A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_{\alpha}(\mathbb R^n)$ depends on the index $\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-V\"ais\"al\"a's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_{\alpha} (\mathbb R^n)$ for any $(\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_{\alpha}(\mathbb R^n)$ is shown to be invariant under inversions for all $0<\alpha<1$.


Introduction
Quasiconformal mappings can be characterized via invariant function spaces. For example, a homeomorphism f : R n → R n , n ≥ 2, is quasiconformal if and only if the composition operator C f (given by C f (u) = u • f ) is bounded on the homogeneous Sobolev spaceẆ 1, n (R n ); see for example [5]. The composition property is easiest seen from the usual analytic definition, according to which a homeomorphism f : R n → R n , n ≥ 2, is quasiconformal if f ∈ W 1, 1 loc (R n ; R n ) and there is a constant Indeed, modulo technicalities, one simply uses the chain rule and a change of variables. It is far less obvious that also the invariance of the Triebel-Lizorkin spacesḞ s n/s, q (R n ) with 0 < s < 1 and n/(n + s) < q < ∞ characterizes quasiconformality, see [10,2,6,4]. The difficulty here is that one has to deal with "fractional derivatives" and thus the inequality from the analytic definition is not immediately helpful. For the off-diagonal Besov spacesḂ s n/s, q (R n ) with q n/s, the situation is different: each homeomorphism f for which C f is bounded onḂ s n/s, q (R n ) has to be quasiconformal and even bi-Lipschitz; these spaces are clearly bi-Lipschitz invariant, see [4]. Recall here that f is bi-Lipschitz if there exists a constant L ≥ 1 such that Furthermore, the John-Nirenberg space BMO(R n ) is invariant under quasiconformal mappings and each sufficiently regular homeomorphism f for which C f is a bounded operator on BMO(R n ) is necessarily quasiconformal; see [7,1].
In their 2000 paper [3], Essen, Jasson, Peng and Xiao introduced the so-called Q-spaces Q α (R n ), 0 < α < 1, that satisfyẆ 1, n (R n ) ⊂Ḟ α n/α, n/α (R n ) ⊆ Q α (R n ) ⊆ BMO(R n ). Each Q α (R n ) consists of all u ∈ L 2 loc (R n ) with The above definition actually makes perfect sense for all −∞ < α < ∞, but the case α ≥ 1 (when n ≥ 2) reduces to constant functions and the case α < 0 to BMO(R n ); see [3]. These spaces have received considerable interest. In [3], five open problems related to the spaces Q α (R n ) were posed. All but the following one of them have by now been solved.
A quasiconformal composition problem for the Q-spaces ( [3,Problem 8.4]): Let f be a quasiconformal mapping. Prove or disprove the boundedness of the composition operator C f on Q α (R n ) with α ∈ (0, 1).
By the above string of inclusions of function spaces, all of which except for the Q-spaces are known to be quasiconformally invariant, suggests that the answer should be in the positive.
We show that, surprisingly, the answer to the above question depends on the quasiconformal mapping in question through the shrinking properties of the mapping. For example, the quasiconformal mapping f (x) = x|x| induces a bounded composition operator for all 0 < α < 1, but if the Jacobian of a quasiconformal mapping decays to zero when we approach a sufficiently large set, then the invariance may fail. Thus, the case of Q-spaces is be very different from the other function spaces that we discussed above.
In order to state our results, we need to introduce some terminology whose analogues have appeared in estimating the upper box-counting dimension of the singular set of a suitable weak solution of the Navier-Stokes system [8].
Definition 1.1. For a set E ⊆ R n and every r > 0, denote by N cov (r, E) the minimal number of cubes with edge length r required to cover E.
where the supremum is taken over all balls where the first supremum is taken over all r ∈ (0, ∞) and the second is over all balls We also need the concept of the local Muckenhoupt class.

Definition 1.2.
For a closed set E ⊆ R n and a nonnegative function w : R n → R, we say that w belongs to the local Muckenhoupt class A 1 (R n ; E) provided there exists a positive constant C such that The main result of this paper is the following theorem. Theorem 1.3. Given n ≥ 2, let f : R n → R n be a quasiconformal mapping with J f ∈ A 1 (R n ; E) for some closed set E ⊆ R n . If E is a bounded set with dim L E ∈ [0, n) or E is an unbounded set with , then C f is bounded on Q α (R n ) for all α ∈ (0, 1). Theorem 1.3 is essentially sharp, see Theorems 1.6 and 1.7 below. As the first important consequence of Theorem 1.3, we have the following result.
Furthermore, for the Tukia-Väisälä quasiconformal extension f : R n → R n of an arbitrary quasiconformal (quasisymmetric) mapping g : R n−p → R n−p , we obtain the second important consequence of Theorem 1.3. Corollary 1.5. Given 1 ≤ p < n, suppose g : R n−p → R n−p is a quasiconformal mapping when n − p ≥ 2, or a quasisymmetric mapping when n − p = 1. Let f : R n → R n be the Tukia-Väisälä's quasiconformal extension of g as in [9]. Then the following hold: The proof of Theorem 1.3 relies on a new characterization of Q-spaces established in Section 3. This technical result allows us to employ our Muckenhoupt assumption and the control on the number of Whitney-type balls guaranteed by our dimension estimate. We expect that our approach will allow one to handle various other function spaces as well.
Our assumption on the control of the fractal size of the degerancy set, whenever which is bounded or unbounded, is necessary in the following sense. Theorem 1.6. Let n ≥ 2 and 0 < α 0 < 1. There is a bounded set E α 0 with dim L E α 0 = n − 2α 0 and a quasiconformal (Lipschitz) mapping f : The main idea in the constructions for Theorem 1.6 is to patch up suitable pieces of radial stretchings in a family of pairwise disjoint balls. In this manner, we also construct an unbounded set E α 0 ⊂ Z n with dim LG E α 0 = n − 2α but dim L E α 0 = 0 and an associated quasiconformal mapping as in Theorem 1.6; see below. This also shows the need for dim LG in Theorem 1.3. Theorem 1.7. Let n ≥ 2 and 0 < α 0 < 1. There exists a unbounded set E α 0 ⊂ Z n with dim LG E α 0 = n− 2α 0 but dim L E α 0 = 0, and a quasiconformal (Lipschitz) mapping f : R n → R n with J f ∈ A 1 (R n ; E α 0 ) for which C f is not bounded on Q α (R n ) for any α ∈ (α 0 , 1). This paper is organized as follows: Section 2 clarifies the relationship between the Minkowski dimension and the local Minkowski dimension dim L or the global Minkowski dimension dim LG and also computes dim L and dim LG for the sets in Theorems 1.6 and 1.7; Section 3 explores a new aspect of Q α (R n ), which will be used in the proof of Theorem 1.3; in Section 4, we prove Theorem 1.3; Section 5 contains the proofs of Corollaries 1.4 and 1.5; Section 6 is devoted to the proofs of Theorems 1.6 and 1.7.
Finally, as the converse of the above open question, given a homeomorphism f : R n → R n for which the composition operator C f is a bounded on Q α (R n ) for some α ∈ (0, 1), one would like to know if f is necessarily quasiconformal. The answer is actually in the positive, at least under suitable regularity assumptions on the homeomorphism in question. Since this requires some work, the details will be given in a forthcoming paper.
Notation. In the sequel, we denote by C a positive constant which is independent of the main parameters, but may vary from line to line. The symbol A B or B A means that A ≤ CB. If A B and B A, we then write A ∼ B. For any locally integrable function u and measurable set X, we denote by -X u the average of u on X, namely, -X u ≡ 1 |X| X u dx. For a set Ω and x ∈ R n , we use d(x, Ω) to denote inf z∈Ω |x − z|, the distance from x to Ω. For λQ, we mean the cube concentric with Q, with sides parallel to the axes, and with length ℓ(λQ) = λℓ(Q); similarly, λB denotes the ball concentric with Q with radius λr B , where r B is the radius of B.

Local and global Minkowski dimensions
In this section, we clarify the relation between the Minkowski dimension and the above dimensions dim L and dim LG . Recall that for a bounded set E ⊂ R n , its Minkowski dimension dim M E is defined by where N cov (r, E) is the minimum number of cubes with edge length r required to cover E.

Lemma 2.1.
(i) For every set E ⊂ R n and every R ≥ 1, we have log(r B /r) .
(ii) For every set E ⊂ R n , we always have where the supremum is taken over all balls in R n .
Proof. (i) From the definition, we always have Towards the reverse inequality, notice that every ball B of radius 1 ≤ r B ≤ R can be covered by c n R n balls B i of radii 1. So and hence for all r < r B /N and r < 1, we have Since the first term on the right-hand side tends to 0 as N → ∞, by the definition of dim L E, we obtain the desired inequality.
The other inequalities follow from the definitions and (i) directly.
(iii) These statements are trivial.
If E is a set of finitely many points, observing that N cov (r, E ∩ B) 1 for every ball B with radius r B ≥ Nr, we obtain where [θk] is the largest integer less than or equal to θk.
for all balls B and all θ ∈ [0, 1] since (2 N θ ) n ∩ B only contains finitely many points.
Proof. We first show that dim L (2 N θ ) n = 0. Observe that each B ⊂ R n with r B ≤ 1 contains at most a uniform number of points in Z n . So for each N ≥ 1 and r ∈ (0, r B /N), we can cover B ∩ (2 N θ ) n by a uniform number of balls of radii r, that is, To show (2.2), we first consider the easy cases dim LG N n = n and dim LG (2 N ) n = 0. Indeed, for every ball B ⊂ R n with r B = N, we have and hence, by Lemma 2.1, dim LG N n = n.
On the other hand, for each N and r > 0, if r ≤ 1 and Nr < r B , we have Hence The proof of dim LG (2 N θ ) n ≤ θn is reduced to verifying that for every large N, all r > 0 and all balls B with r B ≥ Nr, we have Indeed, this implies that To prove (2.3), we consider two cases under the assumption N ≥ 2 5 . Case 1: 0 < r ≤ 1. If r B < 2, then (2 N θ ) n ∩ B contains no more than a uniform number of points and hence ♯((2 N θ ) n ∩ B) 1 (r B /r) θn .

Remark 2.3. Lemma 2.2 indicates that the dimension dim
LG not only measures the local selfsimilarity and local Minkowski size but also measures the global selfsimilarity of E.
For a slight modification of the standard Cantor construction, we obtain E a and its self-similar extension E a so that dim L and dim LG are the same and coincide with dim M E a . Precisely, the sets E a and E a are defined as follows. Let a ∈ (0, 1). Let Notice that E m a consists of 2 mn disjoint cubes {Q m, j } 2 mn j=1 with edge length [(1 − a)/2] m , and E m+1 a ⊂ E m a . Denote by z m, j the center of Q m, j and z 0 = ( 1 2 , · · · , 1 2 ) the center of Q 0 = I n 0 . Denote by E a the closure of the collection of all these centers, that is, In this case, we consider the larger family for all possible k ≥ −m and i = 1, · · · , 2 (m+k)n . Let z m, j be the center of Q m, j . We also have Lemma 2.4. For every a ∈ (0, 1), Proof. By Lemma 2.1, it suffices to show that .
To this end, notice that for each k > m, we have and hence, N cov (r, E a ) ∼ 2 (k r −m)n . In particular, N cov (r, E a ) 2 k r n , which implies that . Hence , as desired.

A characterization of Q-spaces
In this section, we characterize membership in Q-spaces via oscillations. To do so, let us introduce a couple of concepts. Let u be a measurable function. For α ∈ (0, 1), q ∈ (0, ∞), and each ball Also, for every ball B ⊂ R n and each function u on B, set Then u Q α (R n ) = sup B Φ α (u, B) 1/2 , where the supremum is taken over all balls B ⊂ R n .  B(x 0 , 16r)). Consequently, To verify Proposition 3.1, we need the following estimate from [6].

Lemma 3.2.
Let σ ∈ (0, ∞) and u ∈ L σ loc (R n ). Then there is a set E with |E| = 0 such that for each Proof of Proposition 3.1. By Lemma 3.2, we obtain Applying Hölder's inequality and changing the order of summation, we obtain B(x 0 , 8r)).
For J 2 , notice that Then, applying an argument similar to the above estimate for J 1 , we have B(x 0 , 8r)).
Combining the estimates on J 1 and J 2 , we obtain B(x 0 , 8r)).
On the other hand, noticing that for all x ∈ R n , r > 0 and k ≥ 0 one has we utilize q ∈ (0, 2] and the Hölder inequality to achieve Thus, by changing the order of the integrals with respect to dz and dx, u, B(x 0 , 16r)).
This completes the proof of Proposition 3.1.

Proof of Theorem 1.3
Here we only prove Theorem 1.3 under the assumption diam E < ∞. The case diam E = ∞ is similar. Without loss of generality, we may assume that diam E = 1 and E ⊂ B(0, 1). By Proposition 3.1, it suffices to show that We divide the argument into two cases. Thus, Hence we have Observe that J f ∈ A 1 (R n , E) also implies that Therefore, by this and a change of the variables again, which further yields that Moreover, by quasisymmetry of f , for all j ∈ Z and z ∈ R n , we have Recalling that holds for some constant N 2 ≥ 1 (independent of x 0 , r; see [5]), we arrive at which together with Proposition 3.1 gives as desired.
Recall that each domain Ω admits a Whitney decomposition. In particular, for Ω = R n \ E, there exists a collection W Ω = {S j } j∈N of countably many dyadic (closed) cubes such that To see this, by the definition of dim L E there exists constants N 1 ≥ 8 and k 1 ∈ N such that for all which implies that For every δ > 0, denote by N cov (δ, E ∩ 32B) the collection of cubes of edge length δ required to cover E ∩ 32B and ♯N cov (δ, E ∩ 32B) = N cov (δ, E ∩ 32B). For k ≥ −N 0 and S k,i ∈ S k (16B), we have 2 11 √ nS k,i ∩ E ∅ and hence S k,i intersects some cube Q ∈ N cov (2 −k , E ∩ 32B), which implies that S k,i ⊂ 2 13 nQ. Also notice that for each cube Q ∈ N cov (2 −k , 16B ∩ E), the cube 2 13 nQ can only contain a uniformly bounded number of S k,i ∈ S k (16B). We conclude that for k ≥ −N 0 , This together with (4.5) gives that for k ≥ k 1 + k 0 + N 1 , On the other hand, if k 0 − N 0 ≤ k ≤ k 1 + k 0 + N 1 , then by 2 k−k 0 ≤ 2 k 1 +N 1 +N 0 1 we always have This gives the above claim.
By Proposition 3.1, we have For each S k,i , let B k,i be the ball centered at x k,i (x k, i is the center of S k,i ) and radius 2 √ nℓ(S k,i ). Then So applying the above Case 1 to 16B k,i , we have This, together with n − 2α − dim L E − ǫ > 0, gives To estimate P 2 , write Observing that we obtain Therefore, by n − 2α − dim L E − ǫ > 0, one gets Recall that it was proved by Reimann [7] that u • f BMO(R n ) u BMO(R n ) , and also in [3] that u BMO(R n ) u Q α (R n ) . Thus P 2 u 2 Q α (R n ) . Combining the estimates for P 1 and P 2 , we arrive at Ψ α, 2 (u • f, B(x 0 , r)) u 2 Q α (R n ) for all x 0 and r as desired.
Case 3: d(x 0 , E) ≤ 2r and r > 1. Without loss of generality, we may assume that x 0 = 0. Denote by M the minimum number of balls, which are centered in B(0, 1) \ B(0, 1/2) and have radius 2 −9 , required to cover B(0, 1) \ B(0, 1/2). Let {B j } M j=1 be a sequence of such balls and write their centers Notice that Assume that 2 k 0 −1 ≤ r < 2 k 0 . Then k 0 ≥ 1, and B(x 0 , 16r) \ B(x 0 , 2) can be covered by the family j = B(0, 2). Then we have By Proposition 3.1 and the result of Case 1 applied to 32B k, j , we have due to (4.6), and hence For P 4 , an argument similar to P 2 in the Case 2 leads to P 4 u 2 Q α (R n ) . This finishes the proof of Theorem 1.3.

Proofs of Corollaries 1.4 and 1.5
Proof of Corollary 1.4. Notice that if β > 0, then f is a quasiconformal mapping from R n → R n , and that By Theorem 1.3, if β > 0, then C f is bounded on Q α (R n ) for all α ∈ (0, 1). If β < 0, then f is not a quasiconformal mapping from R n → R n ; so we can not apply Theorem 1.3 directly. However, observe that f is a quasiconformal mapping from R n \ {0} to R n with J f (x) ∼ |x| β−1 yielding J f ∈ A 1 (R n , {0}). Thus, an argument similar to but easier than that for Theorem 1.3 will lead to the boundedness of C f on Q α (R n ) for all α ∈ (0, 1). Indeed, let u ∈ Q α (R n ) and B = B(x 0 , r) be an arbitrary ball of R n . If r < |x 0 |/4, then With the help of this and J f ∈ A 1 (R n , {0}), similarly to Case 1 in the proof of Theorem 1.3, we obtain (4.2), (4.3) and (4.4). This implies then by B(x 0 , r) ⊂ B(0, 2r) and Proposition 3.1, we have f, B(0, 32r)). Then B(0, 32r) \ {0} is covered by the family of balls {B k, j : k ≥ 0, 0 ≤ j ≤ M}. Therefore, we obtain Similarly to the estimate on P 3 , we have P 5 u 2 Q α (R n ) ; and similarly to but easier than for P 2 , we obtain P 6 u 2 Q α (R n ) . Putting all together gives , as desired, and hence finishes the proof of Corollary 1.4.
Proof of Corollary 1.5. For our convenience, let R n + = {z = (x, y) : x ∈ R n−1 & y > 0}. We also write H n = R n + \ R n−1 and equip it with the hyperbolic distance d H n -that is - where the infimum is taken over all rectifiable curves γ in H n joining w and w ′ . Suppose that g : R n−1 → R n−1 is a quasiconformal mapping when n ≥ 3, or a quasisymmetric mapping when n = 2. According to Tukia-Väisälä [9, Theorem 3.11], g can be extended to such a quasiconformal mapping f : R n Obviously, such an f can be further extended to a quasiconformal mapping f : R n → R n by reflection, that is, For sake of simplicity, we write f as f , and generally set We equip H n, p with the distance d H n, p , an analog of the hyperbolic distance, via where the infimum is taken over all rectifiable curves γ in H n, p joining w and w ′ . Suppose that g : R n−p → R n−p is a quasiconformal mapping when n − p ≥ 2, or a quasisymmetric mapping when n− p = 1. In accordance with Tukia-Väisälä's [9, Section 3.13], g can be extended to a quasiconformal mapping f : R n → R n such that (i) f | R n−p = g; (ii) f | H n, p is a L-biLipschitz with respect to d H n, p for some constant L ≥ 1.
Notice that both f and f −1 are biLipschitz with respect to d H n, p . We show that C f is bounded; the case of C f −1 is analogous. By Theorem 1.3, it suffices to verify J f ∈ A 1 (R n ; R n−p ). In what follows, we only consider the case p = 1; the argument can easily be modified to handle the case p ≥ 2.
First observe that where d( f (z), R n−1 ) stands for the Euclidean distance from the point f (z) to R n−1 . Indeed, upon taking r > 0 small enough such that which in turn implies .
as desired. Now let B(x 0 , r) be an arbitrary ball with radius r ≤ |y 0 |/2 and z 0 = (x 0 , y 0 ). Obviously, we have Then, it is enough to prove that by making "radial" stretchings with respect to their centers, where |Q m, j | = a[(1 − a)/2] mn . Notice that and J f (x) = |D f (x)| n = 1 otherwise. Thus f is a quasiconformal mapping. Moreover, it is easy to check that Then .

Set also
With each z m, j ∈ E, we associate a ball B m, j such that We make two claims: Assuming that both (6.1) and (6.2) hold for the moment, we arrive at which tends to ∞ as m → ∞ since β > 0 and ℓ ∼ m. This gives Theorem 1.6 under (6.1)-(6.2). Finally, we verify (6.1)-(6.2).
Then f is a quasiconformal mapping and        J f (x) ∼ |x − k| nβ if x ∈ B( k, 1) for some k ∈ (3N) n ; J f (x) = 1 otherwise.
Such a C f is not bounded on Q α (R n ) for all α ∈ (α 0 , 1) and hence satisfies our requirement; we omit the details.