales of mixed-norm Lizorkin-Triebel spaces refine the scales of mixed-norm Sobolev spaces (cf. [2, Proposition  2.10]), and hence the history of these spaces goes far back in time; the reader is referred to [3, Remark  2.3] and [1, Remark  10] for a brief historical overview, which also list some of the ways to define Lizorkin-Triebel spaces.Our exposition uses the Fourier-analytic definition, but first we recall the definition of the anisotropic distance function|
·
|
a
→, where a
→
=
(
a
1
,
…
,
a
n
)
∈
[
1
,
∞
[
n, on ℝ
n and some of its properties. Using the quasihomogeneous dilation t
a
→
x
:
=
(
t
a
1
x
1
,
…
,
t
a
n
x
n
) for t
≥
0, |
x
|
a
→ is for x
∈
ℝ
n
∖
{
0
} defined as the unique t
>
0 such that t
-
a
→
x
∈
S
n
-
1 (|
0
|
a
→
:
=
0), that is,(10)
x
1
2
t
2
a
1
+
⋯
+
x
n
2
t
2
a
n
=
1
.
By the Implicit Function Theorem, |
·
|
a
→ is C
∞ on ℝ
n
∖
{
0
}. We also recall the quasi-homogeneity |
t
a
→
x
|
a
→
=
t
|
x
|
a
→ together with (cf. [1, Section  3])
(11)
|
x
+
y
|
a
→
≤
|
x
|
a
→
+
|
y
|
a
→
,
(12)
max
⁡
(
|
x
1
|
1
/
a
1
,
…
,
|
x
n
|
1
/
a
n
)
≤
|
x
|
a
→
≤
|
x
1
|
1
/
a
1
+
⋯
+
|
x
n
|
1
/
a
n
.The definition ofF
p
→
,
q
s
,
a
→
(
ℝ
n
) uses a Littlewood-Paley decomposition, that is, 1
=
∑
j
=
0
∞
‍
Φ
j
(
ξ
), which (for convenience) is based on a fixed ψ
∈
C
0
∞ such that 0
≤
ψ
(
ξ
)
≤
1 for all ξ, ψ
(
ξ
)
=
1 if |
ξ
|
a
→
≤
1 and ψ
(
ξ
)
=
0 if |
ξ
|
a
→
≥
3
/
2; setting Φ
=
ψ
-
ψ
(
2
a
→
·
), we define
(13)
Φ
0
(
ξ
)
:
=
ψ
(
ξ
)
,
Φ
j
(
ξ
)
:
=
Φ
(
2
-
j
a
→
ξ
)
,
j
=
1,2
,
.Definition 2.
The Lizorkin-Triebel spaceF
p
→
,
q
s
,
a
→
(
ℝ
n
) with s
∈
ℝ, 0
<
p
→
<
∞ and 0
<
q
≤
∞ consists of all u
∈
𝒮
′
(
ℝ
n
) such that
(14)
∥
u
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
≔
∥
(
∑
j
=
0
∞
‍
2
j
s
q
|
ℱ
-
1
(
Φ
j
(
ξ
)
ℱ
u
(
ξ
)
)
(
·
)
|
q
)
1
/
q
∣
L
p
→
(
ℝ
n
)
∥
<
∞
.The numberq is called the sum exponent and the entries in p
→ are integral exponents, while s is a smoothness index. Usually the statements are valid for the full ranges 0
<
p
→
<
∞, 0
<
q
≤
∞, so we refrain from repeating these. Instead we focus on whether s
∈
ℝ is allowed or not. In the isotropic case, that is, a
→
=
(
1
,
…
,
1
), the parameter a
→ is omitted.We will also consider the closely related Besov spaces, recalled using the abbreviation(15)
u
j
(
x
)
:
=
ℱ
-
1
(
Φ
j
(
ξ
)
ℱ
u
(
ξ
)
)
(
x
)
,
x
∈
ℝ
n
,
j
∈
ℕ
0
.Definition 3.
The Besov spaceB
p
→
,
q
s
,
a
→
(
ℝ
n
) consists of all u
∈
𝒮
′
(
ℝ
n
) such that
(16)
∥
u
∣
B
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
:
=
(
∑
j
=
0
∞
‍
2
j
s
q
∥
u
j
∣
L
p
→
(
ℝ
n
)
∥
q
)
1
/
q
<
∞
.In [1, 2] many results on these classes are elaborated, and hence we just recall a few facts. They are quasi-Banach spaces (Banach spaces if min
⁡
(
p
1
,
…
,
p
n
,
q
)
≥
1) and the quasinorm is subadditive, when raised to the power d
:
=
min
⁡
(
1
,
p
1
,
…
,
p
n
,
q
),
(17)
∥
u
+
v
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
≤
∥
u
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
+
∥
v
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
,
u
,
v
∈
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
.
Also the spaces do not depend on the chosen anisotropic decomposition of unity (up to equivalent quasinorms) and there are continuous embeddings
(18)
𝒮
(
ℝ
n
)
↪
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
↪
𝒮
′
(
ℝ
n
)
,
where 𝒮 is dense in F
p
→
,
q
s
,
a
→ for q
<
∞.Since forλ
>
0, the space F
p
→
,
q
s
,
a
→ coincides with F
p
→
,
q
λ
s
,
λ
a
→, cf. [2, Lemma  3.24], most results obtained for the scales when a
→
≥
1 can be extended to the case 0
<
a
→
<
1 (for details we refer to [3, Remark  2.6]).The subspaceL
1
,
loc
⁡
(
ℝ
n
)
⊂
𝒟
′
(
ℝ
n
) of locally integrable functions is equipped with the Fréchet space topology defined from the seminorms u
↦
∫
|
x
|
≤
j
‍
|
u
(
x
)
|
d
x, j
∈
ℕ. By C
b
(
ℝ
n
) we denote the Banach space of bounded, continuous functions, endowed with the supremum norm.Lemma 4.
Lets
∈
ℝ and α
∈
ℕ
0
n be arbitrary. (i)
The differential operatorD
α is bounded F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
a
→
·
α
,
a
→
(
ℝ
n
).
(ii)
Fors
>
∑
ℓ
=
1
n
‍
(
(
a
ℓ
/
p
ℓ
)
-
a
ℓ
)
+ there is an embedding F
p
→
,
q
s
,
a
→
(
ℝ
n
)
↪
L
1
,
loc
⁡
(
ℝ
n
).
(iii)
The embeddingF
p
→
,
q
s
,
a
→
↪
C
b
(
ℝ
n
) holds true whenever s
>
(
a
1
/
p
1
)
+
⋯
+
(
a
n
/
p
n
).Proof.
For part (i) the reader is referred to [2, Lemma  3.22], where a proof using standard techniques for F
p
→
,
q
s
,
a
→ is indicated (though the cross-reference in that proof in [2] should have been to Proposition  3.13 instead of 3.14).
Part (ii) is obtained from the Nikol’skij inequality (cf. [1, Corollary  3.8]), which allows a reduction to the case in which p
j
≥
1 for j
=
1
,
…
,
n, while s
>
0; then the claim follows from the embedding F
p
→
,
1
0
,
a
→
↪
L
1
,
loc
⁡. Part (iii) follows at once from [2, (3.20)].A local maximisation over a ball can be estimated inL
p
→, at least for functions in certain subspaces of C
b
(
ℝ
n
) (cf. Lemma 4(iii)).Lemma 5 (see [3]).
Whens
>
∑
l
=
1
n
‍
a
l
/
min
⁡
(
p
1
,
…
,
p
l
), then for each C
>
0
(19)
∥
sup
⁡
|
x
-
y
|
<
C
|
u
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
≤
c
∥
u
∣
F
p
→
,
q
s
,
a
→
∥
.Next we extend a well-known embedding to the mixed-norm setting. LetC
*
ρ
(
ℝ
n
) denote the Hölder class of order ρ
>
0, which by definition consists of all u
∈
C
k
(
ℝ
n
) satisfying
(20)
∥
u
∥
ρ
:
=
∑
|
α
|
≤
k
‍
sup
⁡
x
∈
ℝ
n
|
D
α
u
(
x
)
|
+
∑
|
α
|
=
k
‍
sup
⁡
x
-
y
∈
ℝ
n
∖
{
0
}
|
D
α
u
(
x
)
-
D
α
u
(
y
)
|
|
x
-
y
|
k
-
ρ
<
∞
,
whereby k is the integer satisfying k
<
ρ
≤
k
+
1.Lemma 6.
Forρ
>
0 and s
∈
ℝ with s
≤
ρ there is an embedding C
*
ρ
(
ℝ
n
)
↪
B
∞
,
∞
s
,
a
→
(
ℝ
n
).Proof.
The claim follows by adapting the proof of [14, Proposition  8.6.1] to the anisotropic case, that is,
(21)
∥
u
∣
B
∞
,
∞
s
,
a
→
∥
=
sup
⁡
j
∈
ℕ
0
2
s
j
sup
⁡
x
∈
ℝ
n
|
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
|
≤
c
ρ
∥
u
∥
ρ
.
The expressions in the Besov norm are for j
≥
1 estimated using that ℱ
-
1
Φ has vanishing moments of arbitrary order,
(22)
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
=
∫
‍
ℱ
-
1
Φ
(
y
)
×
(
u
(
x
-
2
-
j
a
→
y
)
-
∑
|
α
|
≤
k
‍
∂
α
u
(
x
)
α
!
(
-
2
-
j
a
→
y
)
α
)
d
y
.
Using a Taylor expansion of order k
-
1 with k
∈
ℕ chosen such that k
<
ρ
≤
k
+
1 (or directly if k
=
0), we get an estimate of the parenthesis by
(23)
|
∑
|
α
|
=
k
‍
k
α
!
(
-
2
-
j
a
→
y
)
α
×
∫
0
1
‍
(
1
-
θ
)
k
-
1
(
∂
α
u
(
x
-
2
-
j
a
→
θ
y
)
-
∂
α
u
(
x
)
)
d
θ
∑
|
α
|
=
k
‍
k
α
!
|
≤
∑
|
α
|
=
k
‍
k
α
!
|
2
-
j
a
→
y
|
k
∥
u
∥
ρ
|
2
-
j
a
→
y
|
ρ
-
k
∫
0
1
‍
(
1
-
θ
)
k
-
1
d
θ
≤
c
ρ
′
|
2
-
j
a
→
y
|
ρ
∥
u
∥
ρ
.
Now we obtain, since a
→
≥
1,
(24)
sup
⁡
x
∈
ℝ
n
|
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
|
≤
c
ρ
′
2
-
j
ρ
∥
u
∥
ρ
∫
‍
|
ℱ
-
1
Φ
(
y
)
|
|
y
|
ρ
d
y
≤
c
ρ
2
-
j
ρ
∥
u
∥
ρ
.
This bound can also be used for j
=
0, if c
ρ is large enough, so (21) holds for ρ
≥
s.As a tool we also need to know the mapping properties of certain Fourier multipliersλ
(
D
)
u
:
=
ℱ
-
1
(
λ
(
ξ
)
u
^
(
ξ
)
). For generality’s sake, we give the following.Proposition 7.
Whenλ
∈
C
∞
(
ℝ
n
) for some r
∈
ℝ has finite seminorms of the form
(25)
C
α
(
λ
)

≔

sup
⁡
{
2
-
j
(
r
-
a
→
·
α
)
|
D
α
λ
(
2
j
a
→
ξ
)
|
∣
j
∈
ℕ
0
,
1
4
≤
|
ξ
|
a
→
≤
4
}

,
≔
p
{
2
-
j
(
r
-
a
→
·
α
)
|
D
α
λ
(
2
j
a
→
ξ
)
|
∣
j
∈
ℕ
0
,
1
4
≤
|
ξ
|
a
d
→
≤
4
d
}
,
α
∈
ℕ
0
n
,
then λ
(
D
) is continuous on 𝒮
′
(
ℝ
n
) and moreover bounded F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
r
,
a
→
(
ℝ
n
) for all s
∈
ℝ, with operator norm ∥
λ
(
D
)
∥
≤
c
p
→
,
q
∑
|
α
|
≤
N
p
→
,
q
‍
C
α
(
λ
).Proof.
The quasi-homogeneity of|
·
|
a
→ yields that |
D
α
λ
(
ξ
)
|
≤
c
C
α
(
λ
)
(
1
+
|
ξ
|
a
→
)
r
-
a
→
·
α, and hence every derivative is of polynomial growth (cf. (12)), so λ
(
D
) is a well-defined continuous map on 𝒮
′. Boundedness follows as in the proof of [2, Proposition  3.15], mutatis mutandis. In fact, only the last step there needs an adaptation to the symbol λ
(
ξ
), but this is trivial because finitely many of the constants C
α
(
λ
) can enter the estimates.


### 2.3. Lift Operators

The invariance under coordinate transformations will be established below using a somewhat unconventional lift operatorΛ
r, r
∈
ℝ,
(26)
Λ
r
u
=
OP
(
λ
r
(
ξ
)
)
u
=
ℱ
-
1
(
λ
r
(
ξ
)
u
^
(
ξ
)
)
,
λ
r
(
ξ
)
=
∑
k
=
1
n
‍
(
1
+
ξ
k
2
)
r
/
(
2
a
k
)
.
To apply Proposition 7, we derive an estimate uniformly in j
∈
ℕ
0 and over the set 1
/
4
≤
|
ξ
|
a
→
≤
4: while the mixed derivatives vanish, the explicit higher order chain rule in the Appendix yields
(27)
|
D
ξ
l
α
l
(
2
-
j
r
λ
r
(
2
j
a
→
ξ
)
)
|
≤
∑
k
=
1
α
l
‍
c
k
(
2
-
2
j
a
l
+
ξ
l
2
)
(
r
/
2
a
l
)
-
k
2
j
(
α
l
a
l
-
2
k
a
l
)
×
∑
k
=
n
1
+
n
2
α
l
=
n
1
+
2
n
2
‍
(
2
(
2
j
a
l
ξ
l
)
)
n
1
2
n
2
<
∞
.
Indeed, the precise summation range gives α
l
=
n
1
+
2
(
k
-
n
1
), so the harmless power 2
n
1
+
n
2 results. (Note that this means that |
D
α
λ
r
(
2
j
a
→
ξ
)
|
≤
C
α
2
j
(
r
-
a
→
·
α
).)Nowλ
r
(
ξ
) has no zeros, and for λ
r
(
ξ
)
-
1 it is analogous to obtain such estimates uniformly with respect to j of D
α
(
2
j
r
λ
r
(
2
j
a
→
ξ
)
-
1
), using the Appendix and the aforementioned. So Proposition 7 gives both that Λ
r is a homeomorphism on 𝒮
′ (although Λ
r
-
1
≠
Λ
-
r) and the proof of our lemma.Lemma 8.
The mapΛ
r is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
r
,
a
→
(
ℝ
n
) for s
∈
ℝ.In a similar way one also finds the next auxiliary result.Lemma 9.
For anyμ
∈
ℝ, k
∈
{
1
,
…
,
n
} the operator (
1
-
∂
x
k
2
)
μ
u
=
OP
(
(
1
+
ξ
k
2
)
μ
)
u is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
2
μ
a
k
,
a
→
(
ℝ
n
) for all s
∈
ℝ.A standard choice of an anisotropic lift operator is obtained by associating eachξ
∈
ℝ
n with (
1
,
ξ
)
∈
ℝ
1
+
n, which is given the weights (
1
,
a
→
), and by setting
(28)
〈
ξ
〉
a
→
=
|
1
,
ξ
|
(
1
,
a
→
)
.
This is in C
∞, as |
·
|
(
1
,
a
→
) is so outside the origin. (Note the analogy to 〈
ξ
〉
=
1
+
|
ξ
|
2.) Moreover, ∂
α
〈
ξ
〉
a
→
t is for each t
∈
ℝ estimated by powers of |
ξ
|, (cf. [15, Lemma  1.4]). Therefore, there is a linear homeomorphism Ξ
a
→
t
:
𝒮
′
→
𝒮
′ given by
(29)
Ξ
a
→
t
u
:
=
OP
(
〈
ξ
〉
a
→
t
)
u
=
ℱ
-
1
(
〈
ξ
〉
a
→
t
u
^
(
ξ
)
)
,
t
∈
ℝ
.
In our mixed-norm setup, it is a small exercise to show that it restricts to a homeomorphism
(30)
Ξ
a
→
t
:
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
⟶
F
p
→
,
q
s
-
t
,
a
→
(
ℝ
n
)
,
∀
s
∈
ℝ
.
Indeed, invoking Proposition 7, the task is as in (27) to show a uniform bound, and using the elementary properties of 〈
ξ
〉
a
→ (cf. [15, Lemma  1.4]) one finds for t
-
a
→
·
α
≥
0,
(31)
|
D
α
(
2
-
j
t
〈
2
j
a
→
ξ
〉
a
→
t
)
|
=
2
j
(
a
→
·
α
-
t
)
|
D
η
α
〈
η
〉
a
→
t
∣
η
=
2
j
a
→
ξ
|
≤
c
2
j
(
a
→
·
α
-
t
)
〈
2
j
a
→
ξ
〉
a
→
t
-
a
→
·
α
≤
c
〈
ξ
〉
t
-
a
→
·
α
.
When t
-
a
→
·
α
≤
0, then |
ξ
|
a
→
t
-
a
→
·
α is the outcome on the right-hand side. But the uniformity results in both cases, since the estimates pertain to (1
/
4
)
≤
|
ξ
|
a
→
≤
4.We digress to recall that the classical fractional Sobolev spaceH
p
→
s
,
a
→
(
ℝ
n
), for s
∈
ℝ and 1
<
p
→
<
∞, consists of the u
∈
𝒮
′ for which Ξ
a
→
s
u
∈
L
p
→
(
ℝ
n
), with ∥
u
∣
H
p
→
s
,
a
→
∥
:
=
∥
Ξ
a
→
s
u
∣
L
p
→
∥. If m
k
:
=
s
/
a
k
∈
ℕ
0 for all k, then H
p
→
s
,
a
→ coincides (as shown by Lizorkin [16]) with the space W
p
→
(
m
1
,
…
,
m
n
)
(
ℝ
n
) of u
∈
L
p
→ having ∂
x
k
m
k
u in L
p
→ for all k.This characterisation is valid forF
p
→
,
2
s
,
a
→ with 1
<
p
→
<
∞ in view of the identification
(32)
u
∈
H
p
→
s
,
a
→
(
ℝ
n
)
⟺
u
∈
F
p
→
,
2
s
,
a
→
(
ℝ
n
)
,
which by use of Ξ
s reduces to the case L
p
→
=
F
p
→
,
2
0
,
a
→. The latter is a Littlewood-Paley inequality that may be proved with general methods of harmonic analysis (cf. [2, Remark  3.16]).A general reference on mixed-norm Sobolev spaces is the classical book of Besov et al. [13, 17]. Schmeisser and Triebel [18] treated F
p
→
,
q
s
,
a
→ for n
=
2.Remark 10.
Traces on hyperplanes were considered forH
p
→
s
,
a
→
(
ℝ
n
) by Lizorkin [16] and for W
p
→
m
→
(
ℝ
n
) by Bugrov [19], who raised the problem of traces at {
x
j
=
0
} for j
<
n. This was solved by Berkolaĭko, who treated traces in the F
p
→
,
q
s
,
a
→
(
ℝ
n
)-scales for 1
<
p
→
<
∞ in, for example, [20]. The range 0
<
p
→
<
∞ was covered on ℝ
n for j
=
1 and j
=
n in [2], and in our forthcoming paper [9], we carry over the trace results to F
p
→
,
q
s
,
a
→-spaces over a smooth cylindrical domain Ω
×
]
0
,
T
[.Remark 11.
We take the opportunity to correct a minor inaccuracy in [2], where a lift operator (also) called Λ
r unfortunately was defined to have symbol (
1
+
|
ξ
|
a
→
2
)
r
/
2. However, it is not in C
∞
(
ℝ
n
) for a
→
≠
(
1
,
…
,
1
); this can be seen from the example for n
=
2 with a
→
=
(
2,1
) where [15, Example  1.1] gives the explicit formula
(33)
|
ξ
|
a
→
=
2
-
1
/
2
(
ξ
2
2
+
(
ξ
2
4
+
4
ξ
1
2
)
1
/
2
)
1
/
2
.
Here an easy calculation shows that D
ξ
1
|
ξ
|
a
→
2 is discontinuous along the line (
ξ
1
,
0
), which is inherited by the symbol, for example, for r
=
2. The resulting operator is therefore not defined on all of 𝒮
′. However, this is straightforward to avoid by replacing the lift operator in [2] by the better choice Ξ
r given in (30). This gives the space H
p
→
s
,
a
→
(
ℝ
n
) in (32).


### 2.4. Paramultiplication

This section contains a pointwise multiplier assertion for theF
p
→
,
q
s
,
a
→-scales. We consider the densely defined product on 𝒮
′
×
𝒮
′, introduced in [21, Definition  3.1] and in an isotropic setup in [22, Chapter  4],
(34)
u
·
v
≔
lim
⁡
j
→
∞
ℱ
-
1
(
ψ
(
2
-
j
a
→
ξ
)
ℱ
u
(
ξ
)
)
·
ℱ
-
1
(
ψ
(
2
-
j
a
→
ξ
)
ℱ
v
(
ξ
)
)
,
which is considered for those pairs (
u
,
v
) in 𝒮
′
×
𝒮
′ for which the limit on the right-hand side exists in 𝒟
′ and is independent of ψ. Here ψ
∈
C
0
∞ is the function used in the construction of the Littlewood-Paley decomposition (in principle the independence should be verified for all ψ
∈
C
0
∞ equalling 1 near the origin, but this is not a problem here).To illustrate how this product extends the usual one and to prepare for an application below, the following is recalled.Lemma 12 (see [21]).
Whenf
∈
C
∞
(
ℝ
n
) has derivatives of any order of polynomial growth, and when g
∈
𝒮
′
(
ℝ
n
) is arbitrary, then the limit in (34) exists and equals the usual product f
·
g, as defined on C
∞
×
𝒟
′.Using this extended product, we introduce the usual space of multipliers(35)
M
(
F
p
→
,
q
s
,
a
→
)
:
=
{
u
∈
𝒮
′
∣
u
·
v
∈
F
p
→
,
q
s
,
a
→
,
∀
v
∈
F
p
→
,
q
s
,
a
→
}
equipped with the induced operator quasinorm
(36)
∥
u
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
:
=
sup
⁡
{
∥
u
·
v
∣
F
p
→
,
q
s
,
a
→
∥
∣
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
≤
1
}
.As Lemma6 at once yields C
L
∞
∞
⊂
∩
s
>
0
B
∞
,
∞
s
,
a
→ (a well-known result in the isotropic case) for C
L
∞
∞
:
=
{
g
∈
C
∞
∣
∀
α
:
D
α
g
∈
L
∞
}, the next result is in particular valid for u
∈
C
L
∞
∞.Lemma 13.
Lets
∈
ℝ and take s
1
>
s such that also
(37)
s
1
>
∑
ℓ
=
1
n
‍
(
a
ℓ
min
⁡
(
1
,
q
,
p
1
,
…
,
p
ℓ
)
-
a
ℓ
)
-
s
.
Then each u
∈
B
∞
,
∞
s
1
,
a
→ defines a multiplier of F
p
→
,
q
s
,
a
→ and
(38)
∥
u
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
.Proof.
The proof will be brief as it is based on standard arguments from paramultiplication, (cf. [21] and [22, Chapter  4] for details). In particular we will use the decomposition
(39)
u
·
v
=
Π
1
(
u
,
v
)
+
Π
2
(
u
,
v
)
+
Π
3
(
u
,
v
)
.
The exact form of this can also be recalled from the below formulae. In terms of the Littlewood-Paley partition 1
=
∑
j
=
0
∞
‍
Φ
j
(
ξ
) from Definition 2, we set Ψ
j
=
Φ
0
+
⋯
+
Φ
j for j
≥
1 and Ψ
0
=
Φ
0. These are used in Fourier multipliers, now written with upper indices as u
j
=
ℱ
-
1
(
Ψ
j
u
^
).
Note first thats
1
>
0, whence B
∞
,
∞
s
1
,
a
→
↪
L
∞, which is useful since the dyadic corona criterion for F
p
→
,
q
s
,
a
→ (cf. [2, Lemma  3.20]), implies the well-known simple estimate
(40)
∥
Π
1
(
u
,
v
)
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
u
∣
L
∞
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
Furthermore, since
(41)
s
2
:
=
s
1
+
s
>
∑
ℓ
=
1
n
‍
a
ℓ
min
⁡
(
1
,
q
,
p
1
,
…
,
p
ℓ
)
-
|
a
→
|
,
using the dyadic ball criterion for F
p
→
,
q
s
,
a
→ (cf. [2, Lemma  3.19]), we find that
(42)
∥
Π
2
(
u
,
v
)
∣
F
p
→
,
q
s
2
,
a
→
∥
≤
c
∥
2
j
s
2
u
j
v
j
∣
L
p
→
(
ℓ
q
)
∥
≤
c
sup
⁡
k
∈
ℕ
0
2
k
s
1
∥
u
k
∣
L
∞
∥
∥
2
j
s
|
v
j
|
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
To estimate Π
3
(
u
,
v
) we first consider the case s
>
0 and pick t
∈
]
s
,
s
1
[. The dyadic corona criterion together with the formula v
j
=
v
0
+
⋯
+
v
j and a summation lemma, which exploits that t
-
s
1
<
0 (cf. [15, Lemma  3.8]), gives
(43)
∥
Π
3
(
u
,
v
)
∣
F
p
→
,
q
t
,
a
→
∥
≤
c
sup
⁡
k
∈
ℕ
0
2
k
s
1
∥
u
k
∣
L
∞
∥
×
∥
2
(
t
-
s
1
)
j
v
j
-
2
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
2
(
t
-
s
1
)
j
∑
k
=
0
j
‍
|
v
k
|
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
t
-
s
1
,
a
→
∥
.
Since t
-
s
1
<
0
<
s implies F
p
→
,
q
s
,
a
→
↪
F
p
→
,
q
t
-
s
1
,
a
→ and also F
p
→
,
q
t
,
a
→
↪
F
p
→
,
q
s
,
a
→ holds, the above yields
(44)
∥
Π
3
(
u
,
v
)
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
For s
≤
0 the procedure is analogous, except that (43) is derived for t
∈
]
0
,
s
1
+
s
[, which is nonempty by assumption (37) on s; then standard embeddings again give (44).
In closing, we remark that as required the productu
·
v is independent of the test function ψ appearing in the definition. Indeed for q
<
∞ this follows from Lemma 12, which gives the coincidence between this product on 𝒮
′
×
𝒮 and the usual one, hence by density of 𝒮 (cf. (18)) and the above estimates, the map v
↦
u
·
v extends uniquely by continuity to all g
∈
F
p
→
,
q
s
,
a
→. For q
=
∞ the embedding F
p
→
,
∞
s
,
a
→
↪
F
p
→
,
1
s
-
ε
,
a
→ for ε
>
0 yields the independence using the previous case.


## 2.1. Notation

The Schwartz space𝒮
(
ℝ
n
) contains all rapidly decreasing C
∞-functions. It is equipped with the family of seminorms, using D
α
:
=
(
-
i
∂
x
1
)
α
1
⋯
(
-
i
∂
x
n
)
α
n for each multi-index α
=
(
α
1
,
…
,
α
n
) with α
j
∈
ℕ
0
:
=
ℕ
∪
{
0
}, and 〈
x
〉
2
:
=
1
+
|
x
|
2,
(6)
p
M
(
φ
)
≔
sup
⁡
{
〈
x
〉
M
|
D
α
φ
(
x
)
|
∣
x
∈
ℝ
n
,
|
α
|
≤
M
}
,
≔
s
{
〈
x
〉
M
|
D
α
φ
(
x
)
|
∣
x
∈
ℝ
n
,
|
α
|
≤
M
M
}
,
M
∈
ℕ
0
or with
(7)
q
N
,
α
(
ψ
)
:
=
∫
ℝ
n
‍
〈
x
〉
N
|
D
α
ψ
(
x
)
|
d
x
,
N
∈
ℕ
0
,
α
∈
ℕ
0
n
.
The Fourier transformation ℱ
g
(
ξ
)
=
g
^
(
ξ
)
=
∫
ℝ
n
‍
e
-
i
x
·
ξ
g
(
x
)
d
x for g
∈
𝒮
(
ℝ
n
) extends by duality to the dual space 𝒮
′
(
ℝ
n
) of temperate distributions.Inequalities for vectorsp
→
=
(
p
1
,
…
,
p
n
) are understood componentwise; as are functions, for example, p
→
!
=
p
1
!
⋯
p
n
!. Moreover, t
+
:
=
max
⁡
(
0
,
t
) for t
∈
ℝ.For0
<
p
→
≤
∞ the space L
p
→
(
ℝ
n
) consists of all Lebesgue measurable functions such that
(8)
∥
u
∣
L
p
→
(
ℝ
n
)
∥
≔
(
∫
-
∞
∞
‍
(
⋯
(
∫
-
∞
∞
‍
|
u
(
x
1
,
…
,
x
n
)
|
p
1
d
x
1
)
p
2
/
p
1
∫
-
∞
∞
‍
⋯
)
p
n
/
p
n
-
1
d
x
n
)
1
/
p
n
<
∞
,
with the modification of using the essential supremum over x
j in case p
j
=
∞. Equipped with this quasinorm, L
p
→
(
ℝ
n
) is a quasi-Banach space (normed if p
j
≥
1 for all j).Furthermore, for0
<
q
≤
∞ we will use the notation L
p
→
(
ℓ
q
)
(
ℝ
n
) for the space of all sequences {
u
k
}
k
=
0
∞ of Lebesgue measurable functions u
k
:
ℝ
n
→
ℂ such that
(9)
∥
{
u
k
}
k
=
0
∞
∣
L
p
→
(
ℓ
q
)
(
ℝ
n
)
∥
:
=
∥
(
∑
k
=
0
∞
‍
|
u
k
(
·
)
|
q
)
1
/
q
∣
L
p
→
(
ℝ
n
)
∥
<
∞
,
with supremum over k in case q
=
∞. This quasi-norm is often abbreviated to ∥
u
k
∣
L
p
→
(
ℓ
q
)
∥ and when p
→
=
(
p
,
…
,
p
) we simplify L
p
→ to L
p. If max
⁡
(
p
1
,
…
,
p
n
,
q
)
<
∞ sequences of C
0
∞-functions are dense in L
p
→
(
ℓ
q
).Generic constants will primarily be denoted byc or C and when relevant, their dependence on certain parameters will be explicitly stated. B
(
0
,
r
) stands for the ball in ℝ
n centered at 0 with radius r
>
0, and U
¯ denotes the closure of a set U
⊂
ℝ
n.


## 2.2. Anisotropic Lizorkin-Triebel Spaces with Mixed Norms

The scales of mixed-norm Lizorkin-Triebel spaces refine the scales of mixed-norm Sobolev spaces (cf. [2, Proposition  2.10]), and hence the history of these spaces goes far back in time; the reader is referred to [3, Remark  2.3] and [1, Remark  10] for a brief historical overview, which also list some of the ways to define Lizorkin-Triebel spaces.Our exposition uses the Fourier-analytic definition, but first we recall the definition of the anisotropic distance function|
·
|
a
→, where a
→
=
(
a
1
,
…
,
a
n
)
∈
[
1
,
∞
[
n, on ℝ
n and some of its properties. Using the quasihomogeneous dilation t
a
→
x
:
=
(
t
a
1
x
1
,
…
,
t
a
n
x
n
) for t
≥
0, |
x
|
a
→ is for x
∈
ℝ
n
∖
{
0
} defined as the unique t
>
0 such that t
-
a
→
x
∈
S
n
-
1 (|
0
|
a
→
:
=
0), that is,(10)
x
1
2
t
2
a
1
+
⋯
+
x
n
2
t
2
a
n
=
1
.
By the Implicit Function Theorem, |
·
|
a
→ is C
∞ on ℝ
n
∖
{
0
}. We also recall the quasi-homogeneity |
t
a
→
x
|
a
→
=
t
|
x
|
a
→ together with (cf. [1, Section  3])
(11)
|
x
+
y
|
a
→
≤
|
x
|
a
→
+
|
y
|
a
→
,
(12)
max
⁡
(
|
x
1
|
1
/
a
1
,
…
,
|
x
n
|
1
/
a
n
)
≤
|
x
|
a
→
≤
|
x
1
|
1
/
a
1
+
⋯
+
|
x
n
|
1
/
a
n
.The definition ofF
p
→
,
q
s
,
a
→
(
ℝ
n
) uses a Littlewood-Paley decomposition, that is, 1
=
∑
j
=
0
∞
‍
Φ
j
(
ξ
), which (for convenience) is based on a fixed ψ
∈
C
0
∞ such that 0
≤
ψ
(
ξ
)
≤
1 for all ξ, ψ
(
ξ
)
=
1 if |
ξ
|
a
→
≤
1 and ψ
(
ξ
)
=
0 if |
ξ
|
a
→
≥
3
/
2; setting Φ
=
ψ
-
ψ
(
2
a
→
·
), we define
(13)
Φ
0
(
ξ
)
:
=
ψ
(
ξ
)
,
Φ
j
(
ξ
)
:
=
Φ
(
2
-
j
a
→
ξ
)
,
j
=
1,2
,
.Definition 2.
The Lizorkin-Triebel spaceF
p
→
,
q
s
,
a
→
(
ℝ
n
) with s
∈
ℝ, 0
<
p
→
<
∞ and 0
<
q
≤
∞ consists of all u
∈
𝒮
′
(
ℝ
n
) such that
(14)
∥
u
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
≔
∥
(
∑
j
=
0
∞
‍
2
j
s
q
|
ℱ
-
1
(
Φ
j
(
ξ
)
ℱ
u
(
ξ
)
)
(
·
)
|
q
)
1
/
q
∣
L
p
→
(
ℝ
n
)
∥
<
∞
.The numberq is called the sum exponent and the entries in p
→ are integral exponents, while s is a smoothness index. Usually the statements are valid for the full ranges 0
<
p
→
<
∞, 0
<
q
≤
∞, so we refrain from repeating these. Instead we focus on whether s
∈
ℝ is allowed or not. In the isotropic case, that is, a
→
=
(
1
,
…
,
1
), the parameter a
→ is omitted.We will also consider the closely related Besov spaces, recalled using the abbreviation(15)
u
j
(
x
)
:
=
ℱ
-
1
(
Φ
j
(
ξ
)
ℱ
u
(
ξ
)
)
(
x
)
,
x
∈
ℝ
n
,
j
∈
ℕ
0
.Definition 3.
The Besov spaceB
p
→
,
q
s
,
a
→
(
ℝ
n
) consists of all u
∈
𝒮
′
(
ℝ
n
) such that
(16)
∥
u
∣
B
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
:
=
(
∑
j
=
0
∞
‍
2
j
s
q
∥
u
j
∣
L
p
→
(
ℝ
n
)
∥
q
)
1
/
q
<
∞
.In [1, 2] many results on these classes are elaborated, and hence we just recall a few facts. They are quasi-Banach spaces (Banach spaces if min
⁡
(
p
1
,
…
,
p
n
,
q
)
≥
1) and the quasinorm is subadditive, when raised to the power d
:
=
min
⁡
(
1
,
p
1
,
…
,
p
n
,
q
),
(17)
∥
u
+
v
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
≤
∥
u
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
+
∥
v
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
d
,
u
,
v
∈
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
.
Also the spaces do not depend on the chosen anisotropic decomposition of unity (up to equivalent quasinorms) and there are continuous embeddings
(18)
𝒮
(
ℝ
n
)
↪
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
↪
𝒮
′
(
ℝ
n
)
,
where 𝒮 is dense in F
p
→
,
q
s
,
a
→ for q
<
∞.Since forλ
>
0, the space F
p
→
,
q
s
,
a
→ coincides with F
p
→
,
q
λ
s
,
λ
a
→, cf. [2, Lemma  3.24], most results obtained for the scales when a
→
≥
1 can be extended to the case 0
<
a
→
<
1 (for details we refer to [3, Remark  2.6]).The subspaceL
1
,
loc
⁡
(
ℝ
n
)
⊂
𝒟
′
(
ℝ
n
) of locally integrable functions is equipped with the Fréchet space topology defined from the seminorms u
↦
∫
|
x
|
≤
j
‍
|
u
(
x
)
|
d
x, j
∈
ℕ. By C
b
(
ℝ
n
) we denote the Banach space of bounded, continuous functions, endowed with the supremum norm.Lemma 4.
Lets
∈
ℝ and α
∈
ℕ
0
n be arbitrary. (i)
The differential operatorD
α is bounded F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
a
→
·
α
,
a
→
(
ℝ
n
).
(ii)
Fors
>
∑
ℓ
=
1
n
‍
(
(
a
ℓ
/
p
ℓ
)
-
a
ℓ
)
+ there is an embedding F
p
→
,
q
s
,
a
→
(
ℝ
n
)
↪
L
1
,
loc
⁡
(
ℝ
n
).
(iii)
The embeddingF
p
→
,
q
s
,
a
→
↪
C
b
(
ℝ
n
) holds true whenever s
>
(
a
1
/
p
1
)
+
⋯
+
(
a
n
/
p
n
).Proof.
For part (i) the reader is referred to [2, Lemma  3.22], where a proof using standard techniques for F
p
→
,
q
s
,
a
→ is indicated (though the cross-reference in that proof in [2] should have been to Proposition  3.13 instead of 3.14).
Part (ii) is obtained from the Nikol’skij inequality (cf. [1, Corollary  3.8]), which allows a reduction to the case in which p
j
≥
1 for j
=
1
,
…
,
n, while s
>
0; then the claim follows from the embedding F
p
→
,
1
0
,
a
→
↪
L
1
,
loc
⁡. Part (iii) follows at once from [2, (3.20)].A local maximisation over a ball can be estimated inL
p
→, at least for functions in certain subspaces of C
b
(
ℝ
n
) (cf. Lemma 4(iii)).Lemma 5 (see [3]).
Whens
>
∑
l
=
1
n
‍
a
l
/
min
⁡
(
p
1
,
…
,
p
l
), then for each C
>
0
(19)
∥
sup
⁡
|
x
-
y
|
<
C
|
u
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
≤
c
∥
u
∣
F
p
→
,
q
s
,
a
→
∥
.Next we extend a well-known embedding to the mixed-norm setting. LetC
*
ρ
(
ℝ
n
) denote the Hölder class of order ρ
>
0, which by definition consists of all u
∈
C
k
(
ℝ
n
) satisfying
(20)
∥
u
∥
ρ
:
=
∑
|
α
|
≤
k
‍
sup
⁡
x
∈
ℝ
n
|
D
α
u
(
x
)
|
+
∑
|
α
|
=
k
‍
sup
⁡
x
-
y
∈
ℝ
n
∖
{
0
}
|
D
α
u
(
x
)
-
D
α
u
(
y
)
|
|
x
-
y
|
k
-
ρ
<
∞
,
whereby k is the integer satisfying k
<
ρ
≤
k
+
1.Lemma 6.
Forρ
>
0 and s
∈
ℝ with s
≤
ρ there is an embedding C
*
ρ
(
ℝ
n
)
↪
B
∞
,
∞
s
,
a
→
(
ℝ
n
).Proof.
The claim follows by adapting the proof of [14, Proposition  8.6.1] to the anisotropic case, that is,
(21)
∥
u
∣
B
∞
,
∞
s
,
a
→
∥
=
sup
⁡
j
∈
ℕ
0
2
s
j
sup
⁡
x
∈
ℝ
n
|
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
|
≤
c
ρ
∥
u
∥
ρ
.
The expressions in the Besov norm are for j
≥
1 estimated using that ℱ
-
1
Φ has vanishing moments of arbitrary order,
(22)
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
=
∫
‍
ℱ
-
1
Φ
(
y
)
×
(
u
(
x
-
2
-
j
a
→
y
)
-
∑
|
α
|
≤
k
‍
∂
α
u
(
x
)
α
!
(
-
2
-
j
a
→
y
)
α
)
d
y
.
Using a Taylor expansion of order k
-
1 with k
∈
ℕ chosen such that k
<
ρ
≤
k
+
1 (or directly if k
=
0), we get an estimate of the parenthesis by
(23)
|
∑
|
α
|
=
k
‍
k
α
!
(
-
2
-
j
a
→
y
)
α
×
∫
0
1
‍
(
1
-
θ
)
k
-
1
(
∂
α
u
(
x
-
2
-
j
a
→
θ
y
)
-
∂
α
u
(
x
)
)
d
θ
∑
|
α
|
=
k
‍
k
α
!
|
≤
∑
|
α
|
=
k
‍
k
α
!
|
2
-
j
a
→
y
|
k
∥
u
∥
ρ
|
2
-
j
a
→
y
|
ρ
-
k
∫
0
1
‍
(
1
-
θ
)
k
-
1
d
θ
≤
c
ρ
′
|
2
-
j
a
→
y
|
ρ
∥
u
∥
ρ
.
Now we obtain, since a
→
≥
1,
(24)
sup
⁡
x
∈
ℝ
n
|
ℱ
-
1
(
Φ
j
ℱ
u
)
(
x
)
|
≤
c
ρ
′
2
-
j
ρ
∥
u
∥
ρ
∫
‍
|
ℱ
-
1
Φ
(
y
)
|
|
y
|
ρ
d
y
≤
c
ρ
2
-
j
ρ
∥
u
∥
ρ
.
This bound can also be used for j
=
0, if c
ρ is large enough, so (21) holds for ρ
≥
s.As a tool we also need to know the mapping properties of certain Fourier multipliersλ
(
D
)
u
:
=
ℱ
-
1
(
λ
(
ξ
)
u
^
(
ξ
)
). For generality’s sake, we give the following.Proposition 7.
Whenλ
∈
C
∞
(
ℝ
n
) for some r
∈
ℝ has finite seminorms of the form
(25)
C
α
(
λ
)

≔

sup
⁡
{
2
-
j
(
r
-
a
→
·
α
)
|
D
α
λ
(
2
j
a
→
ξ
)
|
∣
j
∈
ℕ
0
,
1
4
≤
|
ξ
|
a
→
≤
4
}

,
≔
p
{
2
-
j
(
r
-
a
→
·
α
)
|
D
α
λ
(
2
j
a
→
ξ
)
|
∣
j
∈
ℕ
0
,
1
4
≤
|
ξ
|
a
d
→
≤
4
d
}
,
α
∈
ℕ
0
n
,
then λ
(
D
) is continuous on 𝒮
′
(
ℝ
n
) and moreover bounded F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
r
,
a
→
(
ℝ
n
) for all s
∈
ℝ, with operator norm ∥
λ
(
D
)
∥
≤
c
p
→
,
q
∑
|
α
|
≤
N
p
→
,
q
‍
C
α
(
λ
).Proof.
The quasi-homogeneity of|
·
|
a
→ yields that |
D
α
λ
(
ξ
)
|
≤
c
C
α
(
λ
)
(
1
+
|
ξ
|
a
→
)
r
-
a
→
·
α, and hence every derivative is of polynomial growth (cf. (12)), so λ
(
D
) is a well-defined continuous map on 𝒮
′. Boundedness follows as in the proof of [2, Proposition  3.15], mutatis mutandis. In fact, only the last step there needs an adaptation to the symbol λ
(
ξ
), but this is trivial because finitely many of the constants C
α
(
λ
) can enter the estimates.


## 2.3. Lift Operators

The invariance under coordinate transformations will be established below using a somewhat unconventional lift operatorΛ
r, r
∈
ℝ,
(26)
Λ
r
u
=
OP
(
λ
r
(
ξ
)
)
u
=
ℱ
-
1
(
λ
r
(
ξ
)
u
^
(
ξ
)
)
,
λ
r
(
ξ
)
=
∑
k
=
1
n
‍
(
1
+
ξ
k
2
)
r
/
(
2
a
k
)
.
To apply Proposition 7, we derive an estimate uniformly in j
∈
ℕ
0 and over the set 1
/
4
≤
|
ξ
|
a
→
≤
4: while the mixed derivatives vanish, the explicit higher order chain rule in the Appendix yields
(27)
|
D
ξ
l
α
l
(
2
-
j
r
λ
r
(
2
j
a
→
ξ
)
)
|
≤
∑
k
=
1
α
l
‍
c
k
(
2
-
2
j
a
l
+
ξ
l
2
)
(
r
/
2
a
l
)
-
k
2
j
(
α
l
a
l
-
2
k
a
l
)
×
∑
k
=
n
1
+
n
2
α
l
=
n
1
+
2
n
2
‍
(
2
(
2
j
a
l
ξ
l
)
)
n
1
2
n
2
<
∞
.
Indeed, the precise summation range gives α
l
=
n
1
+
2
(
k
-
n
1
), so the harmless power 2
n
1
+
n
2 results. (Note that this means that |
D
α
λ
r
(
2
j
a
→
ξ
)
|
≤
C
α
2
j
(
r
-
a
→
·
α
).)Nowλ
r
(
ξ
) has no zeros, and for λ
r
(
ξ
)
-
1 it is analogous to obtain such estimates uniformly with respect to j of D
α
(
2
j
r
λ
r
(
2
j
a
→
ξ
)
-
1
), using the Appendix and the aforementioned. So Proposition 7 gives both that Λ
r is a homeomorphism on 𝒮
′ (although Λ
r
-
1
≠
Λ
-
r) and the proof of our lemma.Lemma 8.
The mapΛ
r is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
r
,
a
→
(
ℝ
n
) for s
∈
ℝ.In a similar way one also finds the next auxiliary result.Lemma 9.
For anyμ
∈
ℝ, k
∈
{
1
,
…
,
n
} the operator (
1
-
∂
x
k
2
)
μ
u
=
OP
(
(
1
+
ξ
k
2
)
μ
)
u is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
-
2
μ
a
k
,
a
→
(
ℝ
n
) for all s
∈
ℝ.A standard choice of an anisotropic lift operator is obtained by associating eachξ
∈
ℝ
n with (
1
,
ξ
)
∈
ℝ
1
+
n, which is given the weights (
1
,
a
→
), and by setting
(28)
〈
ξ
〉
a
→
=
|
1
,
ξ
|
(
1
,
a
→
)
.
This is in C
∞, as |
·
|
(
1
,
a
→
) is so outside the origin. (Note the analogy to 〈
ξ
〉
=
1
+
|
ξ
|
2.) Moreover, ∂
α
〈
ξ
〉
a
→
t is for each t
∈
ℝ estimated by powers of |
ξ
|, (cf. [15, Lemma  1.4]). Therefore, there is a linear homeomorphism Ξ
a
→
t
:
𝒮
′
→
𝒮
′ given by
(29)
Ξ
a
→
t
u
:
=
OP
(
〈
ξ
〉
a
→
t
)
u
=
ℱ
-
1
(
〈
ξ
〉
a
→
t
u
^
(
ξ
)
)
,
t
∈
ℝ
.
In our mixed-norm setup, it is a small exercise to show that it restricts to a homeomorphism
(30)
Ξ
a
→
t
:
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
⟶
F
p
→
,
q
s
-
t
,
a
→
(
ℝ
n
)
,
∀
s
∈
ℝ
.
Indeed, invoking Proposition 7, the task is as in (27) to show a uniform bound, and using the elementary properties of 〈
ξ
〉
a
→ (cf. [15, Lemma  1.4]) one finds for t
-
a
→
·
α
≥
0,
(31)
|
D
α
(
2
-
j
t
〈
2
j
a
→
ξ
〉
a
→
t
)
|
=
2
j
(
a
→
·
α
-
t
)
|
D
η
α
〈
η
〉
a
→
t
∣
η
=
2
j
a
→
ξ
|
≤
c
2
j
(
a
→
·
α
-
t
)
〈
2
j
a
→
ξ
〉
a
→
t
-
a
→
·
α
≤
c
〈
ξ
〉
t
-
a
→
·
α
.
When t
-
a
→
·
α
≤
0, then |
ξ
|
a
→
t
-
a
→
·
α is the outcome on the right-hand side. But the uniformity results in both cases, since the estimates pertain to (1
/
4
)
≤
|
ξ
|
a
→
≤
4.We digress to recall that the classical fractional Sobolev spaceH
p
→
s
,
a
→
(
ℝ
n
), for s
∈
ℝ and 1
<
p
→
<
∞, consists of the u
∈
𝒮
′ for which Ξ
a
→
s
u
∈
L
p
→
(
ℝ
n
), with ∥
u
∣
H
p
→
s
,
a
→
∥
:
=
∥
Ξ
a
→
s
u
∣
L
p
→
∥. If m
k
:
=
s
/
a
k
∈
ℕ
0 for all k, then H
p
→
s
,
a
→ coincides (as shown by Lizorkin [16]) with the space W
p
→
(
m
1
,
…
,
m
n
)
(
ℝ
n
) of u
∈
L
p
→ having ∂
x
k
m
k
u in L
p
→ for all k.This characterisation is valid forF
p
→
,
2
s
,
a
→ with 1
<
p
→
<
∞ in view of the identification
(32)
u
∈
H
p
→
s
,
a
→
(
ℝ
n
)
⟺
u
∈
F
p
→
,
2
s
,
a
→
(
ℝ
n
)
,
which by use of Ξ
s reduces to the case L
p
→
=
F
p
→
,
2
0
,
a
→. The latter is a Littlewood-Paley inequality that may be proved with general methods of harmonic analysis (cf. [2, Remark  3.16]).A general reference on mixed-norm Sobolev spaces is the classical book of Besov et al. [13, 17]. Schmeisser and Triebel [18] treated F
p
→
,
q
s
,
a
→ for n
=
2.Remark 10.
Traces on hyperplanes were considered forH
p
→
s
,
a
→
(
ℝ
n
) by Lizorkin [16] and for W
p
→
m
→
(
ℝ
n
) by Bugrov [19], who raised the problem of traces at {
x
j
=
0
} for j
<
n. This was solved by Berkolaĭko, who treated traces in the F
p
→
,
q
s
,
a
→
(
ℝ
n
)-scales for 1
<
p
→
<
∞ in, for example, [20]. The range 0
<
p
→
<
∞ was covered on ℝ
n for j
=
1 and j
=
n in [2], and in our forthcoming paper [9], we carry over the trace results to F
p
→
,
q
s
,
a
→-spaces over a smooth cylindrical domain Ω
×
]
0
,
T
[.Remark 11.
We take the opportunity to correct a minor inaccuracy in [2], where a lift operator (also) called Λ
r unfortunately was defined to have symbol (
1
+
|
ξ
|
a
→
2
)
r
/
2. However, it is not in C
∞
(
ℝ
n
) for a
→
≠
(
1
,
…
,
1
); this can be seen from the example for n
=
2 with a
→
=
(
2,1
) where [15, Example  1.1] gives the explicit formula
(33)
|
ξ
|
a
→
=
2
-
1
/
2
(
ξ
2
2
+
(
ξ
2
4
+
4
ξ
1
2
)
1
/
2
)
1
/
2
.
Here an easy calculation shows that D
ξ
1
|
ξ
|
a
→
2 is discontinuous along the line (
ξ
1
,
0
), which is inherited by the symbol, for example, for r
=
2. The resulting operator is therefore not defined on all of 𝒮
′. However, this is straightforward to avoid by replacing the lift operator in [2] by the better choice Ξ
r given in (30). This gives the space H
p
→
s
,
a
→
(
ℝ
n
) in (32).


## 2.4. Paramultiplication

This section contains a pointwise multiplier assertion for theF
p
→
,
q
s
,
a
→-scales. We consider the densely defined product on 𝒮
′
×
𝒮
′, introduced in [21, Definition  3.1] and in an isotropic setup in [22, Chapter  4],
(34)
u
·
v
≔
lim
⁡
j
→
∞
ℱ
-
1
(
ψ
(
2
-
j
a
→
ξ
)
ℱ
u
(
ξ
)
)
·
ℱ
-
1
(
ψ
(
2
-
j
a
→
ξ
)
ℱ
v
(
ξ
)
)
,
which is considered for those pairs (
u
,
v
) in 𝒮
′
×
𝒮
′ for which the limit on the right-hand side exists in 𝒟
′ and is independent of ψ. Here ψ
∈
C
0
∞ is the function used in the construction of the Littlewood-Paley decomposition (in principle the independence should be verified for all ψ
∈
C
0
∞ equalling 1 near the origin, but this is not a problem here).To illustrate how this product extends the usual one and to prepare for an application below, the following is recalled.Lemma 12 (see [21]).
Whenf
∈
C
∞
(
ℝ
n
) has derivatives of any order of polynomial growth, and when g
∈
𝒮
′
(
ℝ
n
) is arbitrary, then the limit in (34) exists and equals the usual product f
·
g, as defined on C
∞
×
𝒟
′.Using this extended product, we introduce the usual space of multipliers(35)
M
(
F
p
→
,
q
s
,
a
→
)
:
=
{
u
∈
𝒮
′
∣
u
·
v
∈
F
p
→
,
q
s
,
a
→
,
∀
v
∈
F
p
→
,
q
s
,
a
→
}
equipped with the induced operator quasinorm
(36)
∥
u
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
:
=
sup
⁡
{
∥
u
·
v
∣
F
p
→
,
q
s
,
a
→
∥
∣
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
≤
1
}
.As Lemma6 at once yields C
L
∞
∞
⊂
∩
s
>
0
B
∞
,
∞
s
,
a
→ (a well-known result in the isotropic case) for C
L
∞
∞
:
=
{
g
∈
C
∞
∣
∀
α
:
D
α
g
∈
L
∞
}, the next result is in particular valid for u
∈
C
L
∞
∞.Lemma 13.
Lets
∈
ℝ and take s
1
>
s such that also
(37)
s
1
>
∑
ℓ
=
1
n
‍
(
a
ℓ
min
⁡
(
1
,
q
,
p
1
,
…
,
p
ℓ
)
-
a
ℓ
)
-
s
.
Then each u
∈
B
∞
,
∞
s
1
,
a
→ defines a multiplier of F
p
→
,
q
s
,
a
→ and
(38)
∥
u
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
.Proof.
The proof will be brief as it is based on standard arguments from paramultiplication, (cf. [21] and [22, Chapter  4] for details). In particular we will use the decomposition
(39)
u
·
v
=
Π
1
(
u
,
v
)
+
Π
2
(
u
,
v
)
+
Π
3
(
u
,
v
)
.
The exact form of this can also be recalled from the below formulae. In terms of the Littlewood-Paley partition 1
=
∑
j
=
0
∞
‍
Φ
j
(
ξ
) from Definition 2, we set Ψ
j
=
Φ
0
+
⋯
+
Φ
j for j
≥
1 and Ψ
0
=
Φ
0. These are used in Fourier multipliers, now written with upper indices as u
j
=
ℱ
-
1
(
Ψ
j
u
^
).
Note first thats
1
>
0, whence B
∞
,
∞
s
1
,
a
→
↪
L
∞, which is useful since the dyadic corona criterion for F
p
→
,
q
s
,
a
→ (cf. [2, Lemma  3.20]), implies the well-known simple estimate
(40)
∥
Π
1
(
u
,
v
)
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
u
∣
L
∞
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
Furthermore, since
(41)
s
2
:
=
s
1
+
s
>
∑
ℓ
=
1
n
‍
a
ℓ
min
⁡
(
1
,
q
,
p
1
,
…
,
p
ℓ
)
-
|
a
→
|
,
using the dyadic ball criterion for F
p
→
,
q
s
,
a
→ (cf. [2, Lemma  3.19]), we find that
(42)
∥
Π
2
(
u
,
v
)
∣
F
p
→
,
q
s
2
,
a
→
∥
≤
c
∥
2
j
s
2
u
j
v
j
∣
L
p
→
(
ℓ
q
)
∥
≤
c
sup
⁡
k
∈
ℕ
0
2
k
s
1
∥
u
k
∣
L
∞
∥
∥
2
j
s
|
v
j
|
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
To estimate Π
3
(
u
,
v
) we first consider the case s
>
0 and pick t
∈
]
s
,
s
1
[. The dyadic corona criterion together with the formula v
j
=
v
0
+
⋯
+
v
j and a summation lemma, which exploits that t
-
s
1
<
0 (cf. [15, Lemma  3.8]), gives
(43)
∥
Π
3
(
u
,
v
)
∣
F
p
→
,
q
t
,
a
→
∥
≤
c
sup
⁡
k
∈
ℕ
0
2
k
s
1
∥
u
k
∣
L
∞
∥
×
∥
2
(
t
-
s
1
)
j
v
j
-
2
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
2
(
t
-
s
1
)
j
∑
k
=
0
j
‍
|
v
k
|
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
t
-
s
1
,
a
→
∥
.
Since t
-
s
1
<
0
<
s implies F
p
→
,
q
s
,
a
→
↪
F
p
→
,
q
t
-
s
1
,
a
→ and also F
p
→
,
q
t
,
a
→
↪
F
p
→
,
q
s
,
a
→ holds, the above yields
(44)
∥
Π
3
(
u
,
v
)
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
u
∣
B
∞
,
∞
s
1
,
a
→
∥
∥
v
∣
F
p
→
,
q
s
,
a
→
∥
.
For s
≤
0 the procedure is analogous, except that (43) is derived for t
∈
]
0
,
s
1
+
s
[, which is nonempty by assumption (37) on s; then standard embeddings again give (44).
In closing, we remark that as required the productu
·
v is independent of the test function ψ appearing in the definition. Indeed for q
<
∞ this follows from Lemma 12, which gives the coincidence between this product on 𝒮
′
×
𝒮 and the usual one, hence by density of 𝒮 (cf. (18)) and the above estimates, the map v
↦
u
·
v extends uniquely by continuity to all g
∈
F
p
→
,
q
s
,
a
→. For q
=
∞ the embedding F
p
→
,
∞
s
,
a
→
↪
F
p
→
,
1
s
-
ε
,
a
→ for ε
>
0 yields the independence using the previous case.


## 3. Characterisation by Local Means

Characterisation of Lizorkin-Triebel spacesF
p
,
q
s by local means is due to Triebel, [4, 2.4.6], and it was from the outset an important tool in proving invariance of the scale under diffeomorphisms. An extensive treatment of characterisations of mixed-norm spaces F
p
→
,
q
s
,
a
→ in terms of quasinorms based on convolutions, in particular the case of local means, was given in [3], which to a large extent is based on extensions to mixed norms of inequalities in [8]. For the reader’s convenience we recall the needed results.Throughout this section we consider a fixed anisotropya
→
≥
1 with a
_
:
=
min
⁡
(
a
1
,
…
,
a
n
) and functions ψ
0
,
ψ
∈
𝒮
(
ℝ
n
) that fulfil Tauberian conditions in terms of some ε
>
0 and/or a moment condition of order M
ψ
≥
-
1 (M
ψ
=
-
1 means that the condition is void),
(45)
|
ℱ
ψ
0
(
ξ
)
|
>
0
on
⁡
{
ξ
∣
|
ξ
|
a
→
<
2
ε
}
,
(46)
|
ℱ
ψ
(
ξ
)
|
>
0
on
⁡
{
ξ
∣
ε
/
2
<
|
ξ
|
a
→
<
2
ε
}
,
(47)
D
α
(
ℱ
ψ
)
(
0
)
=
0
for
|
α
|
≤
M
ψ
.
Note by (12) that in case (45) is fulfilled for the Euclidean distance, it holds true also in the anisotropic case, perhaps with a different ε.We henceforth change notation, from (15), to
(48)
φ
j
(
x
)
=
2
j
|
a
→
|
φ
(
2
j
a
→
x
)
,
φ
∈
𝒮
,
j
∈
ℕ
,
which gives rise to the sequence (
ψ
j
)
j
∈
ℕ
0. The nonlinear Peetre-Fefferman-Stein maximal operators induced by (
ψ
j
)
j
∈
ℕ
0 are for an arbitrary vector r
→
=
(
r
1
,
…
,
r
n
)
>
0 and any f
∈
𝒮
′
(
ℝ
n
) given by (dependence on a
→ and r
→ is omitted)
(49)
ψ
j
*
f
(
x
)
=
sup
⁡
y
∈
ℝ
n
|
ψ
j
*
f
(
y
)
|
∏
ℓ
=
1
n
‍
(
1
+
2
j
a
ℓ
|
x
ℓ
-
y
ℓ
|
)
r
ℓ
,
x
∈
ℝ
n
,
j
∈
ℕ
0
.
Later we will also refer to the trivial estimate
(50)
|
ψ
j
*
f
(
x
)
|
≤
ψ
j
*
f
(
x
)
.Finally for an index setΘ, we consider ψ
θ
,
0
,
ψ
θ
∈
𝒮
(
ℝ
n
), θ
∈
Θ, where the ψ
θ satisfy (47) for some M
ψ
θ independent of θ
∈
Θ, and also φ
0
,
φ
∈
𝒮
(
ℝ
n
) that fulfil (45)-(46) in terms of an ε
′
>
0. Setting ψ
θ
,
j
(
x
)
=
2
j
|
a
→
|
ψ
θ
(
2
j
a
→
x
) for j
∈
ℕ, we can state the first result relating different quasinorms.Theorem 14 (see [3]).
Let0
<
p
→
<
∞, 0
<
q
≤
∞, and -
∞
<
s
<
(
M
ψ
θ
+
1
)
a
_. For a given r
→ in (49) and an integer M
≥
-
1 chosen so large that (
M
+
1
)
a
_
-
2
a
→
·
r
→
+
s
>
0, we assume that
(51)
A
∶
=
sup
⁡
θ
∈
Θ
max
⁡
∥
D
α
ℱ
ψ
θ
∣
L
∞
∥
<
∞
,
B
∶
=
sup
⁡
θ
∈
Θ
max
⁡
∥
(
1
+
|
ξ
|
)
M
+
1
D
γ
ℱ
ψ
θ
(
ξ
)
∣
L
1
∥
<
∞
,
C
∶
=
sup
⁡
θ
∈
Θ
max
⁡
∥
D
α
ℱ
ψ
θ
,
0
∣
L
∞
∥
<
∞
,
D
∶
=
sup
⁡
θ
∈
Θ
max
⁡
∥
(
1
+
|
ξ
|
)
M
+
1
D
γ
ℱ
ψ
θ
,
0
(
ξ
)
∣
L
1
∥
<
∞
,
where the maxima are over α such that |
α
|
≤
M
ψ
θ
+
1 or α
≤
⌈
r
→
+
2
⌉, respectively, over γ with γ
j
≤
r
j
+
2. Then there exists a constant c
>
0 such that for f
∈
𝒮
′
(
ℝ
n
),
(52)
∥
{
2
s
j
sup
⁡
θ
∈
Θ
ψ
θ
,
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
(
A
+
B
+
C
+
D
)
∥
{
2
s
j
φ
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
.It is also possible to estimate the maximal function in terms of the convolution appearing in its numerator.Theorem 15 (see [3]).
Letψ
0
,
ψ
∈
𝒮
(
ℝ
n
) satisfy the Tauberian conditions (45)-(46). When 0
<
p
→
<
∞, 0
<
q
≤
∞, -
∞
<
s
<
∞, and
(53)
1
r
l
<
min
⁡
(
q
,
p
1
,
…
,
p
n
)
,
l
=
1
,
…
,
n
,
there exists a constant c
>
0 such that for f
∈
𝒮
′
(
ℝ
n
),
(54)
∥
{
2
s
j
ψ
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∥
{
2
s
j
ψ
j
*
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
.As a consequence of Theorems14 and 15 (the first applied for a trivial index set like Θ
=
{
1
}), we obtain the characterisation of F
p
→
,
q
s
,
a
→-spaces by local means.Theorem 16 (see [3]).
Letk
0
,
k
0
∈
𝒮 such that ∫
‍
k
0
(
x
)
d
x
≠
0
≠
∫
‍
k
0
(
x
)
d
x and set k
(
x
)
=
Δ
N
k
0
(
x
) for some N
∈
ℕ. When 0
<
p
→
<
∞, 0
<
q
≤
∞, and -
∞
<
s
<
2
N
a
_, then a distribution f
∈
𝒮
′
(
ℝ
n
) belongs to F
p
→
,
q
s
,
a
→
(
ℝ
n
) if and only if (cf. (48) for the k
j)
(55)
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
*
:
=
∥
k
0
*
f
∣
L
p
→
∥
+
∥
{
2
s
j
k
j
*
f
}
j
=
1
∞
∣
L
p
→
(
ℓ
q
)
∥
<
∞
.
Furthermore, ∥
f
∣
F
p
→
,
q
s
,
a
→
∥
* is an equivalent quasinorm on F
p
→
,
q
s
,
a
→
(
ℝ
n
).Application of Theorem16 yields a useful result regarding Lizorkin-Triebel spaces on open subsets, when these are defined by restriction:Definition 17.
LetU
⊂
ℝ
n be open. The space F
p
→
,
q
s
,
a
→
(
U
) is defined as the set of all u
∈
𝒟
′
(
U
) such that there exists a distribution f
∈
F
p
→
,
q
s
,
a
→
(
ℝ
n
) satisfying
(56)
f
(
φ
)
=
u
(
φ
)
∀
φ
∈
C
0
∞
(
U
)
.
We equip F
p
→
,
q
s
,
a
→
(
U
) with the quotient quasinorm given by ∥
u
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
=
inf
⁡
r
U
f
=
u
∥
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥; it is normed if p
→
,
q
≥
1.In (56) it is tacitly understood that on the left-hand side φ is extended by 0 outside U. For this we henceforth use the operator notation e
U
φ. Likewise r
U denotes restriction to U, whereby u
=
r
U
f in (56).The Besov spacesB
p
→
,
q
s
,
a
→
(
U
) on U can be defined analogously. The quotient norms have the well-known advantage that embeddings and completeness can be transferred directly from the spaces on ℝ
n. However, the spaces are probably of little interest, if ∂
U does not satisfy some regularity conditions because we then expect (as in the isotropic case) that they do not coincide with those defined intrinsically.Lemma 18.
LetU
⊂
ℝ
n be open and r
>
0. When F
p
→
,
q
s
,
a
→
(
U
) has the infimum quasinorm given by derived from the local means in Theorem 16 fulfilling supp
⁡
k
0
,
supp
⁡
k
⊂
B
(
0
,
r
), and
(57)
dist
⁡
(
supp
⁡
f
,
ℝ
n
∖
U
)
>
2
r
holds for some f
∈
F
p
→
,
q
s
,
a
→
(
U
) with compact support, then
(58)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
=
∥
e
U
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
.
In other words, the infimum is attained at e
U
f for such f.Proof.
For any other extensionf
~
∈
𝒮
′
(
ℝ
n
) the difference g
=
f
~
-
e
U
f is nonzero in 𝒮
′
(
ℝ
n
) and supp
⁡
e
U
f
∩
supp
⁡
g
=
∅. So by the properties of r,
(59)
supp
⁡
(
k
j
*
e
U
f
)
∩
supp
⁡
(
k
j
*
g
)
=
∅
,
j
∈
ℕ
0
.
Since g
≠
0, there is some j such that supp
⁡
(
k
j
*
g
)
≠
∅, and hence k
j
*
g
(
x
)
≠
0 on an open set disjoint from supp
⁡
(
k
j
*
e
U
f
). This term therefore effectively contributes to the L
p
→-norm in (55) and thus ∥
f
~
∣
F
p
→
,
q
s
,
a
→
∥
=
∥
e
U
f
+
g
∣
F
p
→
,
q
s
,
a
→
∥
>
∥
e
U
f
∣
F
p
→
,
q
s
,
a
→
∥, which shows (58).


## 4. Invariance under Diffeomorphisms

The aim of this section is to show thatF
p
→
,
q
s
,
a
→
(
ℝ
n
) is invariant under suitable diffeomorphisms σ
:
ℝ
n
→
ℝ
n and from this deduce similar results in a variety of setups.


### 4.1. Bounded Diffeomorphisms

A one-to-one mappingy
=
σ
(
x
) of ℝ
n onto ℝ
n is here called a diffeomorphism if the components σ
j
:
ℝ
n
→
ℝ have classical derivatives D
α
σ
j for all α
∈
ℕ
n. We set τ
(
y
)
=
σ
-
1
(
y
).For convenienceσ is called a bounded diffeomorphism when σ and τ furthermore satisfy
(60)
C
α
,
σ
:
=
max
⁡
j
∈
{
1
,
…
,
n
}
∥
D
α
σ
j
∣
L
∞
∥
<
∞
,
(61)
C
α
,
τ
:
=
max
⁡
j
∈
{
1
,
…
,
n
}
∥
D
α
τ
j
∣
L
∞
∥
<
∞
.
In this case there are obviously positive constants (when J
σ denotes the Jacobian matrix)
(62)
c
σ
:
=
inf
⁡
x
∈
ℝ
n
|
det
⁡
J
σ
(
x
)
|
>
0
,
c
τ
:
=
inf
⁡
y
∈
ℝ
n
|
det
⁡
J
τ
(
y
)
|
>
0
.
For example, by the Leibniz formula for determinants, c
σ
≥
1
/
(
n
!
∏
|
α
|
=
1
‍
C
α
,
τ
)
>
0.Conversely, whenever aC
∞-map σ
:
ℝ
n
→
ℝ
n fulfils (60) and that c
σ
>
0, then τ is C
∞ (as J
τ
(
y
)
=
1
/
(
det
⁡
J
σ
(
τ
(
y
)
)
)
Adj
J
σ
(
τ
(
y
)
), if Adj denotes the adjugate, each τ
j
τ
k is in C
m if τ is so) and using, for example, the Appendix it is seen by induction over |
α
| that also (61) is fulfilled. Hence such a σ is a bounded diffeomorphism.Recall that for a bounded diffeomorphismσ and a temperate distribution f, the composition f
∘
σ denotes the temperate distribution given by
(63)
〈
f
∘
σ
,
ψ
〉
=
〈
f
,
ψ
∘
τ
|
det
⁡
J
τ
|
〉
for
ψ
∈
𝒮
.
It is continuous 𝒮
′
→
𝒮
′ as the adjoint of the continuous map ψ
↦
ψ
∘
τ
|
det
⁡
J
τ
| on 𝒮: since |
det
⁡
J
τ
| is in C
L
∞
∞, continuity on 𝒮 can be shown using the higher-order chain rule to estimate each seminorm q
N
,
α
(
ψ
∘
τ
), cf. (7), by ∑
|
β
|
≤
|
α
|
‍
q
N
,
β
(
ψ
) (changing variables, 〈
σ
(
·
)
〉 can be estimated using the Mean Value Theorem on each σ
j).We need a few further conditions, due to the anisotropic situation: one can neither expectf
∘
σ to have the same regularity as f, for example, if σ is a rotation, nor that f
∘
σ
∈
L
p
→ when f
∈
L
p
→. On these grounds we first restrict to the situation in which
(64)
a
0
:
=
a
1
=
a
2
=
⋯
=
a
n
-
1
,
p
0
:
=
p
1
=
⋯
=
p
n
-
1
,
(65)
σ
(
x
)
=
(
σ
′
(
x
1
,
…
,
x
n
-
1
)
,
x
n
)
,
∀
x
∈
ℝ
n
.To prepare for Theorem20, which gives sufficient conditions for the invariance of F
p
→
,
q
s
,
a
→ under bounded diffeomorphisms of the type (65), we first show that it suffices to have invariance for sufficiently large s.Proposition 19.
Letσ be a bounded diffeomorphism on ℝ
n on the form in (65). When (64) holds and there exists s
1
∈
ℝ with the property that f
↦
f
∘
σ is a linear homeomorphism of F
p
→
,
q
s
,
a
→
(
ℝ
n
) onto itself for every s
>
s
1, then this holds true for all s
∈
ℝ.Proof.
It suffices to prove fors
≤
s
1 that
(66)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
with some constant independent of f, as the reverse inequality then follows from the fact that the inverse of σ is also a bounded diffeomorphism with the structure in (65).
Firstr
>
s
1
-
s
+
2
a
n is chosen such that d
0
:
=
(
r
/
2
a
0
) is a natural number. Setting d
n
=
(
r
/
2
a
n
) and taking μ
∈
[
0,1
[ such that d
n
-
μ
∈
ℕ, we have that r
μ
:
=
r
-
2
μ
a
n
>
s
1
-
s.
Now Lemma8 yields the existence of h
∈
F
p
→
,
q
s
+
r
,
a
→ such that f
=
Λ
r
h, that is,
(67)
f
=
(
1
-
∂
x
n
2
)
d
n
-
μ
(
1
-
∂
x
n
2
)
μ
h
+
∑
k
=
1
n
-
1
‍
(
1
-
∂
x
k
2
)
d
0
h
.
Setting g
1
=
(
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ and g
0
=
h
∘
σ, we may apply the higher-order chain rule to for example h
=
g
0
∘
τ (using denseness of 𝒮 in 𝒮
′ and the 𝒮
′-continuity of composition in (63), the Appendix  extends to 𝒮
′). Taking into account that τ
(
x
)
=
(
τ
′
(
x
′
)
,
x
n
), and letting prime indicate summation over multi-indices with β
n
=
0,
(68)
f
=
∑
l
=
0
d
n
-
μ
‍
η
n
,
l
∂
x
n
2
l
g
1
∘
τ
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
η
k
,
β
∂
β
g
0
∘
τ
,
where η
n
,
l
:
=
(
-
1
)
l
(
d
n
-
μ
l
) and the η
k
,
β are functions containing derivatives at least of order 1 of τ, and these can be estimated, say by c
∏
1
≤
m
≤
2
d
0
‍
〈
∂
x
k
m
τ
〉
2
d
0. Composing with σ and applying Lemma 4(i) gives for d
:
=
min
⁡
(
1
,
q
,
p
0
,
p
n
), when ∥
·
∥ denotes the F
p
→
,
q
s
,
a
→-norm,
(69)
∥
f
∘
σ
∥
d
≤
∑
l
=
0
d
n
-
μ
‍
|
η
n
,
l
|
d
∥
∂
x
n
2
l
g
1
∥
d
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
∥
η
k
,
β
∘
σ
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
d
∥
∂
β
g
0
∥
d
≤
c
∥
g
1
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
∥
g
0
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
∥
η
k
,
β
∘
σ
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
d
.
According to the remark preceding Lemma 13, the last sum is finite because η
k
,
β
∈
C
L
∞
∞. Finally, since s
+
r
μ
>
s
1 and s
+
r
>
s
1, the stated assumption means that h
↦
g
1 and h
↦
g
0 are bounded, which in view of r
μ
+
2
μ
a
n
=
r and Lemmas 8 and 9 yields
(70)
∥
f
∘
σ
∥
d
≤
c
∥
h
∣
F
p
→
,
q
s
+
r
μ
+
2
μ
a
n
,
a
→
∥
d
+
∥
h
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
d
,
proving the boundedness of f
↦
f
∘
σ in F
p
→
,
q
s
,
a
→ for all s
∈
ℝ.In addition to the reduction in Proposition19, we adopt in Theorem 20 the strategy for the isotropic, unmixed case developed by Triebel [4, 4.3.2], who used Taylor expansions for the inner and outer functions for large s.While his explanation was rather sketchy, our task is to account for the fact that the strategy extends to anisotropies and to mixed norms. Hence we give full details. This will also allow us to give brief proofs of additional results in Sections4.2 and 5.To control the Taylor expansions, it will be crucial for us to exploit both the local means recalled in Theorem16 and the parameter-dependent setup in Theorem 14. This is prepared for with the following discussion.The functionsk
0 and k in Theorem 16 are for the proof of Theorem 20 chosen (as we may) so that N in the definition of k fulfils s
<
2
N
a
_ and so that both are even functions and
(71)
supp
⁡
k
0
,
supp
⁡
k
⊂
{
x
∈
ℝ
n
∣
|
x
|
≤
1
}
.
The set Θ in Theorem 14 is chosen to be the set of (
n
-
1
)
×
(
n
-
1
) matrices ℬ
=
(
b
i
,
k
) that, in terms of the constants c
σ
,
C
α
,
σ in (62) and (60), respectively, satisfy
(72)
|
det
⁡
ℬ
|
≥
c
σ
,
(73)
max
⁡
i
,
k
|
b
i
,
k
|
≤
max
⁡
|
α
|
=
1
C
α
,
σ
=
:
C
σ
.Splittingz
=
(
z
′
,
z
n
), we set g
(
z
)
=
z
′
γ
′
k
(
z
) for some γ
′
∈
ℕ
0
n
-
1 (chosen later) and define
(74)
ψ
θ
(
y
)
=
g
(
𝒜
y
′
,
y
n
)
where θ is identified with 𝒜
-
1
:
=
J
σ
′
(
x
′
), which obviously belongs to Θ (for each x
′).To verify that the above functionsψ
θ, θ
∈
Θ, satisfy the moment condition (47) for an M
ψ
θ such that the assumption s
<
(
M
ψ
θ
+
1
)
a
_ in Theorem 14 is fulfilled, note that
(75)
ψ
^
θ
(
ξ
)
=
|
det
⁡
𝒜
|
-
1
ℱ
g
(
𝒜
t
-
1
ξ
′
,
ξ
n
)
.
Hence D
α
ψ
^
θ vanishes at ξ
=
0 when D
α
g
^
=
D
α
(
-
D
ξ
′
)
γ
′
k
^
(
ξ
) does so. As k
^
(
ξ
)
=
-
|
ξ
|
2
N
k
0
^
(
ξ
) and k
0
^
(
0
)
≠
0, we have D
α
g
^
(
0
)
=
0 for α satisfying |
α
|
+
|
γ
′
|
≤
2
N
-
1. In the course of the proof below (cf. Step  3), we obtain a θ-independent estimate of |
γ
′
|, hence of M
ψ
θ.Moreover, the constantA in Theorem 14 is finite: basic properties of the Fourier transform give the following estimate, where the constant is independent of 𝒜
-
1
∈
Θ:
(76)
∥
D
α
ℱ
ψ
θ
∣
L
∞
∥
≤
∫
‍
|
y
α
g
(
𝒜
y
′
,
y
n
)
|
d
y
=
|
det
⁡
𝒜
-
1
|
∫
‍
|
z
n
α
n
|
|
(
𝒜
-
1
z
′
)
α
′
|
|
g
(
z
)
|
d
z
≤
c
(
α
,
C
σ
)
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
.
To estimate B we exploit that ℱ
:
B
2,1
n
/
2
(
ℝ
n
)
→
L
1
(
ℝ
n
) is bounded according to Szasz’s inequality (cf. [18, Proposition  1.7.5]) and obtain
(77)
∥
(
1
+
|
·
|
)
M
+
1
D
γ
ℱ
ψ
θ
∣
L
1
∥
≤
c
∥
y
γ
g
(
𝒜
y
′
,
y
n
)
∣
B
2,1
M
+
1
+
(
n
/
2
)
∥
≤
c
(
γ
,
C
σ
,
C
τ
)
∥
k
∣
C
0
m
∥
,
when m
∈
ℕ is chosen so large that m
>
M
+
1
+
n
/
2. In fact, the last inequality is obtained using the embeddings C
0
m
↪
H
m
↪
B
2,1
M
+
1
+
n
/
2 and the estimate
(78)
∥
y
γ
ψ
θ
∣
C
0
m
∥
=
sup
⁡
|
∂
α
(
y
γ
(
𝒜
y
′
)
γ
′
k
(
𝒜
y
′
,
y
n
)
)
|
≤
c
(
γ
,
C
σ
,
C
τ
)
∥
k
∣
C
0
m
∥
.
This relies on the higher-order chain rule (cf. the Appendix and the support of k): it suffices to use the supremum over |
α
|
≤
m and {
y
∈
ℝ
n
∣
|
𝒜
y
′
|
2
+
y
n
2
≤
1
}, and for a point in this set |
y
′
|
≤
∥
𝒜
-
1
∥
|
𝒜
y
′
|
≤
c
(
C
σ
), so we need only estimate an 𝒜-independent cylinder.Replacingk by k
0 in the definition of g and setting ψ
θ
,
0
(
y
)
:
=
g
(
𝒜
y
′
,
y
n
), the finiteness of C and D follows analogously. The Tauberian properties follow from ∫
‍
k
0
≠
0
≠
∫
‍
k
0.Hence all assumptions in Theorem14 are satisfied, and we are thus ready to prove our main result.Theorem 20.
Ifσ is a bounded diffeomorphism on ℝ
n on the form in (65), then f
↦
f
∘
σ is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
,
a
→
(
ℝ
n
) for all s
∈
ℝ when (64) holds.Proof.
According to Proposition19, it suffices to consider s
>
s
1, say for
(79)
s
1
:
=
K
0
a
0
+
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
,
where by K
0 is the smallest integer satisfying
(80)
K
0
a
0
>
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
.
We now let s
∈
]
s
1
,
∞
[ be given and take some K
≥
K
0, that is, K solving (80), such that
(81)
K
a
0
+
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
<
s
<
2
K
a
0
.
(The interval thus defined is nonempty by (80), and the left end point is at least s
1.)
Note that (81) yields that every f
∈
F
p
→
,
q
s
,
a
→ is continuous, (cf. Lemma 4(iii)), so are even the derivatives D
β
f for β
=
(
β
1
,
…
,
β
n
-
1
,
0
), |
β
|
≤
K, since s
-
β
·
a
→
=
s
-
|
β
|
a
0
>
a
→
·
1
/
p
→.
Step  1. For the norms ∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥ and ∥
f
∣
F
p
→
,
q
s
,
a
→
∥ in inequality (66), which also here suffices, we use Theorem 16 with 2
N
>
(
K
-
1
)
(
2
K
-
1
)
+
s
/
a
_.
By the symmetry ofk
0 and k in (71), we will estimate
(82)
k
j
*
(
f
∘
σ
)
(
x
)

=

∫
|
z
|
≤
1
‍
k
(
z
)
f
(
σ
(
x
+
2
-
j
a
→
z
)
)
d
z
,
j
∈
ℕ
,
together with the corresponding expression for k
0, where k is replaced by k
0.
First we make a Taylor expansion of the entries inσ
′
(
x
′
)
:
=
(
σ
1
(
x
′
)
,
…
,
σ
n
-
1
(
x
′
)
) to the order 2
K
-
1. So for ℓ
=
1
,
…
,
n
-
1 there exists ω
ℓ
∈
]
0,1
[ such that
(83)
σ
ℓ
(
x
′
+
z
′
)
=
∑
|
α
′
|
<
2
K
‍
∂
α
′
σ
ℓ
(
x
′
)
α
′
!
z
′
α
′
+
∑
|
α
′
|
=
2
K
‍
∂
α
′
σ
ℓ
(
x
′
+
ω
ℓ
z
′
)
α
′
!
z
′
α
′
.
For convenience, we let ∑
‍
α
′ denote summation over multiindices α
∈
ℕ
0
n having α
n
=
0 and define the vector of Taylor polynomials, respectively, entries of a remainder R,
(84)
P
2
K
-
1
(
z
′
)
=
∑
′
|
α
|
≤
2
K
-
1
∂
α
σ
′
(
x
′
)
α
!
z
α
,
R
ℓ
(
z
′
)
=
∑
′
|
α
|
=
2
K
∂
α
σ
ℓ
(
x
′
+
ω
ℓ
z
′
)
α
!
z
α
.
Applying the Mean Value Theorem tof (cf. (81)), now yields an ω
~
∈
]
0,1
[ so that
(85)
|
k
j
*
(
f
∘
σ
)
(
x
)
|
≤
|
∫
|
z
|
≤
1
‍
k
(
z
)
f
(
P
2
K
-
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
d
z
|
+
∑
d
=
1
n
-
1
‍
∫
|
z
|
≤
1
‍
|
(
2
-
j
a
′
z
′
)
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
|
k
j
*
(
f
∘
σ
)
(
)
|
×
R
d
(
2
-
j
a
′
z
′
)
|
d
z
,
when y
′
:
=
P
2
K
-
1
(
2
-
j
a
′
z
′
)
+
ω
~
(
R
1
(
2
-
j
a
′
z
′
)
,
…
,
R
n
-
1
(
2
-
j
a
′
z
′
)
). Using (60) and (83), it is obvious that this y
′ fulfils
(86)
|
σ
(
x
)
-
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
|
≤
|
σ
′
(
x
′
)
-
y
′
|
+
|
2
-
j
a
n
z
n
|
<
C
for each z
∈
supp
⁡
k and some constant C depending only on n and C
α
,
σ with |
α
|
≤
2
K.
Step  2. Concerning the remainder terms in (85) we exploit (86) to get
(87)
∫
|
z
|
≤
1
‍
|
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
R
d
(
2
-
j
a
′
z
′
)
|
d
z
≤
2
-
2
j
K
a
0
(
∑
|
α
′
|
=
2
K
‍
∥
∂
α
′
σ
d
∣
L
∞
∥
α
′
!
)
×
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
∂
x
d
f
(
y
)
|
.
The exponent in 2
-
2
j
K
a
0 is a result of (64) and the chosen Taylor expansion of σ
(
x
+
2
-
j
a
→
z
), and since s
-
2
K
a
0
<
0 the norm of ℓ
q is trivial to calculate, whence
(88)
∥
2
j
s
∫
|
z
|
≤
1
‍
|
(
2
-
j
a
′
z
′
)
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
R
d
×
(
2
-
j
a
′
z
′
)
|
d
z
∣
L
p
→
(
ℓ
q
)
∫
|
z
|
≤
1
‍
∥
≤
c
∥
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
∂
x
d
f
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
.
Now we use that p
1
=
⋯
=
p
n
-
1 to change variables in the resulting integral over ℝ
n
-
1, with τ
′ denoting (
σ
′
)
-
1. Since Lemma 5 in view of (81) applies to ∂
x
d
f, d
=
1
,
…
,
n
-
1, the right-hand side of the last inequality can be estimated, using also Lemma 4(i), by
(89)
c
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
∥
∂
x
d
f
∣
F
p
→
,
q
s
-
a
d
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
Step  3. To treat the first term in (85), we Taylor expand f
(
·
,
x
n
), which is in C
K
(
ℝ
n
-
1
). Setting P
(
z
′
)
=
P
2
K
-
1
(
z
′
)
-
P
1
(
z
′
), expansion at the vector P
1
(
2
-
j
a
′
z
′
) gives
(90)
f
(
P
2
K
-
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
=
∑
′
0
≤
|
β
|
≤
K
-
1
D
β
f
(
P
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
β
!
×
P
(
2
-
j
a
′
z
′
)
β
+
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
P
(
2
-
j
a
′
z
′
)
β
,
where y
′ is a vector analogous to that in (85) and satisfies (86), perhaps with another C.
To deal with the remainder in (90), note that the order was chosen to ensure that, in the powers P
(
2
-
j
a
′
z
′
)
β, the l’th factor is the β
l’th power of a sum of terms each containing a factor 2
-
j
a
0
|
α
′
| with |
α
′
|
≥
2. Hence each |
β
|
=
K in total contributes by O
(
2
-
2
j
K
a
0
). More precisely, as in Step  2 we obtain
(91)
∫
|
z
|
≤
1
‍
|
k
(
z
)
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
P
(
2
-
j
a
′
z
′
)
β
|
d
z
≤
2
-
j
2
K
a
0
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
(
∑
2
≤
|
α
|
≤
2
K
-
1
‍
C
α
,
σ
)
K
×
∑
′
|
β
|
=
K
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
D
β
f
(
y
)
|
.
In view of (81), Lemma 5 barely also applies to D
β
f for |
β
|
=
K, so the above gives
(92)
∥
2
s
j
∫
|
z
|
≤
1
‍
k
(
z
)
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
∑
|
β
|
=
K
‍
×
P
(
2
-
j
a
′
z
′
)
β
d
z
∣
L
p
→
(
ℓ
q
)
∥
≤
c
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
∑
′
|
β
|
=
K
∥
D
β
f
∣
F
p
→
,
q
s
-
β
·
a
→
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
Now it remains to estimate the other terms resulting from (90), that is,
(93)
∑
′
0
≤
|
β
|
≤
K
-
1
∫
|
z
|
≤
1
‍
k
(
z
)
D
β
f
(
P
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
β
!
∑
0
≤
|
β
|
≤
K
-
1
‍
′
∫
|
z
|
≤
1
‍
×
P
(
2
-
j
a
′
z
′
)
β
d
z
.
Using the multinomial formula on the entries in P
(
z
′
)
=
∑
2
≤
|
γ
|
≤
2
K
-
1
′
z
γ
∂
γ
σ
′
(
x
′
)
/
γ
! and the g and ψ
θ discussed in (74), the above task is finally reduced to controlling terms like
(94)
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
≔
2
-
2
j
|
β
|
a
0
∫
|
z
|
≤
1
‍
g
(
z
)
D
β
2
-
22
j
|
β
|
a
0
∫
|
z
|
≤
1
‍
×

f
(
σ
′
(
x
′
)

+

2
-
j
a
0
J
σ
′
(
x
′
)
z
′
,
x
n

+

2
-
j
a
n
z
n
)
d
z
=
2
-
2
j
|
β
|
a
0
|
det
⁡
𝒜
|
∫
‍
ψ
θ
(
y
)
D
β
=
=
2
-
2
j
|
β
|
a
0
|
det
⁡
𝒜
|
∫
‍
×
f
(
σ
′
(
x
′
)

+

2
-
j
a
0
y
′
,
x
n

+

2
-
j
a
n
y
n
)
d
y
.
Note that in g, ψ
θ we have 2
≤
|
γ
|
≤
|
β
|
(
2
K
-
1
) and |
β
|
≤
K
-
1, β
n
=
0
=
γ
n.
Step  4. Before we estimate (94), it is first observed that all previous steps apply in a similar way to the convolution k
0
*
(
f
∘
σ
)—except in this case there is no dilation, so the ℓ
q-norm is omitted and the function ψ
θ is replaced by ψ
θ
,
0.
So, when collecting the terms of the form (94) with finitely many β, γ in both cases (omitting remainders from Steps  2-3), we obtain with two changes of variables and (50),
(95)
∥
∑
′
β
,
γ
I
0
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
∣
L
p
→
∥
+
∥
2
j
s
∑
′
β
,
γ
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∑
′
β
,
γ
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
×
(
∥
∫
‍
ψ
θ
,
0
(
y
)
D
β
f
(
x
-
y
)
d
y
∣
L
p
→
∥
+

∥
2
j
(
s
-
2
|
β
|
a
0
)
∫
‍
ψ
θ
(
y
)
D
β
f
(
x

-

2
-
j
a
→
y
)
d
y
∣
L
p
→
(
ℓ
q
)
∥
)
≤
c
∑
′
β
,
γ
∥
{
2
j
(
s
-
2
|
β
|
a
0
)
sup
⁡
θ
∈
Θ
ψ
θ
,
j
*
D
β
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
.
Here we apply Theorem 14 to the family of functions ψ
θ
,
0
,
ψ
θ with the φ
j chosen as the Fourier transformed of the system in the Littlewood-Paley decomposition, (cf. (13)). Estimating |
γ
|, the ψ
θ satisfy the moment condition (47) with M
ψ
θ
:
=
2
N
-
1
-
(
K
-
1
)
(
2
K
-
1
), which fulfils s
<
(
M
ψ
θ
+
1
)
a
_, because of the choice of N in Step  1. So, by applying Theorem 15 and Lemma 4(i), using s
-
2
|
β
|
a
0
≤
s
-
β
·
a
→, the above is estimated thus
(96)
∥
{
2
j
s
∑
′
β
,
γ
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
(
A
+
B
+
C
+
D
)
×
∑
′
β
,
γ
∥
{
2
j
(
s
-
2
|
β
|
a
0
)
(
ℱ
-
1
Φ
j
)
*
D
β
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∑
′
β
,
γ
∥
D
β
f
∣
F
p
→
,
q
s
-
2
|
β
|
a
0
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
This proves the necessary estimate for the given s
>
s
1.


### 4.2. Groups of Bounded Diffeomorphisms

It is not difficult to see that the proofs in Section4.1 did not really use that x
n is a single variable. It could just as well have been replaced by a whole group of variables x
′′, corresponding to a splitting x
=
(
x
′
,
x
′′
), provided that σ acts as the identity on x
′′.Moreover,x
′ could equally well have been “embedded” into x
′′, that is, x
′′ could contain variables x
k both with k
<
j
0 and with k
>
j
1 when x
′
=
(
x
j
0
,
…
,
x
j
1
) (but no interlacing); in particular the changes of variables yielding (89) would carry over to this situation when p
j
0
=
⋯
=
p
j
1. It is also not difficult to see that Proposition 19 extends to this situation when a
j
0
=
⋯
=
a
j
1 (perhaps with several g
1-terms, each having a value of μ).Thus we may generalise Theorem20 to situations with a splitting into m
≥
2 groups, that is, ℝ
n
=
ℝ
N
1
×
⋯
×
ℝ
N
m where N
1
+
⋯
+
N
m
=
n, namely, when
(97)
p
→
=
(
p
1
,
…
,
p
1
︸
N
1
,
p
2
,
…
,
p
2
︸
N
2
,
…
,
p
m
,
…
,
p
m
︸
N
m
)
,
(98)
a
→
=
(
a
1
,
…
,
a
1
,
a
2
,
…
,
a
2
,
…
,
a
m
,
…
,
a
m
)
,
(99)
σ
(
x
)
=
(
σ
1
′
(
x
(
1
)
)
,
…
,
σ
m
′
(
x
(
m
)
)
)
with arbitrary bounded diffeomorphisms σ
j
′ on ℝ
N
j and x
(
j
)
∈
ℝ
N
j.Indeed, viewingσ as a composition of σ
1
:
=
σ
1
′
⊗
i
d
ℝ
n
-
N
1, and so forth on ℝ
n, the above gives
(100)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥

≤

c
∥
f
∘
σ
m
∘
⋯
∘
σ
2
∣
F
p
→
,
q
s
,
a
→
∥

≤

⋯

≤

c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.Theorem 21.
f
↦
f
∘
σ is a linear homeomorphism on F
p
→
,
q
s
,
a
→ when (97), (98), and (99) hold.


## 4.1. Bounded Diffeomorphisms

A one-to-one mappingy
=
σ
(
x
) of ℝ
n onto ℝ
n is here called a diffeomorphism if the components σ
j
:
ℝ
n
→
ℝ have classical derivatives D
α
σ
j for all α
∈
ℕ
n. We set τ
(
y
)
=
σ
-
1
(
y
).For convenienceσ is called a bounded diffeomorphism when σ and τ furthermore satisfy
(60)
C
α
,
σ
:
=
max
⁡
j
∈
{
1
,
…
,
n
}
∥
D
α
σ
j
∣
L
∞
∥
<
∞
,
(61)
C
α
,
τ
:
=
max
⁡
j
∈
{
1
,
…
,
n
}
∥
D
α
τ
j
∣
L
∞
∥
<
∞
.
In this case there are obviously positive constants (when J
σ denotes the Jacobian matrix)
(62)
c
σ
:
=
inf
⁡
x
∈
ℝ
n
|
det
⁡
J
σ
(
x
)
|
>
0
,
c
τ
:
=
inf
⁡
y
∈
ℝ
n
|
det
⁡
J
τ
(
y
)
|
>
0
.
For example, by the Leibniz formula for determinants, c
σ
≥
1
/
(
n
!
∏
|
α
|
=
1
‍
C
α
,
τ
)
>
0.Conversely, whenever aC
∞-map σ
:
ℝ
n
→
ℝ
n fulfils (60) and that c
σ
>
0, then τ is C
∞ (as J
τ
(
y
)
=
1
/
(
det
⁡
J
σ
(
τ
(
y
)
)
)
Adj
J
σ
(
τ
(
y
)
), if Adj denotes the adjugate, each τ
j
τ
k is in C
m if τ is so) and using, for example, the Appendix it is seen by induction over |
α
| that also (61) is fulfilled. Hence such a σ is a bounded diffeomorphism.Recall that for a bounded diffeomorphismσ and a temperate distribution f, the composition f
∘
σ denotes the temperate distribution given by
(63)
〈
f
∘
σ
,
ψ
〉
=
〈
f
,
ψ
∘
τ
|
det
⁡
J
τ
|
〉
for
ψ
∈
𝒮
.
It is continuous 𝒮
′
→
𝒮
′ as the adjoint of the continuous map ψ
↦
ψ
∘
τ
|
det
⁡
J
τ
| on 𝒮: since |
det
⁡
J
τ
| is in C
L
∞
∞, continuity on 𝒮 can be shown using the higher-order chain rule to estimate each seminorm q
N
,
α
(
ψ
∘
τ
), cf. (7), by ∑
|
β
|
≤
|
α
|
‍
q
N
,
β
(
ψ
) (changing variables, 〈
σ
(
·
)
〉 can be estimated using the Mean Value Theorem on each σ
j).We need a few further conditions, due to the anisotropic situation: one can neither expectf
∘
σ to have the same regularity as f, for example, if σ is a rotation, nor that f
∘
σ
∈
L
p
→ when f
∈
L
p
→. On these grounds we first restrict to the situation in which
(64)
a
0
:
=
a
1
=
a
2
=
⋯
=
a
n
-
1
,
p
0
:
=
p
1
=
⋯
=
p
n
-
1
,
(65)
σ
(
x
)
=
(
σ
′
(
x
1
,
…
,
x
n
-
1
)
,
x
n
)
,
∀
x
∈
ℝ
n
.To prepare for Theorem20, which gives sufficient conditions for the invariance of F
p
→
,
q
s
,
a
→ under bounded diffeomorphisms of the type (65), we first show that it suffices to have invariance for sufficiently large s.Proposition 19.
Letσ be a bounded diffeomorphism on ℝ
n on the form in (65). When (64) holds and there exists s
1
∈
ℝ with the property that f
↦
f
∘
σ is a linear homeomorphism of F
p
→
,
q
s
,
a
→
(
ℝ
n
) onto itself for every s
>
s
1, then this holds true for all s
∈
ℝ.Proof.
It suffices to prove fors
≤
s
1 that
(66)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
with some constant independent of f, as the reverse inequality then follows from the fact that the inverse of σ is also a bounded diffeomorphism with the structure in (65).
Firstr
>
s
1
-
s
+
2
a
n is chosen such that d
0
:
=
(
r
/
2
a
0
) is a natural number. Setting d
n
=
(
r
/
2
a
n
) and taking μ
∈
[
0,1
[ such that d
n
-
μ
∈
ℕ, we have that r
μ
:
=
r
-
2
μ
a
n
>
s
1
-
s.
Now Lemma8 yields the existence of h
∈
F
p
→
,
q
s
+
r
,
a
→ such that f
=
Λ
r
h, that is,
(67)
f
=
(
1
-
∂
x
n
2
)
d
n
-
μ
(
1
-
∂
x
n
2
)
μ
h
+
∑
k
=
1
n
-
1
‍
(
1
-
∂
x
k
2
)
d
0
h
.
Setting g
1
=
(
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ and g
0
=
h
∘
σ, we may apply the higher-order chain rule to for example h
=
g
0
∘
τ (using denseness of 𝒮 in 𝒮
′ and the 𝒮
′-continuity of composition in (63), the Appendix  extends to 𝒮
′). Taking into account that τ
(
x
)
=
(
τ
′
(
x
′
)
,
x
n
), and letting prime indicate summation over multi-indices with β
n
=
0,
(68)
f
=
∑
l
=
0
d
n
-
μ
‍
η
n
,
l
∂
x
n
2
l
g
1
∘
τ
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
η
k
,
β
∂
β
g
0
∘
τ
,
where η
n
,
l
:
=
(
-
1
)
l
(
d
n
-
μ
l
) and the η
k
,
β are functions containing derivatives at least of order 1 of τ, and these can be estimated, say by c
∏
1
≤
m
≤
2
d
0
‍
〈
∂
x
k
m
τ
〉
2
d
0. Composing with σ and applying Lemma 4(i) gives for d
:
=
min
⁡
(
1
,
q
,
p
0
,
p
n
), when ∥
·
∥ denotes the F
p
→
,
q
s
,
a
→-norm,
(69)
∥
f
∘
σ
∥
d
≤
∑
l
=
0
d
n
-
μ
‍
|
η
n
,
l
|
d
∥
∂
x
n
2
l
g
1
∥
d
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
∥
η
k
,
β
∘
σ
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
d
∥
∂
β
g
0
∥
d
≤
c
∥
g
1
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
∥
g
0
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
∥
η
k
,
β
∘
σ
∣
M
(
F
p
→
,
q
s
,
a
→
)
∥
d
.
According to the remark preceding Lemma 13, the last sum is finite because η
k
,
β
∈
C
L
∞
∞. Finally, since s
+
r
μ
>
s
1 and s
+
r
>
s
1, the stated assumption means that h
↦
g
1 and h
↦
g
0 are bounded, which in view of r
μ
+
2
μ
a
n
=
r and Lemmas 8 and 9 yields
(70)
∥
f
∘
σ
∥
d
≤
c
∥
h
∣
F
p
→
,
q
s
+
r
μ
+
2
μ
a
n
,
a
→
∥
d
+
∥
h
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
d
,
proving the boundedness of f
↦
f
∘
σ in F
p
→
,
q
s
,
a
→ for all s
∈
ℝ.In addition to the reduction in Proposition19, we adopt in Theorem 20 the strategy for the isotropic, unmixed case developed by Triebel [4, 4.3.2], who used Taylor expansions for the inner and outer functions for large s.While his explanation was rather sketchy, our task is to account for the fact that the strategy extends to anisotropies and to mixed norms. Hence we give full details. This will also allow us to give brief proofs of additional results in Sections4.2 and 5.To control the Taylor expansions, it will be crucial for us to exploit both the local means recalled in Theorem16 and the parameter-dependent setup in Theorem 14. This is prepared for with the following discussion.The functionsk
0 and k in Theorem 16 are for the proof of Theorem 20 chosen (as we may) so that N in the definition of k fulfils s
<
2
N
a
_ and so that both are even functions and
(71)
supp
⁡
k
0
,
supp
⁡
k
⊂
{
x
∈
ℝ
n
∣
|
x
|
≤
1
}
.
The set Θ in Theorem 14 is chosen to be the set of (
n
-
1
)
×
(
n
-
1
) matrices ℬ
=
(
b
i
,
k
) that, in terms of the constants c
σ
,
C
α
,
σ in (62) and (60), respectively, satisfy
(72)
|
det
⁡
ℬ
|
≥
c
σ
,
(73)
max
⁡
i
,
k
|
b
i
,
k
|
≤
max
⁡
|
α
|
=
1
C
α
,
σ
=
:
C
σ
.Splittingz
=
(
z
′
,
z
n
), we set g
(
z
)
=
z
′
γ
′
k
(
z
) for some γ
′
∈
ℕ
0
n
-
1 (chosen later) and define
(74)
ψ
θ
(
y
)
=
g
(
𝒜
y
′
,
y
n
)
where θ is identified with 𝒜
-
1
:
=
J
σ
′
(
x
′
), which obviously belongs to Θ (for each x
′).To verify that the above functionsψ
θ, θ
∈
Θ, satisfy the moment condition (47) for an M
ψ
θ such that the assumption s
<
(
M
ψ
θ
+
1
)
a
_ in Theorem 14 is fulfilled, note that
(75)
ψ
^
θ
(
ξ
)
=
|
det
⁡
𝒜
|
-
1
ℱ
g
(
𝒜
t
-
1
ξ
′
,
ξ
n
)
.
Hence D
α
ψ
^
θ vanishes at ξ
=
0 when D
α
g
^
=
D
α
(
-
D
ξ
′
)
γ
′
k
^
(
ξ
) does so. As k
^
(
ξ
)
=
-
|
ξ
|
2
N
k
0
^
(
ξ
) and k
0
^
(
0
)
≠
0, we have D
α
g
^
(
0
)
=
0 for α satisfying |
α
|
+
|
γ
′
|
≤
2
N
-
1. In the course of the proof below (cf. Step  3), we obtain a θ-independent estimate of |
γ
′
|, hence of M
ψ
θ.Moreover, the constantA in Theorem 14 is finite: basic properties of the Fourier transform give the following estimate, where the constant is independent of 𝒜
-
1
∈
Θ:
(76)
∥
D
α
ℱ
ψ
θ
∣
L
∞
∥
≤
∫
‍
|
y
α
g
(
𝒜
y
′
,
y
n
)
|
d
y
=
|
det
⁡
𝒜
-
1
|
∫
‍
|
z
n
α
n
|
|
(
𝒜
-
1
z
′
)
α
′
|
|
g
(
z
)
|
d
z
≤
c
(
α
,
C
σ
)
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
.
To estimate B we exploit that ℱ
:
B
2,1
n
/
2
(
ℝ
n
)
→
L
1
(
ℝ
n
) is bounded according to Szasz’s inequality (cf. [18, Proposition  1.7.5]) and obtain
(77)
∥
(
1
+
|
·
|
)
M
+
1
D
γ
ℱ
ψ
θ
∣
L
1
∥
≤
c
∥
y
γ
g
(
𝒜
y
′
,
y
n
)
∣
B
2,1
M
+
1
+
(
n
/
2
)
∥
≤
c
(
γ
,
C
σ
,
C
τ
)
∥
k
∣
C
0
m
∥
,
when m
∈
ℕ is chosen so large that m
>
M
+
1
+
n
/
2. In fact, the last inequality is obtained using the embeddings C
0
m
↪
H
m
↪
B
2,1
M
+
1
+
n
/
2 and the estimate
(78)
∥
y
γ
ψ
θ
∣
C
0
m
∥
=
sup
⁡
|
∂
α
(
y
γ
(
𝒜
y
′
)
γ
′
k
(
𝒜
y
′
,
y
n
)
)
|
≤
c
(
γ
,
C
σ
,
C
τ
)
∥
k
∣
C
0
m
∥
.
This relies on the higher-order chain rule (cf. the Appendix and the support of k): it suffices to use the supremum over |
α
|
≤
m and {
y
∈
ℝ
n
∣
|
𝒜
y
′
|
2
+
y
n
2
≤
1
}, and for a point in this set |
y
′
|
≤
∥
𝒜
-
1
∥
|
𝒜
y
′
|
≤
c
(
C
σ
), so we need only estimate an 𝒜-independent cylinder.Replacingk by k
0 in the definition of g and setting ψ
θ
,
0
(
y
)
:
=
g
(
𝒜
y
′
,
y
n
), the finiteness of C and D follows analogously. The Tauberian properties follow from ∫
‍
k
0
≠
0
≠
∫
‍
k
0.Hence all assumptions in Theorem14 are satisfied, and we are thus ready to prove our main result.Theorem 20.
Ifσ is a bounded diffeomorphism on ℝ
n on the form in (65), then f
↦
f
∘
σ is a linear homeomorphism F
p
→
,
q
s
,
a
→
(
ℝ
n
)
→
F
p
→
,
q
s
,
a
→
(
ℝ
n
) for all s
∈
ℝ when (64) holds.Proof.
According to Proposition19, it suffices to consider s
>
s
1, say for
(79)
s
1
:
=
K
0
a
0
+
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
,
where by K
0 is the smallest integer satisfying
(80)
K
0
a
0
>
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
.
We now let s
∈
]
s
1
,
∞
[ be given and take some K
≥
K
0, that is, K solving (80), such that
(81)
K
a
0
+
(
n
-
1
)
a
0
p
0
+
a
n
min
⁡
(
p
0
,
p
n
)
<
s
<
2
K
a
0
.
(The interval thus defined is nonempty by (80), and the left end point is at least s
1.)
Note that (81) yields that every f
∈
F
p
→
,
q
s
,
a
→ is continuous, (cf. Lemma 4(iii)), so are even the derivatives D
β
f for β
=
(
β
1
,
…
,
β
n
-
1
,
0
), |
β
|
≤
K, since s
-
β
·
a
→
=
s
-
|
β
|
a
0
>
a
→
·
1
/
p
→.
Step  1. For the norms ∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥ and ∥
f
∣
F
p
→
,
q
s
,
a
→
∥ in inequality (66), which also here suffices, we use Theorem 16 with 2
N
>
(
K
-
1
)
(
2
K
-
1
)
+
s
/
a
_.
By the symmetry ofk
0 and k in (71), we will estimate
(82)
k
j
*
(
f
∘
σ
)
(
x
)

=

∫
|
z
|
≤
1
‍
k
(
z
)
f
(
σ
(
x
+
2
-
j
a
→
z
)
)
d
z
,
j
∈
ℕ
,
together with the corresponding expression for k
0, where k is replaced by k
0.
First we make a Taylor expansion of the entries inσ
′
(
x
′
)
:
=
(
σ
1
(
x
′
)
,
…
,
σ
n
-
1
(
x
′
)
) to the order 2
K
-
1. So for ℓ
=
1
,
…
,
n
-
1 there exists ω
ℓ
∈
]
0,1
[ such that
(83)
σ
ℓ
(
x
′
+
z
′
)
=
∑
|
α
′
|
<
2
K
‍
∂
α
′
σ
ℓ
(
x
′
)
α
′
!
z
′
α
′
+
∑
|
α
′
|
=
2
K
‍
∂
α
′
σ
ℓ
(
x
′
+
ω
ℓ
z
′
)
α
′
!
z
′
α
′
.
For convenience, we let ∑
‍
α
′ denote summation over multiindices α
∈
ℕ
0
n having α
n
=
0 and define the vector of Taylor polynomials, respectively, entries of a remainder R,
(84)
P
2
K
-
1
(
z
′
)
=
∑
′
|
α
|
≤
2
K
-
1
∂
α
σ
′
(
x
′
)
α
!
z
α
,
R
ℓ
(
z
′
)
=
∑
′
|
α
|
=
2
K
∂
α
σ
ℓ
(
x
′
+
ω
ℓ
z
′
)
α
!
z
α
.
Applying the Mean Value Theorem tof (cf. (81)), now yields an ω
~
∈
]
0,1
[ so that
(85)
|
k
j
*
(
f
∘
σ
)
(
x
)
|
≤
|
∫
|
z
|
≤
1
‍
k
(
z
)
f
(
P
2
K
-
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
d
z
|
+
∑
d
=
1
n
-
1
‍
∫
|
z
|
≤
1
‍
|
(
2
-
j
a
′
z
′
)
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
|
k
j
*
(
f
∘
σ
)
(
)
|
×
R
d
(
2
-
j
a
′
z
′
)
|
d
z
,
when y
′
:
=
P
2
K
-
1
(
2
-
j
a
′
z
′
)
+
ω
~
(
R
1
(
2
-
j
a
′
z
′
)
,
…
,
R
n
-
1
(
2
-
j
a
′
z
′
)
). Using (60) and (83), it is obvious that this y
′ fulfils
(86)
|
σ
(
x
)
-
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
|
≤
|
σ
′
(
x
′
)
-
y
′
|
+
|
2
-
j
a
n
z
n
|
<
C
for each z
∈
supp
⁡
k and some constant C depending only on n and C
α
,
σ with |
α
|
≤
2
K.
Step  2. Concerning the remainder terms in (85) we exploit (86) to get
(87)
∫
|
z
|
≤
1
‍
|
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
R
d
(
2
-
j
a
′
z
′
)
|
d
z
≤
2
-
2
j
K
a
0
(
∑
|
α
′
|
=
2
K
‍
∥
∂
α
′
σ
d
∣
L
∞
∥
α
′
!
)
×
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
∂
x
d
f
(
y
)
|
.
The exponent in 2
-
2
j
K
a
0 is a result of (64) and the chosen Taylor expansion of σ
(
x
+
2
-
j
a
→
z
), and since s
-
2
K
a
0
<
0 the norm of ℓ
q is trivial to calculate, whence
(88)
∥
2
j
s
∫
|
z
|
≤
1
‍
|
(
2
-
j
a
′
z
′
)
k
(
z
)
∂
x
d
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
R
d
×
(
2
-
j
a
′
z
′
)
|
d
z
∣
L
p
→
(
ℓ
q
)
∫
|
z
|
≤
1
‍
∥
≤
c
∥
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
∂
x
d
f
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
.
Now we use that p
1
=
⋯
=
p
n
-
1 to change variables in the resulting integral over ℝ
n
-
1, with τ
′ denoting (
σ
′
)
-
1. Since Lemma 5 in view of (81) applies to ∂
x
d
f, d
=
1
,
…
,
n
-
1, the right-hand side of the last inequality can be estimated, using also Lemma 4(i), by
(89)
c
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
∥
∂
x
d
f
∣
F
p
→
,
q
s
-
a
d
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
Step  3. To treat the first term in (85), we Taylor expand f
(
·
,
x
n
), which is in C
K
(
ℝ
n
-
1
). Setting P
(
z
′
)
=
P
2
K
-
1
(
z
′
)
-
P
1
(
z
′
), expansion at the vector P
1
(
2
-
j
a
′
z
′
) gives
(90)
f
(
P
2
K
-
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
=
∑
′
0
≤
|
β
|
≤
K
-
1
D
β
f
(
P
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
β
!
×
P
(
2
-
j
a
′
z
′
)
β
+
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
P
(
2
-
j
a
′
z
′
)
β
,
where y
′ is a vector analogous to that in (85) and satisfies (86), perhaps with another C.
To deal with the remainder in (90), note that the order was chosen to ensure that, in the powers P
(
2
-
j
a
′
z
′
)
β, the l’th factor is the β
l’th power of a sum of terms each containing a factor 2
-
j
a
0
|
α
′
| with |
α
′
|
≥
2. Hence each |
β
|
=
K in total contributes by O
(
2
-
2
j
K
a
0
). More precisely, as in Step  2 we obtain
(91)
∫
|
z
|
≤
1
‍
|
k
(
z
)
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
P
(
2
-
j
a
′
z
′
)
β
|
d
z
≤
2
-
j
2
K
a
0
∫
|
z
|
≤
1
‍
|
k
(
z
)
|
d
z
(
∑
2
≤
|
α
|
≤
2
K
-
1
‍
C
α
,
σ
)
K
×
∑
′
|
β
|
=
K
sup
⁡
|
σ
(
x
)
-
y
|
<
C
|
D
β
f
(
y
)
|
.
In view of (81), Lemma 5 barely also applies to D
β
f for |
β
|
=
K, so the above gives
(92)
∥
2
s
j
∫
|
z
|
≤
1
‍
k
(
z
)
∑
′
|
β
|
=
K
D
β
f
(
y
′
,
x
n
+
2
-
j
a
n
z
n
)
β
!
∑
|
β
|
=
K
‍
×
P
(
2
-
j
a
′
z
′
)
β
d
z
∣
L
p
→
(
ℓ
q
)
∥
≤
c
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
∑
′
|
β
|
=
K
∥
D
β
f
∣
F
p
→
,
q
s
-
β
·
a
→
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
Now it remains to estimate the other terms resulting from (90), that is,
(93)
∑
′
0
≤
|
β
|
≤
K
-
1
∫
|
z
|
≤
1
‍
k
(
z
)
D
β
f
(
P
1
(
2
-
j
a
′
z
′
)
,
x
n
+
2
-
j
a
n
z
n
)
β
!
∑
0
≤
|
β
|
≤
K
-
1
‍
′
∫
|
z
|
≤
1
‍
×
P
(
2
-
j
a
′
z
′
)
β
d
z
.
Using the multinomial formula on the entries in P
(
z
′
)
=
∑
2
≤
|
γ
|
≤
2
K
-
1
′
z
γ
∂
γ
σ
′
(
x
′
)
/
γ
! and the g and ψ
θ discussed in (74), the above task is finally reduced to controlling terms like
(94)
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
≔
2
-
2
j
|
β
|
a
0
∫
|
z
|
≤
1
‍
g
(
z
)
D
β
2
-
22
j
|
β
|
a
0
∫
|
z
|
≤
1
‍
×

f
(
σ
′
(
x
′
)

+

2
-
j
a
0
J
σ
′
(
x
′
)
z
′
,
x
n

+

2
-
j
a
n
z
n
)
d
z
=
2
-
2
j
|
β
|
a
0
|
det
⁡
𝒜
|
∫
‍
ψ
θ
(
y
)
D
β
=
=
2
-
2
j
|
β
|
a
0
|
det
⁡
𝒜
|
∫
‍
×
f
(
σ
′
(
x
′
)

+

2
-
j
a
0
y
′
,
x
n

+

2
-
j
a
n
y
n
)
d
y
.
Note that in g, ψ
θ we have 2
≤
|
γ
|
≤
|
β
|
(
2
K
-
1
) and |
β
|
≤
K
-
1, β
n
=
0
=
γ
n.
Step  4. Before we estimate (94), it is first observed that all previous steps apply in a similar way to the convolution k
0
*
(
f
∘
σ
)—except in this case there is no dilation, so the ℓ
q-norm is omitted and the function ψ
θ is replaced by ψ
θ
,
0.
So, when collecting the terms of the form (94) with finitely many β, γ in both cases (omitting remainders from Steps  2-3), we obtain with two changes of variables and (50),
(95)
∥
∑
′
β
,
γ
I
0
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
∣
L
p
→
∥
+
∥
2
j
s
∑
′
β
,
γ
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∑
′
β
,
γ
(
sup
⁡
y
∈
ℝ
n
-
1
|
det
⁡
J
τ
′
(
y
)
|
)
1
/
p
0
×
(
∥
∫
‍
ψ
θ
,
0
(
y
)
D
β
f
(
x
-
y
)
d
y
∣
L
p
→
∥
+

∥
2
j
(
s
-
2
|
β
|
a
0
)
∫
‍
ψ
θ
(
y
)
D
β
f
(
x

-

2
-
j
a
→
y
)
d
y
∣
L
p
→
(
ℓ
q
)
∥
)
≤
c
∑
′
β
,
γ
∥
{
2
j
(
s
-
2
|
β
|
a
0
)
sup
⁡
θ
∈
Θ
ψ
θ
,
j
*
D
β
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
.
Here we apply Theorem 14 to the family of functions ψ
θ
,
0
,
ψ
θ with the φ
j chosen as the Fourier transformed of the system in the Littlewood-Paley decomposition, (cf. (13)). Estimating |
γ
|, the ψ
θ satisfy the moment condition (47) with M
ψ
θ
:
=
2
N
-
1
-
(
K
-
1
)
(
2
K
-
1
), which fulfils s
<
(
M
ψ
θ
+
1
)
a
_, because of the choice of N in Step  1. So, by applying Theorem 15 and Lemma 4(i), using s
-
2
|
β
|
a
0
≤
s
-
β
·
a
→, the above is estimated thus
(96)
∥
{
2
j
s
∑
′
β
,
γ
I
j
,
β
,
γ
(
σ
′
(
x
′
)
,
x
n
)
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
(
A
+
B
+
C
+
D
)
×
∑
′
β
,
γ
∥
{
2
j
(
s
-
2
|
β
|
a
0
)
(
ℱ
-
1
Φ
j
)
*
D
β
f
}
j
=
0
∞
∣
L
p
→
(
ℓ
q
)
∥
≤
c
∑
′
β
,
γ
∥
D
β
f
∣
F
p
→
,
q
s
-
2
|
β
|
a
0
,
a
→
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.
This proves the necessary estimate for the given s
>
s
1.


## 4.2. Groups of Bounded Diffeomorphisms

It is not difficult to see that the proofs in Section4.1 did not really use that x
n is a single variable. It could just as well have been replaced by a whole group of variables x
′′, corresponding to a splitting x
=
(
x
′
,
x
′′
), provided that σ acts as the identity on x
′′.Moreover,x
′ could equally well have been “embedded” into x
′′, that is, x
′′ could contain variables x
k both with k
<
j
0 and with k
>
j
1 when x
′
=
(
x
j
0
,
…
,
x
j
1
) (but no interlacing); in particular the changes of variables yielding (89) would carry over to this situation when p
j
0
=
⋯
=
p
j
1. It is also not difficult to see that Proposition 19 extends to this situation when a
j
0
=
⋯
=
a
j
1 (perhaps with several g
1-terms, each having a value of μ).Thus we may generalise Theorem20 to situations with a splitting into m
≥
2 groups, that is, ℝ
n
=
ℝ
N
1
×
⋯
×
ℝ
N
m where N
1
+
⋯
+
N
m
=
n, namely, when
(97)
p
→
=
(
p
1
,
…
,
p
1
︸
N
1
,
p
2
,
…
,
p
2
︸
N
2
,
…
,
p
m
,
…
,
p
m
︸
N
m
)
,
(98)
a
→
=
(
a
1
,
…
,
a
1
,
a
2
,
…
,
a
2
,
…
,
a
m
,
…
,
a
m
)
,
(99)
σ
(
x
)
=
(
σ
1
′
(
x
(
1
)
)
,
…
,
σ
m
′
(
x
(
m
)
)
)
with arbitrary bounded diffeomorphisms σ
j
′ on ℝ
N
j and x
(
j
)
∈
ℝ
N
j.Indeed, viewingσ as a composition of σ
1
:
=
σ
1
′
⊗
i
d
ℝ
n
-
N
1, and so forth on ℝ
n, the above gives
(100)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
∥

≤

c
∥
f
∘
σ
m
∘
⋯
∘
σ
2
∣
F
p
→
,
q
s
,
a
→
∥

≤

⋯

≤

c
∥
f
∣
F
p
→
,
q
s
,
a
→
∥
.Theorem 21.
f
↦
f
∘
σ is a linear homeomorphism on F
p
→
,
q
s
,
a
→ when (97), (98), and (99) hold.


## 5. Derived results

### 5.1. Diffeomorphisms on Domains

The strategies of Proposition19 and Theorem 20 also give the following local version. For example, for the paraboloid U
=
{
x
|
x
n
>
x
1
2
+
⋯
+
x
n
-
1
2
} we may take σ to consist in a rotation around the x
n-axis (cf. (65)).Theorem 22.
LetU
,
V
⊂
ℝ
n be open and σ
:
U
→
V a C
∞-bijection as in (65). If (64) is fulfilled and f
∈
F
p
→
,
q
s
,
a
→
(
V
) has compact support, then f
∘
σ
∈
F
p
→
,
q
s
,
a
→
(
U
) and
(101)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
holds for a constant c depending only on σ and the set supp
⁡
f.Proof.
Step  1. Let us consider s
>
s
1 (cf. (79)), and adapt the proof of Theorem 20 to the local set-up. We will prove the statement for the f
∈
F
p
→
,
q
s
,
a
→
(
V
) satisfying supp
⁡
f
⊂
K
⊂
V for some arbitrary compact set K. First we fix r
∈
]
0,1
[ so small that
(102)
6
r
<
min
⁡
(
dist
⁡
(
K
,
ℝ
n
∖
V
)
,
dist
⁡
(
σ
-
1
(
K
)
,
ℝ
n
∖
U
)
)
.
Then, by Lemma 18, we have ∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
=
∥
e
U
(
f
∘
σ
)
∣
F
p
→
,
q
s
,
a
→
∥ when Theorem 16 is utilised for k
0
,
k
∈
𝒮, say, so that supp
⁡
k
0
,
supp
⁡
k
⊂
B
(
0
,
r
) (cf. also (71)). Extension by 0 outside U of f
∘
σ is redundant, for it suffices to integrate over x
∈
W
:
=
supp
⁡
(
f
∘
σ
)
+
B
¯
(
0
,
r
). However, to apply the Mean Value Theorem (cf. (85)), we extend f by 0 instead; that is, we consider (82) with integration over |
z
|
≤
r and with f replaced by e
V
f.
Sincee
V
f inherits the regularity of f (cf. Lemma 18) and ∂
α
σ can be estimated on the compact set W, the proof of Theorem 20 carries over straightforwardly. For example, one obtains a variant of (89) where |
det
⁡
J
τ
′
(
x
′
)
|
1
/
p
0 is estimated over {
x
′
∣
∃
x
n
:
(
x
′
,
x
n
)
∈
σ
(
W
)
}, and the integration is then extended to ℝ
n, which by Lemma 18 yields
(103)
∥
sup
⁡
|
x
-
y
|
<
C
|
∂
x
d
e
V
f
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
≤
c
∥
e
V
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
=
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
.
To estimate the first term in (85) in this local version, the argumentation there is modified as above and the set Θ is chosen to be the set of all (
n
-
1
)
×
(
n
-
1
) matrices satisfying (72) with infimum over x
∈
W and (73) with C
σ
:
=
max
⁡
1
≤
j
≤
n
,
|
α
|
=
1
sup
⁡
x
∈
W
|
D
α
σ
j
(
x
)
|.
Before applying Theorem14 to the new estimate (95), the integration is extended to ℝ
n (using e
V
f). Then application of Theorems 14 and 15 together with Lemma 18 finishes the proof for s
>
s
1.
Step  2. For s
≤
s
1 we use Lemma 8 to write e
V
f
=
Λ
r
h for some h
∈
F
p
→
,
q
s
+
r
,
a
→
(
ℝ
n
); hence the identity (67) holds in 𝒟
′
(
ℝ
n
) for e
V
f and h. Applying r
V to both sides and using that it commutes with differentiation on C
0
∞, hence on 𝒟
′, we obtain (68) as an identity in 𝒟
′
(
V
) for the new g
0
:
=
(
r
V
h
)
∘
σ and g
1
:
=
(
r
V
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ.
Composing withσ we obtain an identity in 𝒟
′
(
U
), when η
k
,
β
∘
σ is treated using cut-off functions. For example, we can take χ
,
χ
1
∈
C
0
∞
(
U
) with χ
≡
1 on supp
⁡
(
f
∘
σ
)
+
B
¯
(
0
,
r
)
=
:
W
r and supp
⁡
χ
⊂
W
2
r, while χ
1
≡
1 on W
3
r and supp
⁡
χ
1
⊂
W
4
r. This entails
(104)
χ
·
f
∘
σ
=
∑
l
=
0
d
n
-
μ
‍
η
n
,
l
χ
∂
x
n
2
l
(
χ
1
g
1
)
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
η
k
,
β
∘
σ
·
χ
∂
β
(
χ
1
g
0
)
.
Using e
U on both sides (and omitting ℝ
n in the spaces), Lemmas 18 and 13 imply
(105)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
d
≤
c
∑
l
=
0
d
n
-
μ
‍
∥
e
U
(
∂
x
n
2
l
(
χ
1
g
1
)
)
∣
F
p
→
,
q
s
,
a
→
∥
d
+
c
∑
′
|
β
|
≤
2
d
0
∥
e
U
(
∂
β
(
χ
1
g
0
)
)
∣
F
p
→
,
q
s
,
a
→
∥
d
.
As e
U and differentiation commute on ℰ
′
(
U
)
∋
χ
1
g
j, Lemma 4(i) leads to an estimate from above. But Lemma 18 applies since the supports are in W
4
r, so with χ
~
1
:
=
χ
1
∘
τ we find that the above is less than or equal to
(106)
c
∥
e
U
(
χ
1
g
1
)
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
c
∥
e
U
(
χ
1
g
0
)
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
=
c
∥
(
χ
~
1
·
r
V
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ
∣
F
p
→
,
q
s
+
r
μ
,
a
→
(
U
)
∥
d
+
c
∥
(
χ
~
1
·
r
V
h
)
∘
σ
∣
F
p
→
,
q
s
+
r
,
a
→
(
U
)
∥
d
.
Using Step  1 and Lemmas 13, 9, 8, and 18, this entails
(107)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
d
≤
c
∥
(
1
-
∂
x
n
2
)
μ
h
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
c
∥
h
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
Λ
r
-
1
e
V
f
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
d
.
This shows the local theorem for s
≤
s
1.There is also a local version of Theorem21, with similar proof, namely the following.Theorem 23.
Letσ
j
:
U
j
→
V
j, j
=
1
,
…
,
m, be C
∞ bijections, where U
j
,
V
j
⊂
ℝ
N
j are open. When a
→
,
p
→ fulfill (97)-(98) and when f
∈
F
p
→
,
q
s
,
a
→
(
U
1
×
⋯
×
U
m
) has compact support, then (101) holds true for U
=
U
1
×
⋯
×
U
m and V
=
V
1
×
⋯
×
V
m.As a preparation for our coming work [9], we include a natural extension to the case of an infinite cylinder, where supp
⁡
f is only required to be compact on cross-sections.Theorem 24.
Letσ
:
U
×
ℝ
→
V
×
ℝ be a C
∞-bijection on the form in (65), and U
,
V
⊂
ℝ
n
-
1 open. If (64) holds and f
∈
F
p
→
,
q
s
,
a
→
(
V
×
ℝ
) has supp
⁡
f
⊂
K
×
ℝ, whereby K
⊂
V is compact, then f
∘
σ
∈
F
p
→
,
q
s
,
a
→
(
U
×
ℝ
) and
(108)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
×
ℝ
)
∥
≤
c
(
supp
⁡
f
,
σ
)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
.Proof.
We adapt the proof of Theorem22: in Step  1 we take r
∈
]
0,1
[ so small that 6
r is less than both dist
⁡
(
K
,
ℝ
n
-
1
∖
V
) and dist
⁡
(
σ
′
-
1
(
K
)
,
ℝ
n
-
1
∖
U
). Since the extension by zero e
V
×
ℝ
f is well defined, as K
⊂
V is compact, it is an immediate corollary to the proof of Lemma 18 that
(109)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
×
ℝ
)
∥
=
∥
e
V
×
ℝ
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
.
Then the proof for s
>
s
1 follows that of Theorem 22, with W
:
=
(
σ
′
-
1
(
K
)
+
B
¯
(
0
,
r
)
)
×
ℝ.
Fors
≤
s
1 we have e
V
×
ℝ
f
=
Λ
r
h for some h
∈
F
p
→
,
q
s
+
r
,
a
→
(
ℝ
n
) (cf. Lemma 8). Hence (68) holds as an identity in 𝒟
′
(
V
×
ℝ
) for g
1
:
=
(
r
V
×
ℝ
(
1
-
∂
x
n
+
1
2
)
μ
h
)
∘
σ and g
0
:
=
(
r
V
×
ℝ
h
)
∘
σ.
Theη
k
,
β
∘
σ are controlled using cut-off functions χ
,
χ
1
∈
C
L
∞
∞
(
U
) with similar properties in terms of the sets W
r
=
(
σ
′
-
1
(
K
)
+
B
¯
(
0
,
r
)
)
×
ℝ. Thus we obtain (104) in 𝒟
′
(
U
×
ℝ
).
Now, as in (109) it is seen that f
∘
σ and e
U
×
ℝ
(
χ
·
f
∘
σ
) have identical norms, so the estimates in Step  2 of the proof of Theorem 22 finish the proof, mutatis mutandis.


### 5.2. Isotropic Spaces

Going to the other extreme, when alsoa
n
=
a
0 and p
n
=
p
0, then the Lizorkin-Triebel spaces are invariant under any bounded diffeomorphism (i.e., without (65)), since in that case we can just change variables in all coordinates, in particular in (88)-(89). Moreover, we can adapt Proposition 19 by taking d
n
=
d
0 and μ
=
0 in the proof; and the set-up prior to Theorem 20 is also easily modified to the isotropic situation. Hence we obtain the following.Corollary 25.
Whenσ
:
ℝ
n
→
ℝ
n is any bounded diffeomorphism, then f
↦
f
∘
σ is a linear homeomorphism of F
p
,
q
s
(
ℝ
n
) onto itself for all s
∈
ℝ.This is known from work of Triebel [4, Theorem  4.3.2], which also contains a corresponding result for Besov spaces. (It is this proof we extended to mixed norms in the previous section.) The result has also been obtained recently by Scharf [23], who covered all s
∈
ℝ by means of an extended notion of atomic decompositions.In an analogous way, we also obtain an isotropic counterpart to Theorem22.Corollary 26.
Whenσ
:
U
→
V is a C
∞-bijection between open sets U
,
V
⊂
ℝ
n, then f
∘
σ
∈
F
p
,
q
s
(
U
) for every f
∈
F
p
,
q
s
(
V
) having compact support and
(110)
∥
f
∘
σ
∣
F
p
,
q
s
(
U
)
∥
≤
c
(
supp
⁡
f
,
σ
)
∥
f
∣
F
p
,
q
s
(
V
)
∥
.


## 5.1. Diffeomorphisms on Domains

The strategies of Proposition19 and Theorem 20 also give the following local version. For example, for the paraboloid U
=
{
x
|
x
n
>
x
1
2
+
⋯
+
x
n
-
1
2
} we may take σ to consist in a rotation around the x
n-axis (cf. (65)).Theorem 22.
LetU
,
V
⊂
ℝ
n be open and σ
:
U
→
V a C
∞-bijection as in (65). If (64) is fulfilled and f
∈
F
p
→
,
q
s
,
a
→
(
V
) has compact support, then f
∘
σ
∈
F
p
→
,
q
s
,
a
→
(
U
) and
(101)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
holds for a constant c depending only on σ and the set supp
⁡
f.Proof.
Step  1. Let us consider s
>
s
1 (cf. (79)), and adapt the proof of Theorem 20 to the local set-up. We will prove the statement for the f
∈
F
p
→
,
q
s
,
a
→
(
V
) satisfying supp
⁡
f
⊂
K
⊂
V for some arbitrary compact set K. First we fix r
∈
]
0,1
[ so small that
(102)
6
r
<
min
⁡
(
dist
⁡
(
K
,
ℝ
n
∖
V
)
,
dist
⁡
(
σ
-
1
(
K
)
,
ℝ
n
∖
U
)
)
.
Then, by Lemma 18, we have ∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
=
∥
e
U
(
f
∘
σ
)
∣
F
p
→
,
q
s
,
a
→
∥ when Theorem 16 is utilised for k
0
,
k
∈
𝒮, say, so that supp
⁡
k
0
,
supp
⁡
k
⊂
B
(
0
,
r
) (cf. also (71)). Extension by 0 outside U of f
∘
σ is redundant, for it suffices to integrate over x
∈
W
:
=
supp
⁡
(
f
∘
σ
)
+
B
¯
(
0
,
r
). However, to apply the Mean Value Theorem (cf. (85)), we extend f by 0 instead; that is, we consider (82) with integration over |
z
|
≤
r and with f replaced by e
V
f.
Sincee
V
f inherits the regularity of f (cf. Lemma 18) and ∂
α
σ can be estimated on the compact set W, the proof of Theorem 20 carries over straightforwardly. For example, one obtains a variant of (89) where |
det
⁡
J
τ
′
(
x
′
)
|
1
/
p
0 is estimated over {
x
′
∣
∃
x
n
:
(
x
′
,
x
n
)
∈
σ
(
W
)
}, and the integration is then extended to ℝ
n, which by Lemma 18 yields
(103)
∥
sup
⁡
|
x
-
y
|
<
C
|
∂
x
d
e
V
f
(
y
)
|
∣
L
p
→
(
ℝ
x
n
)
∥
≤
c
∥
e
V
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
=
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
.
To estimate the first term in (85) in this local version, the argumentation there is modified as above and the set Θ is chosen to be the set of all (
n
-
1
)
×
(
n
-
1
) matrices satisfying (72) with infimum over x
∈
W and (73) with C
σ
:
=
max
⁡
1
≤
j
≤
n
,
|
α
|
=
1
sup
⁡
x
∈
W
|
D
α
σ
j
(
x
)
|.
Before applying Theorem14 to the new estimate (95), the integration is extended to ℝ
n (using e
V
f). Then application of Theorems 14 and 15 together with Lemma 18 finishes the proof for s
>
s
1.
Step  2. For s
≤
s
1 we use Lemma 8 to write e
V
f
=
Λ
r
h for some h
∈
F
p
→
,
q
s
+
r
,
a
→
(
ℝ
n
); hence the identity (67) holds in 𝒟
′
(
ℝ
n
) for e
V
f and h. Applying r
V to both sides and using that it commutes with differentiation on C
0
∞, hence on 𝒟
′, we obtain (68) as an identity in 𝒟
′
(
V
) for the new g
0
:
=
(
r
V
h
)
∘
σ and g
1
:
=
(
r
V
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ.
Composing withσ we obtain an identity in 𝒟
′
(
U
), when η
k
,
β
∘
σ is treated using cut-off functions. For example, we can take χ
,
χ
1
∈
C
0
∞
(
U
) with χ
≡
1 on supp
⁡
(
f
∘
σ
)
+
B
¯
(
0
,
r
)
=
:
W
r and supp
⁡
χ
⊂
W
2
r, while χ
1
≡
1 on W
3
r and supp
⁡
χ
1
⊂
W
4
r. This entails
(104)
χ
·
f
∘
σ
=
∑
l
=
0
d
n
-
μ
‍
η
n
,
l
χ
∂
x
n
2
l
(
χ
1
g
1
)
+
∑
k
=
1
n
-
1
‍
∑
′
|
β
|
≤
2
d
0
η
k
,
β
∘
σ
·
χ
∂
β
(
χ
1
g
0
)
.
Using e
U on both sides (and omitting ℝ
n in the spaces), Lemmas 18 and 13 imply
(105)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
d
≤
c
∑
l
=
0
d
n
-
μ
‍
∥
e
U
(
∂
x
n
2
l
(
χ
1
g
1
)
)
∣
F
p
→
,
q
s
,
a
→
∥
d
+
c
∑
′
|
β
|
≤
2
d
0
∥
e
U
(
∂
β
(
χ
1
g
0
)
)
∣
F
p
→
,
q
s
,
a
→
∥
d
.
As e
U and differentiation commute on ℰ
′
(
U
)
∋
χ
1
g
j, Lemma 4(i) leads to an estimate from above. But Lemma 18 applies since the supports are in W
4
r, so with χ
~
1
:
=
χ
1
∘
τ we find that the above is less than or equal to
(106)
c
∥
e
U
(
χ
1
g
1
)
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
c
∥
e
U
(
χ
1
g
0
)
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
=
c
∥
(
χ
~
1
·
r
V
(
1
-
∂
x
n
2
)
μ
h
)
∘
σ
∣
F
p
→
,
q
s
+
r
μ
,
a
→
(
U
)
∥
d
+
c
∥
(
χ
~
1
·
r
V
h
)
∘
σ
∣
F
p
→
,
q
s
+
r
,
a
→
(
U
)
∥
d
.
Using Step  1 and Lemmas 13, 9, 8, and 18, this entails
(107)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
)
∥
d
≤
c
∥
(
1
-
∂
x
n
2
)
μ
h
∣
F
p
→
,
q
s
+
r
μ
,
a
→
∥
d
+
c
∥
h
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
Λ
r
-
1
e
V
f
∣
F
p
→
,
q
s
+
r
,
a
→
∥
d
≤
c
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
)
∥
d
.
This shows the local theorem for s
≤
s
1.There is also a local version of Theorem21, with similar proof, namely the following.Theorem 23.
Letσ
j
:
U
j
→
V
j, j
=
1
,
…
,
m, be C
∞ bijections, where U
j
,
V
j
⊂
ℝ
N
j are open. When a
→
,
p
→ fulfill (97)-(98) and when f
∈
F
p
→
,
q
s
,
a
→
(
U
1
×
⋯
×
U
m
) has compact support, then (101) holds true for U
=
U
1
×
⋯
×
U
m and V
=
V
1
×
⋯
×
V
m.As a preparation for our coming work [9], we include a natural extension to the case of an infinite cylinder, where supp
⁡
f is only required to be compact on cross-sections.Theorem 24.
Letσ
:
U
×
ℝ
→
V
×
ℝ be a C
∞-bijection on the form in (65), and U
,
V
⊂
ℝ
n
-
1 open. If (64) holds and f
∈
F
p
→
,
q
s
,
a
→
(
V
×
ℝ
) has supp
⁡
f
⊂
K
×
ℝ, whereby K
⊂
V is compact, then f
∘
σ
∈
F
p
→
,
q
s
,
a
→
(
U
×
ℝ
) and
(108)
∥
f
∘
σ
∣
F
p
→
,
q
s
,
a
→
(
U
×
ℝ
)
∥
≤
c
(
supp
⁡
f
,
σ
)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
.Proof.
We adapt the proof of Theorem22: in Step  1 we take r
∈
]
0,1
[ so small that 6
r is less than both dist
⁡
(
K
,
ℝ
n
-
1
∖
V
) and dist
⁡
(
σ
′
-
1
(
K
)
,
ℝ
n
-
1
∖
U
). Since the extension by zero e
V
×
ℝ
f is well defined, as K
⊂
V is compact, it is an immediate corollary to the proof of Lemma 18 that
(109)
∥
f
∣
F
p
→
,
q
s
,
a
→
(
V
×
ℝ
)
∥
=
∥
e
V
×
ℝ
f
∣
F
p
→
,
q
s
,
a
→
(
ℝ
n
)
∥
.
Then the proof for s
>
s
1 follows that of Theorem 22, with W
:
=
(
σ
′
-
1
(
K
)
+
B
¯
(
0
,
r
)
)
×
ℝ.
Fors
≤
s
1 we have e
V
×
ℝ
f
=
Λ
r
h for some h
∈
F
p
→
,
q
s
+
r
,
a
→
(
ℝ
n
) (cf. Lemma 8). Hence (68) holds as an identity in 𝒟
′
(
V
×
ℝ
) for g
1
:
=
(
r
V
×
ℝ
(
1
-
∂
x
n
+
1
2
)
μ
h
)
∘
σ and g
0
:
=
(
r
V
×
ℝ
h
)
∘
σ.
Theη
k
,
β
∘
σ are controlled using cut-off functions χ
,
χ
1
∈
C
L
∞
∞
(
U
) with similar properties in terms of the sets W
r
=
(
σ
′
-
1
(
K
)
+
B
¯
(
0
,
r
)
)
×
ℝ. Thus we obtain (104) in 𝒟
′
(
U
×
ℝ
).
Now, as in (109) it is seen that f
∘
σ and e
U
×
ℝ
(
χ
·
f
∘
σ
) have identical norms, so the estimates in Step  2 of the proof of Theorem 22 finish the proof, mutatis mutandis.


## 5.2. Isotropic Spaces

Going to the other extreme, when alsoa
n
=
a
0 and p
n
=
p
0, then the Lizorkin-Triebel spaces are invariant under any bounded diffeomorphism (i.e., without (65)), since in that case we can just change variables in all coordinates, in particular in (88)-(89). Moreover, we can adapt Proposition 19 by taking d
n
=
d
0 and μ
=
0 in the proof; and the set-up prior to Theorem 20 is also easily modified to the isotropic situation. Hence we obtain the following.Corollary 25.
Whenσ
:
ℝ
n
→
ℝ
n is any bounded diffeomorphism, then f
↦
f
∘
σ is a linear homeomorphism of F
p
,
q
s
(
ℝ
n
) onto itself for all s
∈
ℝ.This is known from work of Triebel [4, Theorem  4.3.2], which also contains a corresponding result for Besov spaces. (It is this proof we extended to mixed norms in the previous section.) The result has also been obtained recently by Scharf [23], who covered all s
∈
ℝ by means of an extended notion of atomic decompositions.In an analogous way, we also obtain an isotropic counterpart to Theorem22.Corollary 26.
Whenσ
:
U
→
V is a C
∞-bijection between open sets U
,
V
⊂
ℝ
n, then f
∘
σ
∈
F
p
,
q
s
(
U
) for every f
∈
F
p
,
q
s
(
V
) having compact support and
(110)
∥
f
∘
σ
∣
F
p
,
q
s
(
U
)
∥
≤
c
(
supp
⁡
f
,
σ
)
∥
f
∣
F
p
,
q
s
(
V
)
∥
.



---

*Source: 964794-2014-03-02.xml*