HOMEOMORPHISMS WHICH ACT ON BESOV SPACES

Gérard Bourdaud and Winfried Sickel

1. Introduction

Every homeomorphism φ of the real line gives rise to a linear operator

T_{φ}: f ↦ f ∘ φ.

Our aim is to characterize the homeomorphisms for which T_{φ} is a bounded operator on the homogeneous Besov space Ḃ_{p}^{s,q} (0 < s < 1) —or its inhomogeneous counterpart B_{p}^{s,q}.

A lot of work has been done on this problem by the Siberian school, including important results on the higher dimensional case. In some sense, the present paper is the continuation of Vodop’yanov’s work [16].

Mostly, we shall have to reject pathological homeomorphisms. We consider the set H of all increasing homeomorphisms φ from ℝ onto ℝ such that

|E| = 0 ⟹ |φ(E)| = 0,

for all Borel sets E. If φ ∈ H, then one has

(1)  ∫_{ℝ} f(φ(x))φ′(x) dx = ∫_{ℝ} f(x) dx,

for all non-negative Lebesgue-measurable functions f. Similarly, we denote by H^{−1} the set of homeomorphisms φ such that φ^{−1} ∈ H.

Let us recall that the homogeneous Besov space Ḃ_{p}^{s,q} (0 < s < 1, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞) is defined as the set of functions f such that

∥f∥Ḃ_{p}^{s,q} = (∫_{ℝ} (∫_{ℝ} |f(x + h) − f(x)|^{p} dx)^{q/p} \frac{dh}{|h|^{1+sq}} )^{1/q} < ∞

(with the usual modifications for p = ∞ and/or q = ∞). The quotient space of Ḃ_{p}^{s,q} by the constant functions is a Banach space with respect to the above norm. The inhomogeneous Besov space is given by

B_{p}^{s,q} = Ḃ_{p}^{s,q} ∩ L^{p}.

This space turns into a Banach space if it is equipped with the norm

∥f∥_{B_{p}^{s,q}} = ∥f∥Ḃ_{p}^{s,q} + ∥f∥_{p}.

We shall also use the classical Sobolev spaces

Ẇ_{p}^{1} = {f: f′ ∈ L^{p}}

and W_{p}^{1} = Ẇ_{p}^{1} ∩ L^{p}. For more details about these function spaces like equivalent norms, embeddings etc., see e.g. [2], [4], [11], [14], [15].

Definition. A homeomorphism φ of the real line is quasi-conformal if there exists an M ≥ 1 such that

\frac{1}{M} ≤ \frac{φ(x + h) − φ(x)}{φ(x) − φ(x − h)} ≤ M,

for all x ∈ ℝ and h > 0.

Let us begin with the results of Vodop’yanov ([16, Theorems 2.1, 2.2]).

Theorem 1. Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, 0 < s < 1, s ≠ 1/p, and let φ be a homeomorphism of the real line. Then T_{φ} is an automorphism of the Besov space Ḃ_{p}^{s,q} (resp. B_{p}^{s,q}) if and only if φ and φ^{−1} are Lipschitz mappings.

Theorem 2. Let 2 ≤ p < ∞ and let φ be a homeomorphism of the real line. Then T_{φ} is an automorphism of the critical homogeneous Besov space Ḃ_{p}^{1/p,p} if and only if φ is quasi-conformal.

Our results are the following:

Theorem 3. Let 1 ≤ p < ∞, 1 ≤ q ≤ ∞, 0 < s < 1/p, and φ ∈ H. Then the following three statements are equivalent:

• T_{φ} is a bounded operator on Ḃ_{p}^{s,q};

• T_{φ} is a bounded operator on B_{p}^{s,q};

• φ^{−1} is a Lipschitz mapping.

Theorem 4. Let 1 < p ≤ ∞, 1 ≤ q ≤ ∞, 1/p < s < 1, and φ ∈ H. Then the following two statements are equivalent:

• T_{φ} is a bounded operator on Ḃ_{p}^{s,q};

• φ is a Lipschitz mapping.

Theorem 5. Let 1 < p < ∞, 1 ≤ q ≤ ∞, 1/p < s < 1, and φ ∈ H. Then the following two statements are equivalent:

• T_{φ} is a bounded operator on B_{p}^{s,q};

• φ is a Lipschitz mapping and sup_{x∈ℝ}(φ^{−1}(x + 1) − φ^{−1}(x)) < ∞.

Theorem 6. Let 1 < p < ∞ and φ ∈ H ∩ H^{−1}. Then T_{φ} is a bounded operator on the critical homogeneous Besov space Ḃ_{p}^{1/p,p} if and only if φ is quasi-conformal.

Our paper is organized as follows. In the Section 2 we prove the necessity of the conditions given above and in the Section 3 the sufficiency. Then we examine the critical inhomogeneous Besov space, for which we have only partial results. As usual, C denotes a constant whose exact value is not important. We shall often say that “φ acts on a space E” if T_{φ} is a bounded operator on E. If 1 ≤ p ≤ +∞, p′ is the conjugate exponent, given by p′ = p/(p − 1).

2. Proofs: necessary conditions

2.1. The case 0 < s < 1/p.

Let us consider a < b and

f = χ_{[a,b]}.

It is well known that f belongs to Ḃ_{p}^{s,q} and that a homogeneity argument gives

(2)  ∥f∥Ḃ_{p}^{s,q} ≤ C(b − a)^{1/p−s}

and

∥f∥_{p} ≤ C(b − a)^{1/p}.

For 0 < b − a ≤ 1, one arrives at

∥f∥Ḃ_{p}^{s,q} ≤ ∥f∥_{B_{p}^{s,q}} ≤ C(b − a)^{1/p−s}.

The boundedness of T_{φ} implies

(3) ∫_{ℝ} (∫_{ℝ} |f(φ(x + h)) − f(φ(x))|^{p} dx)^{q/p} \frac{dh}{|h|^{1+sq}} ≤ C(b − a)^{q(1/p−s)}.

For x ∈ [φ^{−1}(a), φ^{−1}(b)] and h > φ^{−1}(b) − φ^{−1}(a), we have |f(φ(x + h)) − f(φ(x))| = 1 and the estimates (2) and (3) give

(φ^{−1}(b) − φ^{−1}(a))^{q(1/p−s)} ≤ C(b − a)^{q(1/p−s)}.

Thus

\frac{φ^{−1}(b) − φ^{−1}(a)}{b − a} ≤ C.

We leave to the reader the easy modification of the last proof for the cases p = ∞ or q = ∞.

2.2. The case 1/p < s < 1; necessity of φ′ ∈ L^{∞}.

We consider the family (ψ_{ε})_{ε>0} of continuous compactly-supported functions such that

ψ′_{ε} = \frac{1}{ε}(χ_{[−1−ε,−1]} − χ_{[1,1+ε]}).

These functions have the following properties:

• supp ψ_{ε} ⊂ [−ε − 1, ε + 1],

• ψ_{ε}(x) = 1 for x ∈ [−1, +1],
• ∥ψ_{ε}∥Ḃ_{p}^{s,q} ≤ Cε^{1/p−s}.

In order to prove the third property, one may use the embedding

Ḃ_{1}^{s−1/p+1,q} ⊂ Ḃ_{p}^{s,q}

(cf. [2, (6.3.1)]).

We obtain

∥ψ_{ε}∥Ḃ_{p}^{s,q} ≤ C∥ψ′_{ε}∥Ḃ_{1}^{s−1/p,q},

and the inequality (2) leads to

∥ψ′_{ε}∥Ḃ_{1}^{s−1/p,q} ≤ Cε^{1/p−s}.

On the other hand, one has

∥ψ_{ε}∥_{p} ≤ (2 + 2ε)^{1/p};

thus, for ε ≤ 1, the inhomogeneous norm satisfies the same estimate as does the homogeneous one.

Let us consider three numbers a < b < c such that

(4)  φ(c) − φ(b) ≤ \frac{φ(b) − φ(a)}{2} ≤ 1.

Define

r = \frac{φ(b) − φ(a)}{2}, x_{0} = \frac{φ(b) + φ(a)}{2}, ε = \frac{φ(c) − φ(b)}{r},

and

f(x) = ψ_{ε}(\frac{x − x_{0}}{r} ).

The condition (4) implies

∥f∥_{B_{p}^{s,q}} ≤ C(φ(c) − φ(b))^{1/p−s}.

The boundedness of T_{φ} leads to

∫_{c−b}^{c−a} ( ∫_{c−h}^{b} dx)^{q/p} \frac{dh}{h^{1+sq}} ≤ C(φ(c) − φ(b))^{q(1/p−s)}.

By the change of the variables z = (c − b)^{−1}h we derive from this

(5)  ∫1^{\frac{c−a}{c−b}} (z − 1)^{q/p} \frac{dz}{z^{1+sq}} ≤ C (\frac{φ(c) − φ(b)}{c − b})^{q(1/p−s)} .

Let us choose a, b, c in the following manner: b is a Lebesgue point of φ′, a and c are sufficiently close to b in order to guarantee (4) and

c − b ≤ \frac{c − a}{2}.

Then the term on the left-hand side of (5) is greater than the positive constant

∫_{1}^{2}(z − 1)^{q/p} \frac{dz}{z^{1+sq}};

by taking c → b+, one gets φ′(b) ≤ C.

2.3. The inhomogeneous space Ḃ_{p}^{s,q}.

It remains to demonstrate the necessity of

sup_{x∈ℝ}(φ^{−1}(x + 1) − φ^{−1}(x)) < ∞.

We test T_{φ} on the functions

f_{a}(x) = ψ_{1}(x − a), a ∈ ℝ.

Obviously, the family {f_{a}}_{a∈ℝ} is uniformly bounded in all spaces under consideration here. The assumed boundedness of T_{φ} results in

∥f_{a} ∘ φ∥_{p} ≤ C

for all a ∈ ℝ. Integrating |f_{a} ∘ φ|^{p} over the interval [φ^{−1}(a), φ^{−1}(a + 1)] yields

φ^{−1}(a + 1) − φ^{−1}(a) ≤ C.

2.4. The critical homogeneous space.

Following an idea of L. Ahlfors [1], we reduce the problem to the normalized homeomorphisms, i.e., the homeomorphisms φ such that φ(0) = 0 and φ(1) = 1.

Let G be the set of the affine transforms

x ↦ rx + a

(r ≠ 0, a ∈ ℝ). Let us observe that, for all g ∈ G, T_{g} is an isometry of Ḃ_{p}^{1/p,q} onto itself; this implies:

Lemma 1. Let φ be a homeomorphism such that T_{φ} is a bounded operator on Ḃ_{p}^{1/p,q}. Then, for every g ∈ G, T_{φ∘g} and T_{g∘φ} are also bounded operators on Ḃ_{p}^{1/p,q}; moreover, one has

∥T_{φ}∥ = ∥T_{φ∘g}∥ = ∥T_{g∘φ}∥,

where ∥ · ∥ denotes the operator norm in L(Ḃ_{p}^{1/p,q}).

Lemma 2. There exists a constant C(p) > 0 such that for every normalized homeomorphism φ acting on Ḃ_{p}^{1/p,p} one has

φ^{−1}(\frac{1}{2}) ≤ 1 − exp(−1 − C(p)∥T_{φ}∥^{p}).

Proof. Let us consider

f(x) = ψ_{2}(4x − 1).

We have f(x) = 1 for 0 < x < 1/2 and f(x) = 0 for x > 1; if a = φ^{−1}(1/2), we obtain

∫_{1−a}^{1} (∫_{1−h}^{a} dx) \frac{dh}{h^{2}} ≤ ∥T_{φ}∥^{p}∥ψ_{2}∥p ; Ḃ_{p}^{1/p,p}.

It is then easy to arrive at the desired inequality. □

Lemma 3. Let 1 < p < ∞ and φ ∈ H ∩ H^{−1}. If T_{φ} is a bounded operator on Ḃ_{p}^{1/p,p}, then T_{φ^{−1}} is a bounded operator on Ḃ_{p′}^{1/p′,p′}, with the same operator norm.

Proof. Let f be a C^{∞} function in Ḃ_{p′}^{1/p′,p′}. One has

∥T_{φ^{−1}}(f)∥Ḃ_{p′}^{1/p′,p′} = ∥T_{φ^{−1}}(f)′∥Ḃ_{p′}^{−1/p,p′} = ∥(f′ ∘ φ^{−1})(φ^{−1})′∥Ḃ_{p′}^{−1/p,p′} .

Then we notice that Ḃ_{p′}^{−1/p,p′} is the dual space of Ḃ_{p}^{1/p,p}; this gives

∥T_{φ^{−1}}(f)∥Ḃ_{p′}^{1/p′,p′} = sup {|∫ (f′ ∘ φ^{−1})(φ^{−1})′g dx|: ∥g∥Ḃ_{p}^{1/p,p} ≤ 1}
= sup {|∫ f′(g ∘ φ) dx|: ∥g∥Ḃ_{p}^{1/p,p} ≤ 1}
≤ ∥T_{φ}∥ ∥f′∥Ḃ_{p′}^{−1/p,p′} ≤ ∥T_{φ}∥ ∥f∥Ḃ_{p′}^{1/p′,p′} .

Hence we deduce

∥T_{φ^{−1}}∥_{L(Ḃ_{p′}^{1/p′,p′})} ≤ ∥T_{φ}∥_{L(Ḃ_{p}^{1/p,p})}.

We obtain the equality of the two norms by replacing φ by φ^{−1} in the above proof. □

Lemma 4. There exists a function δ_{p}: (0, +∞) → (0, 1) such that for every normalized homeomorphism φ acting on Ḃ_{p}^{1/p,p} one has

δ_{p}(∥T_{φ}∥) ≤ φ(\frac{1}{2}) ≤ 1 − δ_{p}(∥T_{φ}∥).

Proof. By Lemma 3, we can apply Lemma 2 to φ^{−1}; we obtain

(6)  φ(\frac{1}{2}) ≤ 1 − δ_{p}(∥T_{φ}∥),

where

δ_{p}(t) = exp(−1 − C(p′)t^{p′}).

Then we consider σ ∈ G such that σ(x) = 1 − x; by Lemma 1, we can apply (6) to the normalized homeomorphism σ ∘ φ ∘ σ; this gives

φ(\frac{1}{2}) ≥ δ_{p}(∥T_{φ}∥).
□

Now we consider φ ∈ H ∩ H^{−1} such that T_{φ} is a bounded operator on Ḃ_{p}^{1/p,p}. We are going to prove that φ is quasi-conformal. For any interval [a, b], we define g_{1}, g_{2} ∈ G by the following formulae:

g_{1}(t) = (b − a)t + a, g_{2}(t) = \frac{t − φ(a)}{φ(b) − φ(a)} .

By Lemma 1, φ_{0} = g_{2} ∘ φ ∘ g_{1} is a normalized homeomorphism such that ∥T_{φ_{0}}∥ = ∥T_{φ}∥. If we apply Lemma 4 to φ_{0}, we obtain

δ_{p}(∥T_{φ}∥)(φ(b) − φ(a)) ≤ φ(\frac{a + b}{2}) − φ(a) ≤ (1 − δ_{p}(∥T_{φ}∥))(φ(b) − φ(a)),

from which we deduce

\frac{δ}{1 − δ} (φ(\frac{a + b}{2}) − φ(a)) ≤ φ(b) − φ(\frac{a + b}{2})

≤ \frac{1 − δ}{δ} (φ(\frac{a + b}{2}) − φ(a)) .

3. Proofs: sufficient conditions

Our main tool will be the interpolation theory. Therefore, we begin with a collection of obvious results.

• By means of the formula

∫_{ℝ} |f′(φ(x))|^{p}φ′(x)^{p} dx = ∫_{ℝ} |f′(z)|^{p}φ′(φ^{−1}(z))^{p−1} dz

we see that φ′ ∈ L^{∞} is sufficient for T_{φ} to be a bounded operator on Ẇ_{p}^{1} (1 < p < ∞).

• Again by the above formula it follows that there is no additional condition in case of Ẇ_{1}^{1}.

• The formula (1) yields the sufficiency of (φ^{−1})′ ∈ L^{∞} in case of L^{p}-spaces (1 ≤ p < ∞).

• Using the two preceding statements we get that (φ^{−1})′ ∈ L^{∞} is a sufficient condition in case of W_{1}^{1}.

• φ′ ∈ L^{∞} is a sufficient condition in case of the Hölder spaces Ḃ_{∞}^{s,∞} (0 < s < 1).

3.1. The case 0 < s < 1/p.

We first deal with homogeneous spaces. We employ the real interpolation formula

(Ẇ_{1}^{1}, L^{1})_{Θ,q} = Ḃ_{1}^{s,q}, s = 1 − Θ, 0 < Θ < 1

(cf. [4, (6.2.4)]). Then the statements 2 and 3 yield the boundedness of T_{φ} on Ḃ_{1}^{s,q} (0 < s < 1, 1 ≤ q ≤ ∞) provided that (φ^{−1})′ ∈ L^{∞}. We continue with the help of the complex interpolation formula

[Ḃ_{1}^{s_{0},1}, L^{p_{1}}]_{Θ} = Ḟ_{p}^{s,q}, s = (1 − Θ)s_{0},

\frac{1}{p} = (1 − Θ) + \frac{Θ}{p_{1}} , \frac{1}{q} = (1 − Θ) + \frac{Θ}{2}, 0 < Θ < 1

(cf. [6]). (The Triebel-Lizorkin spaces Ḟ_{p}^{s,q} will not play an important role here, cf. [14], [15], [6] for details.) If s_{0} ↑ 1 and p_{1} ↑ ∞, then each s < 1/p is admissible: we obtain the boundedness of T_{φ} on Ḟ_{p}^{s,q} for 1 < p < ∞, 0 < s < 1/p and some q, which depends on s and p. To switch to Besov spaces and to remove this dependence on q we employ

(Ḟ_{p}^{s_{0},q_{0}}, L^{p})_{Θ,q} = Ḃ_{p}^{s,q}, s = (1 − Θ)s_{0}, s_{0} > 0, 0 < Θ < 1

(cf. [5]).

Now, consider the inhomogeneous spaces. But there is no principal difference. Instead of the statement 2 one starts with the statement 4. For the complex interpolation formula we refer the reader to [14, (2.4.2)] and [2, (6.4)].

3.2. The homogeneous space for 1/p < s < 1.

We proceed as above. We first employ

[Ḃ_{∞}^{s_{0},∞}, Ẇ_{p_{1}}^{1}]_{Θ} = Ḟ_{p}^{s,q}, s = (1 − Θ)s_{0} + Θ, (7)

\frac{1}{p} = \frac{Θ}{p_{1}} , \frac{1}{q} = \frac{Θ}{2}, 0 < Θ < 1

(cf. [6]), if 1 < p_{1} < ∞. Further, we need the interpolation formula

(8)  (Ḟ_{p}^{s_{0},q_{0}}, Ẇ_{p}^{1})_{Θ,q} = Ḃ_{p}^{s,q}, s = (1 − Θ)s_{0} + Θ, 0 < Θ < 1

for 1 < p < ∞, 0 < s_{0} < 1 (cf. [6]). Here our strategy looks as follows. We first apply (7) with s_{0} ↓ 0 and p_{1} ↓ 1. Afterwards we use (8).

3.3. The inhomogeneous space for s > 1/p.

What we need is the following inequality

(9)  ∑_{j∈ℤ}( sup_{j≤x≤j+1} |f(x)|)^{p} ≤ C∥f∥_{B_{p}^{s,q}}^{p}.

Let us assume (9) for a while: we get

∥f ∘ φ∥_{p}^{p} = ∑_{j∈ℤ} ∫_{j}^{j+1} |f(x)|^{p}(φ^{−1})′(x) dx

≤ ∑_{j∈ℤ}( sup_{j≤x≤j+1} |f(x)|^{p})(φ^{−1}(j + 1) − φ^{−1}(j)) ≤ C∥f∥_{B_{p}^{s,q}}^{p}.

Together with Subsection 3.2, this proves sufficiency for inhomogeneous Besov spaces. It remains to prove (17). Using the embedding

B_{p}^{s,q} ⊂ B_{p}^{s−ε,p},

it suffices to prove it with p = q. We start from the so-called ℓ^{p}-localization property of B_{p}^{s,p}:

(10)  ∑_{j∈ℤ} ∥θ(· − j)f∥_{B_{p}^{s,p}}^{p} ≤ C(θ)∥f∥_{B_{p}^{s,p}}^{p},

for all θ ∈ D(ℝ) (cf. for instance [3]). The hypothesis s > 1/p leads to

∑_{j∈ℤ} ∥θ(· − j)f∥_{∞}^{p} ≤ C(θ)∥f∥_{B_{p}^{s,p}}^{p}.

A suitable choice of θ gives (9).

3.4. The critical homogeneous space. Step 1: 2 ≤ p < +∞. For the sake of completeness, we give here Vodop’- yanov’s proof. Let φ be a quasi-conformal homeomorphism of the real line. Following Ahlfors [1, Chapter IV, Theorem 2], there exists a quasi-conformal homeomorphism φ̃ of the whole plane ℝ^{2} such that φ̃(x, 0) = (φ(x), 0) (x ∈ ℝ). Next we consider the bounded trace operators:

T r: Ḃ_{p}^{2/p,p}(ℝ^{2}) → Ḃ_{p}^{1/p,p}(ℝ) (p > 2), T r: Ẇ_{2}^{1}(ℝ^{2}) → Ḃ_{2}^{1/2,2}(ℝ).

These operators satisfy

T r(f)(x) = f(x, 0) (x ∈ ℝ).

The existence of the bounded extension operators E, such that T r∘E = Id, is a classical property of the Besov and Sobolev spaces. Thus we have reduced the problem to proving that the quasi-conformal mapping φ̃ defines a bounded operator Tφ̃ on both Ḃ_{p}^{2/p,p}(ℝ^{2}) and Ẇ_{2}^{1}(ℝ^{2}).

For the Sobolev space Ẇ_{2}^{1}(ℝ^{2}) this follows from the estimate

|\frac{∂φ̃}{∂x}(x, y)|^{2} + |\frac{∂φ̃}{∂y}(x, y)|^{2} ≤ C det Jφ̃(x, y)) (a.e.),

where Jφ̃(x, y) is the Jacobian matrix of φ̃ (see [9], [7]). For the Besov space, we use the interpolation formula

Ḃ_{p}^{2/p,p}(ℝ^{2}) = (BMO(ℝ^{2}), Ẇ_{2}^{1}(ℝ^{2}))_{2/p,p}

(see [12]) and Reimann’s theorem [13], which asserts that Tφ̃ is a bounded operator on BMO(ℝ^{2}). Step 2: 1 < p < 2. We use the following two classical properties: as φ̃ is a quasi-conformal mapping of the plane, the same is true for φ^{−1} ([1, Chapter II]); the trace of φ^{−1} on the real line, i.e. φ^{−1}, is a quasi-conformal mapping ([1, Chapter IV, Theorem 1]). Thus we can apply the previous step to φ^{−1}, then we conclude with the help of Lemma 3.

Remark. It is a nice idea of Vodop’yanov to transfer the problem from the real line to the plane. Interpolation with BMO(ℝ) is an alternative method, but it does not give the optimal result. Let us explain why. We start with the interpolation formula

(BMO(ℝ), Ḟ_{1}^{1,2}(ℝ))_{1/p,p}= Ḃ_{p}^{1/p,p}(ℝ).

Recall that

Ḟ_{1}^{1,2}(ℝ) = {f: f′ ∈ Ḟ_{1}^{0,2}(ℝ)}

and that Ḟ_{1}^{0,2}(ℝ) is the Hardy space whose dual space is BMO(ℝ). By duality—as in Lemma 3—and interpolation, we obtain the following result: if φ acts on BMO(ℝ), then φ acts on Ḃ_{p}^{1/p,p}(ℝ). P. Jones [8] studied the homeomorphisms which act on BMO(ℝ): they are characterized by φ′ ∈ A_{∞} (the weight class of Muckenhoupt). But φ′ ∈ A_{∞} is a stronger property than quasi-conformality.

4. Remarks on the critical inhomogeneous space

As far as we know, the full characterization of admissible changes of variables in B_{p}^{1/p,p} is an open problem. First we recall that the condition

(11)  sup_{x∈ℝ}(φ^{−1}(x + 1) − φ^{−1}(x)) < ∞

is necessary for T_{φ} to act on B_{p}^{1/p,p} (the proof given in (2.3) is true for all s > 0).

Proposition 1 ([16]). Let s > 0. If φ is a homeomorphism acting on Ḃ_{p}^{s,p} and satisfying the condition (11), then φ acts also on B_{p}^{s,p}.

Proof. We shall follow Vodop’yanov once again. Let θ be a positive C^{∞} function such that

∑_{j∈ℤ} θ(x − j) = 1

and—say— supp θ ⊂ [−1/4, 5/4]. Then we have

∑_{j∈ℤ} θ(φ(x) − j)^{p} ≥ 2^{−p},

which implies

∥f ∘ φ∥_{p}^{p} ≤ 2^{p} ∑_{j} ∫_{ℝ} |f_{j} ∘ φ|^{p} dx,

where f_{j}(x) = f(x)θ(x − j). Now we consider

α = sup_{x∈ℝ}(φ^{−1}(x + 1) − φ^{−1}(x)).

HOMEOMORPHISMS WHICH ACT ON BESOV SPACES 13

Let us recall the Friedrichs inequality ∥f∥_{p} ≤ C(I)∥f∥Ḃ_{p}^{s,q},

which is satisfied by all functions f with support in an open interval I (see e.g. [10, p. 350]). We apply it to the function

(f_{j} ∘ φ)(x + φ^{−1}(j))

and the interval (−α, 2α); this leads to

∥f ∘ φ∥_{p}^{p} ≤ C ∑_{j} ∥f_{j} ∘ φ∥p ; Ḃ_{p}^{s,p} ≤ C ∑_{j} ∥f_{j}∥_{B_{p}^{s,p}}^{p} ≤ C∥f∥_{B_{p}^{s,p}}^{p}

(the last inequality follows from (10)). □

Proposition 1 and Theorem 6 imply the following: Proposition 2. Let 1 < p < +∞ and φ ∈ H∩H^{−1}. If φ is quasi-conformal and satisfies (11), then φ acts on B_{p}^{1/p,p}.

Vodop’yanov conjectured that the above condition is also necessary for φ to act on B_{p}^{1/p,p} (see [16, Theorem 2.2]). We shall show that it is false: Proposition 3. Let 1 < p < +∞ and φ ∈ H ∩ H^{−1}. If φ has the two following properties:

• there exists q > 1 such that

(12)  sup_{a∈ℝ} ∫_{a}^{a+1}(φ^{−1})′(x)^{q} dx < +∞,

• there exist some constants C > 0 and δ > 0 such that, for almost all (x, y) with |x − y| < δ, one has

(13)  (\frac{φ(x) − φ(y)}{x − y})^{2} \frac{1}{φ′(x)} \frac{1}{φ′(y)} ≤ C,

then T_{φ} is a bounded operator on B_{p}^{1/p,p}. Proof. Recall that

∥f∥_{p} + (∫∫_{|x−y|<δ}(x − y)^{−2}|f(x) − f(y)|^{p} dx dy)^{1/p}

is an equivalent norm in B_{p}^{1/p,p} (cf. [15, (2.6.1)]). In order to estimate ∥f ∘ φ∥_{p}, we take

r = \frac{qp}{q − 1};

the inequality p ≤ r < +∞ yields the embeddings

B_{p}^{1/p,p} ⊂ B_{r}^{1/r,p} ⊂ L^{r};

then (10) leads to

∑_{j∈ℤ} (∫_{j}^{j+1} |f(x)|^{r} dx)^{p/r} ≤ C∥f∥_{B_{p}^{1/p,p}}^{p};

this implies

∥f ∘ φ∥_{p}^{p} = ∑_{j∈ℤ} ∫_{j}^{j+1} |f(x)|^{p}(φ^{−1})′(x) dx

≤ ∑_{j∈ℤ} (∫_{j}^{j+1} |f(x)|^{r} dx)^{p/r} (∫_{j}^{j+1}(φ^{−1})′(x)^{q} dx)^{1−p/r}

≤ C∥f∥_{B_{p}^{1/p,p}}^{p}.

On the other hand, (13) implies

∫∫_{|x−y|<δ}(x − y)^{−2}|(f ∘ φ)(x) − (f ∘ φ)(y)|^{p} dx dy

≤ C ∫∫_{|x−y|<δ} \frac{φ′(x)φ′(y)}{(φ(x) − φ(y))^{2}} |f(φ(x)) − f(φ(y))|^{p} dx dy

≤ C ∫∫ (u − v)^{−2}|f(u) − f(v)|^{p} du dv.

□

It may be convenient to replace (13) by a more manageable condition. That is the purpose of the next proposition.

Proposition 4. For every φ ∈ H, the following statement implies (13): there exist some δ > 0 and C > 0 such that

(14)  φ′(y) ≤ Cφ′(x)

for almost all x, y satisfying |x − y| < δ.

Proof. A classical theorem on product measures is the following: if N is a negligible subset of ℝ^{2}, then, for almost every x ∈ ℝ, one has (x, y) ∉ N for almost every y ∈ ℝ. Thus we get a negligible set M ⊂ ℝ and, for each x ∉ M, a negligible A_{x} ⊂ ℝ such that

φ′(y) ≤ Cφ′(x),

as soon as y ∉ A_{x} and |y − x| < δ. Then for all x ∉ M and y ∈ (x, x + δ) we have

\frac{φ(y) − φ(x)}{y − x} = \frac{1}{y − x} ∫_{x}^{y} φ′(z) dz ≤ Cφ′(x),

and the same is true for x − δ < y < x. We have obtained

(\frac{φ(y) − φ(x)}{y − x})^{2} ≤ C^{2}φ′(x)φ′(y),

for all (x, y) ∈ (ℝ \ M)^{2} such that |x − y| < δ. □

Proposition 5. Let 1 < p < ∞.

• There exists a C^{1}-diffeomorphism φ which acts on Ḃ_{p}^{1/p,p} but not on B_{p}^{1/p,p}.

• There exists a C^{1}-diffeomorphism φ which acts on B_{p}^{1/p,p} but not on Ḃ_{p}^{1/p,p}.

Proof. We first consider the diffeomorphism φ such that

φ′(x) = min (1, \frac{1}{\sqrt{|x|}});

it is easy to see that φ is quasi-conformal—indeed, φ′ belongs to Muckenhoupt’s A_{∞} class—and then defines a bounded operator on Ḃ_{p}^{1/p,p}; on the other hand, one has

φ^{−1}(x + 1) − φ^{−1}(x) = \frac{x}{2} + \frac{3}{4} (x ≥ 1),

so φ does not satisfy (11) and T_{φ} is not bounded on B_{p}^{1/p,p}.

Next we consider the odd function φ such that

φ(x) = e^{x} − 1, x ≥ 0.

One has (φ^{−1})′ ∈ L^{∞} then, a fortiori, (12); moreover, it is easily seen that φ satisfies the condition (14), hence leads to a bounded operator on B_{p}^{1/p,p}. On the other hand, we have

\frac{φ(2x) − φ(x)}{φ(x) − φ(0)} = e^{x} − 1 (x > 0),

which implies that φ is not quasi-conformal. □

References

[1] L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton, 1966.

[2] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer, Berlin, 1976.

[3] G. Bourdaud, Localisations des espaces de Besov, Studia Math. 90 (1988), 153–163.

[4] R. de Vore and G. G. Lorentz, Constructive approximation, Springer, Berlin, 1993.

[5] M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, Function spaces and applications (Lund, 1986), Lecture Notes in Math., vol. 1302, Springer, Berlin-New York, 1988, pp. 223–246.

[6] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Functional Anal. 93 (1990), 34–170.

[7] V. M. Gol’dshtein and Yu. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Kluwer, Dordrecht-Boston-London, 1990.

[8] P. Jones, Homeomorphisms of the line which preserve BMO, Arkiv Math. 21 (1983), 229–231.

[9] V. G. Maz’ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions (1985), Pitman, Boston-London.

[10] V. G. Maz’ya, Sobolev spaces, Springer, 1985.

[11] Y. Meyer, Ondelettes et opérateurs, Herman, Paris, 1990.

[12] J. Peetre and E. Svensson, On the generalized Hardy’s inequality of McGehee, Pigno and Smith and the problem of interpolation between BMO and a Besov space, Math. Scand. 54 (1984), 221–241.

[13] H. M. Reimann, Functions of bounded mean oscillation and quasi-conformal mappings, Comment. Math. Helv. 49 (1974), 260–276.

[14] H. Triebel, Theory of function spaces, Birkhäuser, Basel, 1983.

[15] H. Triebel, Theory of function spaces. II, Birkhäuser, Basel, 1992.

[16] S. K. Vodop’yanov, Mappings of homogeneous groups and imbeddings of functional spaces, Sibirsk. Math. Zh. 30 (1989), 25–41.