4.1. Bounded Diffeomorphisms
A one-to-one mapping
y
=
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 convenience
σ
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 a
C
∞
-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 diffeomorphism
σ
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 expect
f
∘
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 Theorem 20, 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.
Let
σ
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 for
s
≤
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).
First
r
>
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 Lemma 8 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 Proposition 19, 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 Sections 4.2 and 5.
To control the Taylor expansions, it will be crucial for us to exploit both the local means recalled in Theorem 16 and the parameter-dependent setup in Theorem 14. This is prepared for with the following discussion.
The functions
k
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
.
Splitting
z
=
(
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 functions
ψ
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 constant
A
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.
Replacing
k
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 Theorem 14 are satisfied, and we are thus ready to prove our main result.
Theorem 20.
If
σ
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 Proposition 19, 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 of
k
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 in
σ
′
(
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 to
f
(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 Section 4.1 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 Theorem 20 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, viewing
σ
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.