Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

Anal. Geom. Metr. Spaces 2017; 5:98–115

Research Article Open Access

Eero Saksman and Tomás Soto* Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces https://doi.org/10.1515/agms-2017-0006 Received August 22, 2017; accepted November 2, 2017 Abstract: We establish trace theorems for function spaces dened on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the rst order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces dened intrinsically on F. Our method employs the denitions of the function spaces via hyperbolic llings of the underlying metric space.

Keywords: Trace theorems, Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces, hyperbolic lling

MSC: Primary: 46E35, 42B35

1 Introduction

1,p d A classical fact, originally due to Gagliardo [9], states that the traces of the Sobolev space W (R ), p ∈ d−1 1−1/p d−1 (1, ∞), on the hyperplane R × {0} lie in the Besov space Bp,p (R ) and, conversely, any function in 1−1/p d−1 1,p d Bp,p (R ) is a trace of some function in W (R ). This important result has been generalized to many s s other function spaces, most notably to the Triebel-Lizorkin spaces Fp,q and the Besov spaces Bp,q. Loosely speaking, we have

s d s−1/p d−1 s d s−1/p d−1 B ( ) d−1 = B ( ) and F ( ) d−1 = B ( ) p,q R |R p,q R p,q R |R p,p R for p ≥ 1 and s > 1/p. We refer [20, 31, 32, 37] and the references therein for these facts, as well as generaliza- d tions to some classes of subdomains of R . The Besov spaces, and later on the Triebel-Lizorkin spaces, have been studied in the fairly general setting of doubling metric measure spaces; we refer to e.g. [3, 8, 13, 14, 26] and the references therein, although this list is by no means comprehensive. Especially the full scales of these spaces in the setting of doubling metric spaces were introduced in [26], and in this paper we shall work with the equivalent denitions given in [2, 35] in terms of the “hyperbolic llings” of the underlying metric space – the actual denitions are given in the next section. In order to describe our results, let Z := (Z, d, µ) be a Q-Ahlfors regular metric measure space for some

Q > 0, and let F ⊂ Z be a closed λ-Ahlfors regular subset, where λ ∈ (0, Q]. We equip F with the metric d|F and the Hausdor λ-measure.

*Corresponding Author: Tomás Soto: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FI-00014 Helsinki, Finland, E-mail: tomas.soto@helsinki. Eero Saksman: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FI-00014 Helsinki, Finland, E-mail: eero.saksman@helsinki.

Open Access. © 2017 Eero Saksman and Tomás Soto, published by De Gruyter Open. This work is licensed under the Creative Com- mons Attribution-Non-Commercial-NoDerivs 4.0 License. Traces of Function Spaces Ë 99

Theorem 1.1. Let F ⊂ Z be a closed λ-Ahlfors regular subset. Suppose that 0 < s < 1, max Q/(λ + s), (Q − λ)/s < p < ∞ and 0 < q ≤ ∞. Then there exist bounded linear operators

s− Q−λ s− Q−λ ˙ s ˙ p ˙ p ˙ s R: Bp,q(Z) → Bp,q (F) and E: Bp,q (F) → Bp,q(Z) such that ˙ s (i) Rf = f|F for all continuous functions f in Bp,q(Z), and s− Q−λ ˙ p (ii) R Ef ) = f for all f ∈ Bp,q (F).

We refer to Remark 3.3 below for a concrete explanation of the range of the parameter p, as well as an alternative way to interpret part (i) of the statement. A similar result in the range p > 1 and q ≥ 1 has very recently been obtained in [29] using interpolation techniques. To formulate the two other trace theorems, we need a minor additional condition on the subset F.

Denition 1.2. The closed set F ⊂ Z is porous if there exists a constant c ∈ (0, 1) such that for all balls B ⊂ Z with radius r < diam Z such that B ∩ F ≠ ∅, there exists ξ ∈ Z such that B(ξ , cr) ⊂ B\F.

Remark 1.3. For porous λ-Ahlfors regular sets F ⊂ Z, it follows that λ < Q; see e.g. [23, Proposition 3.4]. On the other hand, if F ⊂ Z is λ-Ahlfors regular with λ < Q, it follows that F is porous subset of Z [23, Theorem 5.3].

Our trace theorem for the Triebel-Lizorkin spaces reads as follows.

Theorem 1.4. Let F ⊂ Z be a closed and porous λ-Ahlfors regular subset. Suppose that 0 < s < 1, max Q/(λ + s), (Q − λ)/s < p < ∞ and Q/(Q + s) < q ≤ ∞. Then there exist bounded linear operators

s− Q−λ s− Q−λ ˙ s ˙ p ˙ p ˙ s R: Fp,q(Z) → Bp,p (F) and E: Bp,p (F) → Fp,q(Z) such that ˙ s (i) Rf = f|F for all continuous functions f in Fp,q(Z), and s− Q−λ ˙ p (ii) R Ef ) = f for all Bp,p (F).

Finally, our trace theorem for the Hajłasz-Sobolev spaces M˙ 1,p reads as follows. A similar but much weaker result was established in [7].

Theorem 1.5. Let F ⊂ Z be a closed and porous λ-Ahlfors regular subset. Suppose that max Q/(λ+1), Q−λ < p < ∞. Then there exist bounded linear operators

1− Q−λ 1− Q−λ ˙ 1,p ˙ p ˙ p ˙ 1,p R: M (Z) → Bp,p (F) and E: Bp,p (F) → M (Z) such that ˙ 1,p (i) Rf = f|F for all continuous functions f in M (Z), and 1− Q−λ ˙ p (ii) R Ef ) = f for all Bp,p (F).

One should observe that Theorems 1.1, 1.4 and 1.5 are exactly of the same form as the classical results in the Euclidean setting, and that Theorems 1.4 and 1.5 (along with their non-homogeneous counterparts below) are completely new in this generality. A (very incomplete) list of previous results in the setting where F is a subset of an Euclidean space includes [4, 12, 19, 21, 22, 33, 34, 36] – the trace spaces appearing in these papers are sometimes dened in a non-intrinsic manner, however e.g. [19, 33, 34] employ intrinsic characterizations in terms of optimal polynomial approximations. Recently trace theorems for BV functions in the setting of metric measure spaces were also obtained in [27, 28]. We further refer to [15] for metric results concerning the 100 Ë Eero Saksman and Tomás Soto restriction and extension of Besov and Triebel-Lizorkin functions to subsets which are suciently “thick”, i.e. have positive µ-measure and satisfy a certain measure density condition. We point out that something like porosity needs to be assumed in Theorems 1.4 and 1.5 – F can not be properly Q-dimensional e.g. in terms of the measure density condition considered in [15].

Example 1.6. Let us give a simple application of our results illustrating a curious phenomenon concerning smoothness spaces dened on fractal subsets of an Euclidean space. d Consider a self-similar fractal subset Z of R in the sense of Hutchinson [18] generated by a collection d d of similitudes Si : R → R , 1 ≤ i ≤ N, satisfying the so-called open set condition. Let F be a sub-fractal generated by a proper subcollection of (Si)1≤i≤N . Then by [30, Theorem 4.14] and [23, Theorem 5.3], Z and F d are respectively Q-Ahlfors regular and λ-Ahlfors regular with 0 < λ < Q < d, Z is a porous subset of R and F d is a porous subset of both Z and R . σ −1 In particular, our results imply that the function space B˙ p,p(F) with λ max(0, p − 1) < σ ≤ 1 − (d − λ)/p d can be realized as the trace space of an appropriate Sobolev or Triebel-Lizorkin space dened either on R or on Z!

It is further natural to ask whether analogous trace results hold for non-homogeneous versions of these func- s s tion spaces. In Section 5 we dene the non-homogeneous function spaces Bp,q(Z) and Fp,q(Z), establish some of their basic properties and show that we have the following counterpart for our homogeneous trace theorems.

Theorem 1.7. The Theorems 1.1, 1.4 and 1.5 hold with B, F and M in place of B˙ , F˙ and M˙ respectively.

We nally point out that the Ahlfors regularity of the spaces Z and F is not strictly needed in these results, and a property known in previous literature as Ahlfors co-regularity of F with respect to Z would suce. Section A is an appendix where we elaborate on this, as well as on some technicalities that are needed in the proofs of our main results.

2 Preliminaries

In this section we give the denitions of the relevant function spaces and some details concerning them. In the rest of this section, Z := (Z, d, µ) is assumed to be a doubling metric measure space such that the measure µ is Borel regular and every ball B(ξ , r) := {η ∈ Z : d(η, ξ) < r} has positive and nite µ-measure. The doubling assumption means that there exists a constant c ∈ [1, ∞) such that µB(ξ , 2r) ≤ cµB(ξ , r) for all ξ ∈ Z and r > 0. It follows from this assumption that there exist constants C ≥ 1 and Q > 0 such that the measure µ satises R Q µB(ξ , R) ≤ C µB(ξ , r) (2.1) r for all ξ ∈ Z, r > 0 and R > r. Q is in a sense an upper bound for the dimension of Z, and it will be xed from now on. Let us introduce some notation conventions that will be used in this section, as well as in the later sections with obvious modications. For an arbitrary ball B ⊂ Z with a distinguished center point ξ ∈ Z and radius r > 0, we write tB := B(ξ , tr) for t > 0. If f is a complex-valued function on Z and E ⊂ Z, we write 1 − fdµ := fdµ ˆE µ(E) ˆE 1 whenever the latter quantity is well-dened. The notation Lloc(Z) will (instead of the usual one) stand for the space of complex-valued µ-measurable functions on Z that are integrable on every ball B ⊂ Z. Finally, we will use the notations ., & and ≈ when dealing with unimportant multiplicative constants. More precisely, when f and g are non-negative functions with the same domain, the notations f . g or g & f mean that there exists Traces of Function Spaces Ë 101 a positive constant c, usually independent of some parameters obvious from the context, such that f ≤ cg on the domain of f and g. The notation f ≈ g means that f . g and g . f . The construction referred to above as the hyperbolic lling of Z is roughly speaking a graph (X, E) such that if Z is nice enough and (X, E) is endowed with its natural path metric, (X, E) is hyperbolic in the sense of Gromov and its boundary at innity coincides with Z. We refer to the introduction section of [2] for a detailed explanation of the motivation of this construction in the context of function spaces. We shall next explain the actual construction of (X, E).

For all n ∈ Z, let (ξx)x∈Xn , where Xn is a suitable index set, be a maximal set of points in Z such that −n−1 0 −n d(ξx , ξx0 ) ≥ 2 for all pairwise distinct x, x ∈ Xn. Write B(x) := B(ξx , 2 ) for all x ∈ Xn. It is easily seen −1 that the balls 2 B(x), x ∈ Xn, cover Z, and the doubling assumption implies that the balls B(x), x ∈ Xn, have bounded overlap (uniformly in n). Write |x| := n for all x ∈ Xn (the “level” of x). We then consider the disjoint F 0 union X := Xn, and denote by (X, E) the graph such that the vertices x, x ∈ X are joined by an edge in n∈Z E if and only if x ≠ x0, ||x| − |x0|| ≤ 1 and B(x) ∩ B(x0) ≠ ∅; in this case we write x ∼ x0. The “Poisson extension” 1 Pf : X → C of a function f ∈ Lloc(Z) is dened by

Pf(x) := − fdµ ˆB(x) for all x ∈ X. 0 We equip the edges in E with an orientation and denote by ex,x0 the directed edge from x to x for any two neighbors x and x0 in X. The orientation is chosen so that if x ∼ x0 and |x| < |x0|, then x0 is the endpoint of 0 the edge joining x and x . For an edge e ∈ E, denote by e− the starting point and by e+ the endpoint of e. For a sequence u : X → C, we dene the discrete derivative du : E → C by du(e) = u(e+) − u(e−) for all e ∈ E. Finally, we write |e| := min |e−|, |e+| (the “level” of e) and B(e) := B(e−) ∪ B(e+) for all e ∈ E. s s 1,p The denitions of the spaces B˙ p,q, F˙ p,q and M˙ then read as follows.

s Denition 2.1. (i) Let 0 < s ≤ 1, Q/(Q + s) < p ≤ ∞ and 0 < q ≤ ∞. Then Ip,q(E) is the quasi-normed space of sequences u : E → C such that 1/q X ksq X q kuk s := 2 |u(e)|χ (2.2) Ip,q(E) B(e) Lp(Z) k∈Z |e|=k s (standard modication for q = ∞) is nite. Furthermore, the homogeneous Besov space B˙ p,q(Z) is the quasi- 1 normed space of functions f ∈ Lloc(Z) such that

kf k ˙ s := d(Pf) s Bp,q(Z) Ip,q(E) is nite. s (ii) Let 0 < s ≤ 1, Q/(Q + s) < p < ∞ and Q/(Q + s) < q ≤ ∞. Then Jp,q(E) is the quasi-normed space of sequences u : E → C such that

p/q 1/p X |e|s q kukJs (E) := 2 |u(e)| χB(e)(ξ) dµ(ξ) (2.3) p,q ˆ Z e∈E s (standard modication for q = ∞) is nite. Furthermore, the homogeneous Triebel-Lizorkin space F˙ p,q(Z) is the 1 quasi-normed space of functions f ∈ Lloc(Z) such that

kf k ˙ s := d(Pf) s Fp,q(Z) Jp,q(E) is nite. (iii) Let 0 < s ≤ 1 and 0 < p < ∞. The homogeneous Hajłasz-Sobolev space M˙ s,p(Z) is dened as the class p of µ-measurable functions f : Z → C such that there exists a function g : Z → [0, ∞] in L (Z) such that |f (ξ) − f(η)| ≤ d(ξ , η)sg(ξ) + g(η) ˙ s,p for all ξ, η ∈ Z. The quasi-norm kf kM˙ s,p(Z) of a function f ∈ M is obtained as the inmum of kgkLp over all admissible g. 102 Ë Eero Saksman and Tomás Soto

s s s,p Remark 2.2. (i) Strictly speaking the spaces B˙ p,q(Z), F˙ p,q(Z) and M˙ (Z) become quasi-normed spaces after dividing out the functions f such that kf k = 0, i.e. the functions that are constant µ-almost everywhere. In s the sequel we shall abuse notation by writing f ∈ B˙ p,q(Z) for both functions f and equivalence classes f satisfying kf k ˙ s < ∞, and similarly for the other two families of function spaces introduced above. The Bp,q(Z) precise meaning will be obvious from context. s s (ii) The spaces F˙ p,q(Z) and B˙ p,q(Z) were introduced in [2] and [35] respectively. We refer to these papers for all basic properties concerning these spaces. Let us only mention here that these two function spaces are quasi-Banach spaces for all admissible values of the parameters, and reexive Banach spaces for 1 < p, q < s s ∞. While the sequence spaces Jp,q(E) and Ip,q(E) obviously depend on the choice of the hyperbolic lling s s (X, E), the spaces F˙ p,q(Z) and B˙ p,q(Z) do not – we refer to Remark 2.3 below for more information. s s (iii) The Besov spaces N˙ p,q(Z) and the Triebel-Lizorkin spaces M˙ p,q(Z) were introduced in [26] in the generality of all metric measure spaces. Under our assumptions and in the parameter ranges given in the def- s s inition above, they coincide with B˙ p,q(Z) and F˙ p,q(Z) respectively; see [2, Propostion 3.1] and [35, Proposition ˙ s d ˙ s d 3.1]. In particular, Bp,q(R ) and Fp,q(R ) with 0 < s < 1 coincide with the standard Fourier-analytically de- d ˙ s ned Besov and Triebel-Lizorkin spaces on R for all admissible values of the parameters. The spaces Np,q(Z) s 1 and M˙ p,q(Z) with s ≥ 1 are often trivial [8, Theorem 4.1], but there is one exception important for us: F˙ p,∞(Z) = 1 1,p 1,p M˙ p,∞(Z) = M˙ (Z), where M˙ is the standard rst order Hajłasz-Sobolev space, for Q/(Q + 1) < p < ∞. (iv) The spaces M˙ 1,p(Z) were introduced in [10]; see also [11]. For 1 < p < ∞ they are one of the more well-known generalizations of the standard Sobolev spaces to the setting of metric measure spaces. In [24] it 1,p d was shown that M˙ (R ) for d/(d + 1) < p ≤ 1 coincides with the homogeneous Hardy-Sobolev space with the same indices. s s (v) We have the restriction p > Q/(Q + s) in the denitions of the spaces B˙ p,q(Z) and F˙ p,q(Z). This is because the denitions of these spaces require a priori local integrability, and generally speaking it is for d p > d/(d + s) that the Besov and Triebel-Lizorkin distributions on R are locally integrable functions. We also s impose a similar restriction on the parameter q for the spaces F˙ p,q(Z) because of certain technical reasons which are common in the study of these spaces; see e.g. the proofs in [2] or [37] for more information. s (vi) It will be useful to note that we have the following equivalent quasinorm on the space Ip,q(E) for all admissible parameters:

q/p1/q X ksq X p kuk s ≈ 2 µ B(e) |u(e)| (2.4) Ip,q(E) k∈Z |e|=k (obvious modications for p = ∞ and/or q = ∞). This follows easily from (2.2) and the fact that the sets B(e), |e| = k, have bounded overlap uniformly in k ∈ Z. (vii) Let (Ae)e∈E be a collection of measurable subsets of Z such that Ae ⊂ tB(e−) ∪ tB(e+) for some s s uniform t ≥ 1 and infe∈E µ(Ae)/µ B(e) > 0. Then we get equivalent quasinorms on Ip,q(E) and Jp,q(E) by replacing χB(e) by χAe in (2.2) and (2.3) respectively. This can be proven by a standard maximal function argument; see [2, Proposition 2.2]

Remark 2.3. As mentioned above, the choice of the hyperbolic lling (X, E) is not unique, but this has no s s essential bearing on the classes B˙ p,q(Z) and F˙ p,q(Z) or their quasi-norms – any two admissible choices yield equivalent quasi-norms for both spaces, with the equivalence constants independent of these two choices. However, in this paper we shall need even more exibility in the choice of (X, E) – we want to choose the hyperbolic lling of Z in such a way that a hyperbolic lling of a xed subspace F is obtained in a natural way as the “restriction” of (X, E) to the edges corresponding to balls that lie “above” F. This choice is formulated as Lemma 2.4 below. 0 −n To elaborate on the admissible exibility, it is enough that we have d(x, x ) ≥ c12 for all distinct x, 0 −n x ∈ Xn, that the radii rx of the balls B(x) := B(ξx , rx) (x ∈ Xn) are comparable to 2 uniformly in n, and that the balls c2B(x) cover Z; here the constants c1 > 0 and c2 ∈ (0, 1) are uniform in n. Then (X, E) can x∈Xn s s be constructed exactly as explained above, and the resulting spaces B˙ p,q(Z) and F˙ p,q(Z) have quasinorms Traces of Function Spaces Ë 103 essentially independent of the precise choice of the hyperbolic lling. We refer to [2, Remark 2.8] and [35, Remark 2.8] for details.

Z Z Lemma 2.4. Suppose that F is a closed subset of Z equipped with the metric d|F. Let (X , E ) be an admissible hyperbolic lling of Z, and write XF for the set of vertices x ∈ XZ such that B(x) ∩ F ≠ ∅. Then (XZ , EZ) can be chosen such that the following properties hold. F F (i) The balls corresponding to the vertices in X are centered in F, i.e. ξx ∈ F for all x ∈ X ; (ii) ( ) ( F , F) ( , ) The balls B x |F x∈XF , generate an admissible hyperbolic lling X E of the metric space F d|F . In particular, we have XF ⊂ XZ and EF ⊂ EZ in a natural way.

−n−1 −n Proof. For n ∈ , let (ξ ) 0 be a maximal 2 -separated subset of {ξ ∈ Z : dist (ξ , F) ≥ 2 }, where Z x x∈Xn 0 −n 00 X is a suitable index set. Furthermore, let (ξ ) 00 be a maximal 2 -separated subset of F, with again X n x x∈Xn n −n 0 −n+2 00 a suitable index set. Let B(x) := B(ξx , 2 ) ⊂ Z for x ∈ Xn and B(x) := B(ξ , 2 ) ⊂ Z for x ∈ Xn . Writing Z 0 00 Z F Z −1 X := X ∪ X and X := X , we are in the situation of Remark 2.3 (with c1 = c2 = 2 ), so the resulting n n n n∈Z n graph (XZ , EZ) is an admissible hyperbolic lling of Z. Z 00 In addition, for x ∈ Xn we have B(x) ∩ F ≠ ∅ if and only if x ∈ Xn (hence also ξx ∈ F), and the balls 00 F F B(x)|F ⊂ F corresponding to the vertices x ∈ Xn obviously generate an admissible hyperbolic lling (X , E ) for the metric space (F, d|F).

3 Traces of Besov spaces

In this section, we shall work with the assumptions of Theorem 1.1. In other words, the metric measure space (Z, d, µ) is assumed to be Q-Ahlfors regular (with Q as in (2.1)), which means that the measure µ satises µB(ξ , r) ≈ rQ uniformly in ξ ∈ Z and 0 < r < diam Z. We assume that F is a closed subset of Z of Hausdor dimension λ ∈ (0, Q], equipped with the metric d|F. We denote its λ-Hausdor measure by ν, and assume it to be λ-Ahlfors regular. Write (XZ , EZ) for the hyperbolic lling of Z, and similarly with F in place of Z. The F Z hyperbolic llings are chosen so that Xn is in a natural way a subset of Xn for all n ∈ Z; see Lemma 2.4 above. s To make sense of the trace spaces of the Besov spaces B˙ p,q(Z), let us begin by recalling some very basic properties of locally integrable functions.

For a measurable function f : Z → C, denote by Λf the set of points ξ ∈ Z such that there exists a number cξ ,f ∈ C so that

lim − f − cξ ,f dµ = 0. r→0ˆB(ξ ,r) 1 It is well known ([17, Theorem 2.7]) that the doubling property (2.1) implies that if f ∈ Lloc(Z), then Λf has full µ-measure (namely it contains the Lebesgue points of f) and that it does not depend on the precise rep- resentative of f (with respect to equality µ-almost everywhere). The point is that f as a function is essentially well-dened in Λf , and that under a fractional smoothness assumption on f, the set Z \ Λf turns out to have a relatively small Hausdor dimension. This is quantied in the following lemma, which is well known in the Euclidean setting, so we have only included an outline of a proof that is easily adapted to our setting.

s Lemma 3.1. Suppose that f is a function in F˙ p,q(Z) with 0 < s ≤ 1, Q/(Q + s) < p ≤ Q/s and Q/(Q + s) < q ≤ ∞, ˙ s or f ∈ Bp,q(Z) with 0 < s < 1, Q/(Q + s) < p ≤ Q/s and 0 < q ≤ ∞. Then the Hausdor dimension of Z \ Λf is at most Q − ps.

Proof. We rst consider the case of Triebel-Lizorkin functions. With the parameters as in the statement, we s s s,p have F˙ p,q(Z) ⊂ F˙ p,∞(Z) = M˙ (Z) [2, Proposition 3.1], so it suces to verify the statement for the latter space. Let g ∈ Lp(Z) be a Hajłasz s-gradient of a function f ∈ M˙ s,p(Z), and x ϵ ∈ (0, s). Taking ξ ∈ Z and k k+1 0 < r1 < r2 < 1, and k ∈ N0 such that 2 r1 < r2 ≤ 2 r1, the doubling condition and the weak (1, p)-Poincaré 104 Ë Eero Saksman and Tomás Soto inequality satied by the functions of M˙ s,p(Z) (see [11, Theorem 8.7] and [25, Lemma 4.1]) yield

k X − ≤ n+1 − n + k+1 − fB(ξ ,r2) fB(ξ ,r1) fB(ξ ,2 r1) fB(ξ ,2 r1) fB(ξ ,2 r1) fB(ξ ,r2) n=0 k+1 1/p X ϵ 2n r sup r(s−ϵ)p−Q gp dµ . 1 ˆ n=0 r∈(0,4) B(ξ ,r) 1/p ϵ (s−ϵ)p−Q p ≈ (r2) sup r g dµ . r∈(0,4) ˆB(ξ ,r)

− Now ξ ∈ Z\Λf only if the latter supremum is innite (otherwise one can take cξ ,f = limr→0 B(ξ ,r) fdµ), and since g ∈ Lp, a standard covering argument shows that this happens in a set of Hausdor (Q−sp´ +ϵp)-content zero, so the Hausdor dimension of Z\Λf is at most Q−sp+ϵp. Letting ϵ → 0 yields the desired upper bound. Let us now consider the case of Besov spaces. Suppose that the parameters are as in the statement, and s s take ϵ ∈ (0, s) arbitrarily close to s. We have B˙ p,q(Z) = N˙ p,q(Z) (see [26] for the denition of the latter space), s ϵ,p and it is easily seen that for any f ∈ N˙ p,q(Z), we have f ∈ M˙ (B) for all balls B ⊂ Z. By the rst part of the proof, this means that the Hausdor dimension of Z\Λf is at most Q − ϵp, and taking ϵ → s yields the desired upper bound.

s s Remark 3.2. When p > Q/s, the functions f in F˙ p,q(Z) and B˙ p,q(Z) coincide with Hölder continuous func- ˙ s tions µ-almost everywhere, which means that the set Z\Λf is empty. For the spaces Fp,q(Z), this follows again s s,p s,p from the embedding F˙ p,q(Z) ⊂ M˙ and a Hajłasz’s Sobolev-type embedding theorem for the spaces M˙ ([11, s s Theorem 8.7] and [25, Lemma 4.1]). For the spaces B˙ p,q(Z), we may again note that the functions in N˙ p,q(Z) are locally in M˙ ϵ,p for all ϵ ∈ (0, s) such that p > Q/ϵ. We also note that the Ahlfors regularity of Z is not strictly speaking needed here; a closer examination of the proof shows that the doubling condition (2.1) suces.

With this in mind, we are in a position to give the proof of Theorem 1.1. Before the proof, we still make some remarks concerning the actual statement and dene a collection of auxiliary functions.

Remark 3.3. Let us elaborate on the precise assumptions and the statement of Theorem 1.1. Firstly, the condition p > Q/(λ + s) is equivalent with the requirements that p > Q/(Q + s) and p > λ/(λ + (s − (Q − λ)/p), so that the functions in the Besov spaces in question can be expected to be locally integrable in the rst place. The condition p > (Q − λ)/s comes from the requirement that s − (Q − λ)/p > 0, and by Lemma 3.1 and Remark s 3.2, this also means that the functions in B˙ p,q(Z) are essentially well-dened ν-almost everywhere in F, and s in this way part (i) of the statement also makes sense for all (not necessarily continuous) f ∈ B˙ p,q(Z). Part (ii) should be interpreted pointwise ν-almost everywhere in F.

Z Z Z Denition 3.4. (i) Let (ψx )x∈XZ be a collection of Lipschitz functions ψx : Z → [0, 1] such that ψx is sup- Z |x| Z ported on B(x) for all x, Lip ψ 2 for all |x| and (ψ ) Z is a partition of unity of Z for all n ∈ . Dene x . x x∈Xn Z F the collection of functions (ψx )x∈XF in the same way with F in place of Z. Z Z (ii) For u : X → C, dene Tn u : Z → C for all n ∈ Z by

Z X Z Tn u = u(x)ψx . Z x∈Xn

F F Dene Tn u for u : X → C and n ∈ Z analogously.

1 Z For f ∈ Lloc(Z) we obviously have limn→∞ Tn (Pf ) = f pointwise µ-almost everywhere (e.g. at the Lebesgue s Z points of f). It also turns out that for f ∈ F˙ p,q(Z) with suitable indices, Tn (Pf) approximates f in the quasinorm s s of F˙ p,q(Z) as n → ∞, and a similar result holds in the scale B˙ p,q(Z); see [2, Theorem 3.3] and [35, Theorem 3.2]. All this of course holds with F in place of Z. Traces of Function Spaces Ë 105

Proof of Theorem 1.1. We will rst construct the trace operator R. In fact, as explained in Remark 3.3 above, we could take part (i) of the statement as the denition of R, but we shall construct the operator in a slightly more roundabout way so that the boundedness becomes evident. ˙ s Letting f ∈ Bp,q(Z), we have kf k ˙ s = kd(Pf)kIs ( Z ), and consequently (by (2.4) and the Ahlfors Bp,q(Z) p,q E regularity of the spaces Z and F),

d(Pf )|EF s−γ/p F ≈ d(Pf )|EF s Z ≤ kd(Pf )kIs (EZ ) = kf k ˙ s < ∞. (3.1) Ip,q (E ) Ip,q(E ) p,q Bp,q(Z)

We then need to estimate kf|Fk ˙ s−γ/p in terms of the leftmost quantity above. To this end, write Bp,q (F)

F X F F In u := u(ey,y0 )ψy ψy0 0 F F 0 (y,y )∈(Xn ×Xn+1), y∼y

F for all sequences u dened on E and integers n ∈ Z, and x ξ0 ∈ F. We have that N −1 F X F X F I d(Pf)|EF := lim In d(Pf)|EF (·) − In d(Pf)|EF (ξ0) N→∞ n=−N n=−N

1 converges in Lloc(F) and pointwise ν-almost everywhere (see Lemma A.1 in the Appendix below). According ˙ s−γ/p to [35, Proposition 4.3] (see also [2, Proposition 6.3]), the Bp,q (F)-norm of the limit function is bounded from above by a constant times the leftmost quantity in (3.1), and hence by a constant times kf k ˙ s . Bp,q(Z) F F F Note that we have In (du) = Tn+1u − Tn u for all sequences u : X → C (by denition), and since d(Pf )|EF F (as a sequence on E ) is simply obtained as the discrete derivative of (Pf)|XF , we get

F F F In d(Pf)|EF = Tn+1 (Pf)|XF − Tn (Pf)|XF (3.2) for all n. By Lemma A.1 in the Appendix below, we also have

F F lim TM (Pf )|XF (ξ) − TM (Pf )|XF (ξ0) = 0 (3.3) M→−∞ for all ξ ∈ F. Combining (3.2) and (3.3) with the fact that ν-almost every point of F is a Lebesgue point of f , we get

F F for ν-almost all ξ ∈ F, where T0 (Pf)|XF (ξ0) is a constant. We can therefore take Rf := I d(Pf)|EF + F T0 (Pf)|XF (ξ0). We shall next construct the extension operator E with the additional assumption that q < ∞. In this ˙ s−γ/p ˙ s case, it suces to construct a bounded linear operator E: Bp,q (F) → Bp,q(Z) satisfying (ii) for Lipschitz ˙ s−γ/p ˙ s−γ/p functions f ∈ Bp,q (F) with bounded support, since these functions form a dense subspace of Bp,q (F) ˙ s−γ/p F [35, Corollary 3.3]. Taking f ∈ Bp,q (F), u := d(Pf ) is a priori dened as a sequence on E , but it can be Z Z F extended to E simply by dening u(e) = 0 for all e ∈ E \E . Now kukIs (EZ ) ≈ kd(Pf)k s−γ/p F < ∞, so p,q Ip,q (E )

N −1 Z Z X Z X Z Z Ef := I u + T0 (Pf)(ξ0) = lim In u(·) − In u(ξ0) + T0 (Pf)(ξ0), N→∞ n=−N n=−N 1 ˙ s where ξ0 is a xed point of F, converges in Lloc(Z) and pointwise µ-almost everywhere to a function in Bp,q(Z) with norm bounded from above by a constant times kf k ˙ s−γ/p . Bp,q (F) ˙ s−γ/p To verify the condition (ii) for Lipschitz functions f ∈ Bp,q (F) with bounded support, we rst show that Z in this case the series dening I u converges everywhere in Z to a continuous function. By the Lipschitz con- Z −n P Z tinuity of f , we have supξ∈Z |In u(ξ)| . 2 Lip (f) for all n ∈ Z, so n≥0 In u converges uniformly in Z. Further- Z Z Z Z n more, by the Lipschitz continuity of the functions ψx , x ∈ X , we have |In u(ξ)−In u(ξ0)| . 2 d(ξ , ξ0)kf kL∞(F), 106 Ë Eero Saksman and Tomás Soto

P Z Z so the series n<0 In u(·) − In u(ξ0) converges uniformly on bounded subsets of Z. All in all, the series den- Z ing I u converges uniformly on bounded subsets of Z, and the limit function must hence be continuous in Z. Now since u is not in general obtained as a discrete derivative of a sequence on XZ, we do not have an analog of (3.2) on Z. However, by the choices of the hyperbolic llings (see Lemma 2.4), B(x) ∩ F = ∅ for Z F x ∈ X \X . Hence for all ξ ∈ F and n ∈ Z, Z X Z Z In u(ξ) = u(ey,y0 )ψy (ξ)ψy0 (ξ) 0 Z Z 0 (y,y )∈(Xn ×Xn+1), y∼y X Z Z = u(ey,y0 )ψy (ξ)ψy0 (ξ) 0 F F 0 (y,y )∈(Xn ×Xn+1), y∼y X 0 Z Z = Pf (y ) − Pf(y) ψy (ξ)ψy0 (ξ) 0 F F 0 (y,y )∈(Xn ×Xn+1), y∼y Z Z = Tn+1(Pf )(ξ) − Tn (Pf)(ξ). Z By continuity, all points of Z are Lebesgue points of I u and all points of F are Lebesgue points of f. Combining this with the formula above and Lemma A.1 in the Appendix below we get Z Z Z Z Z I u(ξ) = lim TN+1(Pf)(ξ) − T−N (Pf)(ξ) − T0 (Pf)(ξ0) + T−N (Pf)(ξ0) N→∞ Z = f (ξ) − T0 (Pf)(ξ0) for ξ ∈ F; note that Lemma A.1 applies here since the Ahlfors regularity of Z means that either diam (Z) < ∞ or µ(Z) = ∞. Altogether, Z Z Z Z R E(f) = R I u + R T0 (Pf )(ξ0) = f − T0 (Pf)(ξ0) + T0 (Pf)(ξ0) = f ν-almost everywhere in F. ˙ s−γ/p Finally, if q = ∞, the operator E constructed above extends to a bounded linear operator from Bp,∞ (F) to ˙ s ˙ s−γ/p ˙ s0−γ/p Bp,∞(Z) satisfying (ii), since Bp,∞ (F) is obtained as a real interpolation space between the spaces Bp,p (F) ˙ s1−γ/p ˙ s and Bp,p (F) with s0 < s < s1 and s1 − s0 1, and similarly for the space Bp,∞(Z) [15, Theorem 4.3].

4 Traces of Triebel-Lizorkin and Sobolev spaces

In this section we shall give the proofs of Theorems 1.4 and 1.5. The assumptions on the metric measure spaces are as in the statements of these theorems – Z := (Z, d, µ) is Q-Ahlfors regular, and F is a closed and porous subset of Z equipped with the metric d|F and the λ-Hausdor measure ν, which is assumed to be λ-Ahlfors regular. The hyperbolic llings (XZ , EZ) and (XF , EF) are chosen as in the previous section. s Z The main observation concerning the porosity of F is that now the Jp,q(E )-norm of a sequence living “above” F is essentially independent of q. In [6, Theorem 13.7], a similar phenomenon was observed for sequence spaces corresponding to the Fourier-analytically dened Triebel-Lizorkin spaces in the Euclidean setting, under a slightly weaker condition called NST instead of porosity.

0 Z Lemma 4.1. Let s ∈ (0, ∞), p ∈ (0, ∞) and q, q ∈ (0, ∞]. Then for all sequences u : E → C supported on EF we have

kukJs (EZ ) ≈ kukJs (EZ ), p,q p,q0 with the implied constants independent of u.

Z Z Proof. The porosity of F means that for all e ∈ E such that B(e) ∩ F ≠ ∅, we can take xe ∈ X such that |e| − σ ≤ |xe| ≤ |e| for some xed σ ≥ 0 and 2B(xe) ⊂ B(e)\F. By Remark 2.2 (vii) we thus have p/q 1/p X |e|s q kuk s Z ≈ 2 |u(e)| χ (ξ) dµ(ξ) Jp,q(E ) B(xe) ˆZ e∈EZ : B(e)∩F=∅̸ Traces of Function Spaces Ë 107

(obvious modication for q = ∞), and now it suces to show that for each ξ ∈ Z, only a uniformly nite 0 number of terms in the latter sum are nonzero. Now suppose that B(xe) ∩ B(xe0 ) ≠ ∅ with e and e like in the σ0 sum, and without loss of generality |xe0 | ≤ |xe|. By assumption we have 2 B(xe) ⊃ B(e) for some (universal) 0 σ0 σ0 0 σ > 0, so 2 B(xe) ∩ F ≠ ∅, which further means that 2 B(xe) is not contained in 2B(xe). By construction 0 00 00 Z this means that |xe| ≤ |xe| + σ for some uniform σ ≥ 0. All in all, #{e ∈ E : B(e) ∩ F ≠ ∅ and B(xe) 3 ξ} is bounded uniformly in ξ ∈ Z, completing the proof. With this in mind we can give the proof of Theorem 1.4. Remark 3.3, with obvious modications, holds here as well.

s Proof of Theorem 1.4. Lemma 4.1 tells us that for f ∈ F˙ p,q(Z), we have

kf k ˙ s & d(Pf)|EF s Z ≈ d(Pf )|EF s Z = d(Pf)|EF s Z . Fp,q(Z) Jp,q(E ) Jp,p(E ) Ip,p(E ) ˙ s−(Q−λ)/p R can thus be constructed as in the proof of Theorem 1.1. To construct E, note that for all f ∈ Bp,p (F), the sequence u := d(Pf) dened a priori on EF can be extended as zero on EZ\EF, so by Lemma 4.1 we have

kf k ˙ s−(Q−λ)/p = kd(Pf)k s−(Q−λ)/p ≈ kukIs (EZ ) ≈ kukJs (EZ ), Bp,p (F) Ip,p (F) p,p p,q ˙ s−(Q−λ)/p and since Lipschitz functions with bounded support are dense in Bp,p (F), we may proceed as in the proof of Theorem 1.1.

Theorem 1.5 can be proven by utilizing the same idea. Again, Remark 3.3 with obvious modications holds here as well.

1,p 1 Proof of Theorem 1.5. We have M˙ (Z) = F˙ p,∞(Z) [2, Proposition 3.1]. The operator R can thus be constructed exactly as in the proof of Theorem 1.4 above – here it is important remember that the porosity assumption on F means that λ < Q (see Remark 1.3), which in turn implies 1 − (Q − λ)/p < 1, which one needs to guarantee F ˙ 1−(Q−λ)/p that the series dening I d(Pf)|EF converges to a function in Bp,p (F) [35, Proposition 4.3]. ˙ 1−(Q−λ)/p To construct the operator E, suppose that f ∈ Bp,p (F). Again, extending the sequence u := d(Pf) as zero on EZ\EF, we have

1 kf k 1− Q−λ ≈ kukJ1 (EZ ) ≈ kukJ (EZ ). ˙ p p,p p,1 Bp,p (F) By Lemma A.1 in the Appendix below, we therefore have that

N −1 Z X Z X Z I u := lim In u(·) − In u(ξ0) , N→∞ n=−N n=−N

1 where ξ0 is a xed point of F, converges in Lloc(Z) and pointwise µ-almost everywhere (but not to a function ˙ 1 in Fp,1(Z), since [2, Proposition 6.3] requires s < 1). For all ξ and η outside of a set of zero µ-measure we can Z however use the Lipschitz continuity of the functions ψy to obtain

Z Z X X Z Z Z Z |I u(ξ) − I u(η)| ≤ |u(ey,y0 )| ψy (ξ)ψy0 (ξ) − ψy (η)ψy0 (η) n∈ 0 Z Z 0 Z (y,y )∈(Xn ×Xn+1), y∼y X n X d(ξ , η) 2 |u(e 0 )| χ (ξ) + χ (η) . y,y B(ey,y0 ) B(ey,y0 ) n∈ 0 Z Z 0 Z (y,y )∈(Xn ×Xn+1), y∼y X |e| X |e| . d(ξ , η) 2 |u(e)|χB(e)(ξ) + 2 |u(e)|χB(e)(η) , e∈EZ e∈EZ so p 1/p Z X |e| 1 kI ukM˙ 1,p(Z) . 2 |u(e)|χB(e)(ξ) dµ(ξ) = kukJ (EZ ) ≈ kf k 1− Q−λ , ˆZ p,1 B˙ p (F) e∈EZ p,p ˙ 1−(Q−λ)/p and since Lipschitz functions with bounded support are dense in Bp,p (F), we may proceed as in the proof of Theorem 1.1. 108 Ë Eero Saksman and Tomás Soto

5 Traces of non-homogeneous function spaces

s s In this section we provide a denition of the non-homogeneous function spaces Bp,q and Fp,q in terms of the discrete derivatives and a xed-level trace approximation of the Poisson extension Pf of a function 1 f ∈ Lloc. This denition is of independent interest, and we shall apply it to prove Theorem 1.7, i.e. the non- homogeneous counterpart of our main theorems. A theory of non-homogeneous Besov and Triebel-Lizorkin spaces in the context of reverse doubling metric measure spaces using spaces of test functions has previously been developed e.g. in [13]. Our basic assumption here is that the space (Z, d, µ) satises the doubling condition (2.1). The operator Z T0 is as in Denition 3.4.

s Denition 5.1. (i) Suppose that 0 < s ≤ 1, Q/(Q+s) < p ≤ ∞ and 0 < q ≤ ∞. Then Bp,q(Z) is the quasi-normed 1 space of functions f ∈ Lloc(Z) such that

1/q Z X ksq X q kf k s := kT (Pf)k p + 2 |d(Pf)(e)|χ Bp,q(Z) 0 L (Z) B(e) Lp(Z) k≥0 |e|=k

(standard modication for q = ∞) is nite. s (ii) Suppose that 0 < s ≤ 1, Q/(Q + s) < p < ∞ and Q/(Q + s) < q ≤ ∞. Then Fp,q(Z) is the quasi-normed 1 space of functions f ∈ Lloc(Z) such that

p/q 1/p Z X |e|s q kf kFs (Z) := kT0 (Pf )kLp(Z) + 2 |d(Pf )(e)| χB(e)(ξ) dµ(ξ) p,q ˆ Z |e|≥0

(standard modication for q = ∞) is nite.

Note that unlike in the homogeneous case, k · k s and k · k s are honest quasi-norms (modulo equality Bp,q Fp,q µ-almost everywhere). Many properties of the homogeneous spaces also hold for the spaces B and F, as ev- idenced by the alternative characterization of the non-homogeneous quasi-norms given in the Proposition below – let us simply mention here that they are quasi-Banach spaces, and a standard argument shows that they are reexive Banach spaces when 1 < p, q < ∞. The Proposition below also shows that in the smoothness range 0 < s < 1, these spaces coincide with the non-homogeneous spaces considered in e.g. [15] and [16], d and with the standard Fourier-analytically dened non-homogeneous spaces when Z = R . The Proposition 1 1,p 1,p further shows that we have Fp,∞(Z) = M (Z) for Q/(Q + 1) < p < ∞, where the space M is as dened in [10, 11]

s s p Proposition 5.2. (i) Let s, p and q be admissible parameters for the space Bp,q(Z). Then Bp,q(Z) = L (Z) ∩ s B˙ p,q(Z), and we have

kf kBs (Z) ≈ kf kLp(Z) + kf k ˙ s p,q Bp,q(Z) 1 for all f ∈ Lloc(Z), with the implied constants independent of f. s s p s (ii) Let s, p and q be admissible parameters for the space Fp,q(Z). Then Fp,q(Z) = L (Z) ∩ F˙ p,q(Z), and we have

kf kFs (Z) ≈ kf kLp(Z) + kf k ˙ s p,q Fp,q(Z) 1 for all f ∈ Lloc(Z), with the implied constants independent of f.

Before proving this, let us formulate the following auxiliary result, which we shall use both in the proof of this Proposition as well as in the proof of Theorem 1.7 for the cases with p < 1. The proof of this result is given s s 1 in the Appendix below. The Lemma essentially gives a local embedding of Bp,q(Z) and Fp,q(Z) into L (Z) as long as p > Q/(Q + s). Traces of Function Spaces Ë 109

Lemma 5.3. Suppose that 0 < p < 1 and ϵ > 0 is dened by p = Q/(Q + ϵ). Z (i) Let x ∈ X0 . Then p p−1 Z p X kϵp X p k k 1 ( ) | ( )| + 2 | ( )( )| f L (B(x)) . µ B x T0 Pf dµ d Pf e χB(e) Lp(σB(x)) ˆ ( ) B x k≥0 |e|=k

1 Z Z for all f ∈ Lloc(Z), where σ ≥ 1 is a constant depending only on the choice of the hyperbolic lling (X , E ) (see Remark 2.3) and the implied constant is independent of x and f . (ii) Let B ⊂ Z be a ball with radius r ≥ 1. Then p Q−Qp p−1 Z p X kϵp X p kf k 1 r µ(B) |T0 (Pf )| dµ + 2 |d(Pf )(e)|χ p L (B) . ˆ B(e) L (σB) σB k≥0 |e|=k

1 for all f ∈ Lloc(Z), where σ ≥ 1 is a constant depending only on the choice of the hyperbolic lling, and the implied constant is independent of r, B and f.

Proof of Proposition 5.2. We shall only consider the case of Besov spaces, since the case of Triebel-Lizorkin spaces can be handled in a similar manner. 1 Z k→∞ Suppose that f ∈ Lloc(Z) and take ε ∈ (0, s) (or ε = 0 if p ≤ 1). Since Tk (Pf ) −→ f pointwise µ-almost everywhere in Z, we get

1/p Z X kεp Z Z p k − ( )k p 2 k ( ) − ( )k (5.1) f T0 Pf L (Z) . Tk+1 Pf Tk Pf Lp(Z) k≥0 X X p 1/p 2kεp | ( )( )| . d Pf e χB(e) Lp(Z) k≥0 |e|=k

X X q 1/q 2ksq | ( )( )| . d Pf e χB(e) Lp(Z) k≥0 |e|=k

(obvious modications for p = ∞ and/or q = ∞). The latter quantity appears on both sides of the desired estimate, so it remains to verify that the part of kf k ˙ s corresponding to edges at levels k < 0 is bounded Bp,q(Z) by kf k s . To this end, for each e with |e| < 0 choose a ball Be that contains B(e) and has radius uniformly Bp,q(Z) comparable to 2−|e| (without loss of generality we may also assume that this radius is always ≥ 1). The quantity we wish to estimate is thus

1/q q/p1/q X ksq X q X ksq X 1−p p 2 |d(Pf)(e)|χ p 2 µ(Be) kf k 1 =: I. B(e) L (Z) . L (Be) k<0 |e|=k k<0 |e|=k

1−p p If p ≥ 1, the desired estimate follows easily by using Hölder’s inequality for each term µ(Be) kf k 1 and L (Be) noting that the balls Be at each xed level have uniformly bounded overlap:

Z Z I kf k p ≤ kf − T (Pf)k p + kT (Pf)k p kf k s . . L (Z) 0 L (Z) 0 L (Z) . Bp,q(Z) If on the other hand Q/(Q + s) < p < 1, take ϵ ∈ (0, s) such that p = Q/(Q + ϵ). Then part (ii) of Lemma 5.3 above yields

X 1−p p −k(Q−Qp) Z p X kϵp X p µ(Be) kf k 1 . 2 |T0 (Pf )| dµ + 2 |d(Pf )(e)|χB(e) p L (Be) ˆ L (Z) |e|=k Z k≥0 |e|=k −k(Q−Qp) p = 2 kf k ϵ Bp,p(Z) for each k < 0. As ϵ < s, it is easily seen that kf k ϵ kf k s , and since Q + s − Q/p > 0, we arrive at Bp,p(Z) . Bp,q(Z)

1/q X k(Q+s−Q/p)q I 2 kf k s ≈ kf k s . . Bp,q(Z) Bp,q(Z) k<0 110 Ë Eero Saksman and Tomás Soto

We will further need the following density result, which also is of independent interest. A similar result has been recently established in [16].

Proposition 5.4. Suppose that 0 < s < 1, Q/(Q + s) < p ≤ ∞ and 0 < q < ∞. s Z s (i) If f ∈ Bp,q(Z), then Tn (Pf ) → f in the quasinorm of Bp,q(Z) as n → ∞. s Z (ii) Suppose that in addition p < ∞ and q > Q/(Q + s). If f ∈ Fp,q(Z), then Tn (Pf) → f in the quasinorm of s Fp,q(Z) as n → ∞.

Proof. The analogous results for the homogeneous versions of these spaces can be found in [2, Theorem 3.3] Z and [35, Theorem 3.2]. By Proposition 5.2, it thus suces to verify that the sequence (Tn (Pf))n≥0 converges to f in the quasinorm of Lp(Z). For simplicity we shall only consider the situation of part (i). In fact, this is essentially contained in the estimate (5.1): taking ε ∈ (0, s), we get

1/p Z X kεp X p k − ( )k p 2 | ( )( )| f Tn Pf L (Z) . d Pf e χB(e) Lp(Z) k≥n |e|=k

X X q 1/q 2ksq | ( )( )| , . d Pf e χB(e) Lp(Z) k≥n |e|=k and since q < ∞, the latter quantity tends to zero as n → ∞. With Propositions 5.2 and 5.4 in mind, we are in a position to give the proof the non-homogeneous versions of our main results.

Proof of Theorem 1.7. For brevity we shall only consider the traces and extensions of Besov functions, as the analogs for Theorems 1.4 and 1.5 will then follow with more or less obvious modications of the original proofs. The hyperbolic llings (XZ , EZ) and (XF , EF) are again chosen as in the proof of Theorem 1.1, i.e so F Z Z that Xn is in a natural way a subset of Xn for all n ∈ Z. When necessary, we shall use the notations P f, PF f , BZ(x), BF(x) etc. with the obvious meanings to distinguish between the relevant operators and balls on dierent spaces. s First, suppose that the parameters s, p and q are admissible and f ∈ Bp,q(Z). We will construct the trace of f to F using a modied version of the operator R which is better suited for non-homogeneous case. Denote F F E+ := {e ∈ E : |e| ≥ 0} and write ∞ * F F X F F R f := d(Pf ) F + T (Pf ) F = I d(Pf) F + T (Pf) F I |E+ 0 |X n |E 0 |X n=0 ∞ X F F F = Tn+1 (Pf)|XF − Tn (Pf)|XF + T0 (Pf)|XF . n=0 F As before, the series dening d(Pf) F converges ν-almost everywhere in F, and by Proposition 5.2 the I |E+ ˙ s−(Q−λ)/p limit function has Bp,q (F)-quasinorm bounded by a constant times

d(Pf)|EF s−(Q−λ)/p F kd(Pf )kIs (EZ ) = kf k ˙ s kf kBs (Z). (5.2) + Ip,q (E ) . p,q Bp,q(Z) . p,q F F * We further have d(Pf) F = f − T (Pf) F , and hence R f = f , ν-almost everywhere in F. We thus I |E+ |F 0 |X |F have to show that F d(Pf ) F kf k s (5.3) I |E+ Lp(F) . Bp,q(Z) and that F F T0 (Pf )|XF P + T0 (Pf)|XF ˙ s−(Q−λ)/p kf kBs (Z). (5.4) L (F) Bp,q (F) . p,q For (5.3), take ε ∈ (0, s − (Q − λ)/p) (or ε = 0 if p ≤ 1). Then for ν-almost all ξ ∈ F we have F X X Z Z 0 d(Pf ) F (ξ) ≤ |P f(y) − P f (y )|χ F (ξ)χ F 0 (ξ) I |E+ B (y) B (y ) n≥0 0 F F 0 (y,y )∈(Xn ×Xn+1), y∼y Traces of Function Spaces Ë 111

1/p X nεp X Z p . 2 |d(P f)(e)| χBF (e)(ξ) . n≥0 e∈EF , |e|=n The Ahlfors regularity of Z and F implies that

νBF(e) ≈ 2|e|(Q−λ)µBZ(e) (5.5)

F for all e ∈ E+, so we further get

1/p F X F Z p ( ) F ( ) | ( )| (5.6) T0 Pf |X LP(F) . ν B x P f x F x∈X0 X 1−p 1/p Z( ) k kp . . µ B x f L1(BZ (x),µ) F x∈X0

If p ≥ 1, the latter quantity is easily estimated by kf kLp(Z), and if Q/(Q + s) < p < 1, we can use part (i) of Lemma 5.3 to get an estimate in terms of kf k s . Bp,q(Z) ˙ s−(Q−λ)/p Finally to estimate the Bp,q (F)-quasinorm in (5.4), note that the Lipschitz continuity of the functions F (ψx ) F implies that x∈X0 F F X Z T (Pf) F (ξ) − T (Pf ) F (η) min 1, d(ξ , η) |P f(x)| χ F (ξ) + χ F (η) , 0 |X 0 |X . Bx Bx F x∈X0 ˙ s−(Q−λ)/p so a fairly straightforward computation – which we leave to the reader¹ – gives an estimate of the Bp,q - F p quasinorm of T0 (Pf|XF ) in terms of the L -quasinorm of the function

X Z |P f(x)|χ F , Bx F x∈X0 which is essentially the quantity on the right-hand side of (5.6). s−(Q−λ)/p Now to construct the extension of a function f ∈ B (F), we extend the sequences u := d(Pf) F p,q |E+ Z F Z F and v := (Pf ) F as zero on E \ E+ and X \ X respectively and put |X0 0 ∞ * Z Z X Z Z E f := I u + T0 (v) = In u + T0 (v). n=0

* 1 * s−(Q−λ)/p s That E f converges in Lloc(Z) and the operator E is a bounded from Bp,q (F) into Bp,q(Z) follows easily from an appropriate reformulation of the estimates established above for the operator R*. * * s−(Q−λ)/p To verify that R (E f) = f for f ∈ Bp,q (F), we rst consider the case with q < ∞. Then by Proposition 5.4, it suces to consider functions f that are Lipschitz continuous on every bounded subset of F. In this case it is easily veried that the series dening E*f converges uniformly on bounded subsets of Z, so the limit function is continuous on Z, and hence we plainly have E*f (ξ) = f(ξ) for all ξ ∈ F. For the case q = ∞, one concludes again by using interpolation [15, Theorem 4.3].

˙ s 1 A similar computation also works for the spaces Fp,q(Z) with either s < 1 or s = 1 and q = ∞. 112 Ë Eero Saksman and Tomás Soto

A Appendix

In this section we shall make some additional remarks concerning the main results of this paper and elaborate on some less interesting technicalities that were used in their proofs.

A.1 On the regularity and dimensions of Z and F

In Theorems 1.1 through 1.7, we assumed that the metric measure space (Z, d, µ) and (F, d|F , ν) were respectively Q-Ahlfors regular and λ-Ahlfors regular. Upon examining the proofs, it is easy to see that the only essential way this assumption is used is in the estimates (3.1) and (5.5), i.e. to guarantee that µ BZ(ξ , r) γ ≈ r (A.1) ν BF(ξ , r) for all ξ ∈ F and 0 < r < diam (F), where γ = Q − λ. Therefore this kind of a condition, known in previous literature (see e.g. [28]) as Ahlfors co-dimension γ regularity of F with respect to Z, suces for our main results. To elaborate on this, suppose that the space (Z, d, µ) satises the doubling condition (2.1), and ν is merely a doubling measure on the metric space (Z, d|F) satisfying (A.1) for some γ ∈ (0, Q). From (A.1) and (2.1) it then easily follows that R Q−γ νB (ξ , R) νB (ξ , r) F . r F for all ξ ∈ F, r > 0 and R > r, i.e. λ := Q − γ is an upper bound for the dimension of the space (F, d|F , ν). With these assumptions and notations, Theorems 1.1, 1.4, 1.5 and 1.7 continue to hold. A closer examination of the proof of Theorem 1.7 in fact shows that for the non-homogeneous results, it suces to assume (A.1) for r ≤ 1. However, now have no precise information on the Hausdor dimension of F, so Remark 3.3 does not hold. Therefore part (i) of each Theorem has to be interpreted as originally stated.

A.2 Convergence of the operator I

1 Here we shall establish the Lloc convergence of the operator I, as well as the formula (3.3). The assumption for the Lemma below is that the space (Z, d, µ) satises the doubling property (2.1), and the notation is as in Denition 3.4.

s s Lemma A.1. (i) Suppose that u ∈ Ip,q(E) ∪ Jp,q(E) with 0 < s < 1, Q/(Q + s) < p < ∞ and 0 < q ≤ ∞, or 1 u ∈ Jp,q(E) with Q/(Q + 1) < p < ∞ and 0 < q ≤ 1. Then the limit

N −1 Z X Z X Z I u := lim In u(·) − In u(ξ0) , N→∞ n=−N n=−N

1 where ξ0 ∈ Z is a xed point, exists in Lloc(Z) and pointwise µ-almost everywhere. Z s s (ii) Suppose that u : X → C is a sequence such that du ∈ Ip,q(E) ∪ Jp,q(E) with 0 < s < 1, 0 < p < ∞ and 0 < q ≤ ∞. Then Z Z lim TM u(ξ) − TM u(η) = 0 M→−∞ for all ξ, η ∈ Z. If either diam (Z) < ∞ or µ(Z) = ∞, the same holds for s = 1.

Proof. (i) This is essentially included in the proofs of [2, Proposition 6.3] and [35, Proposition 4.3], but since those results are only formulated in the case 0 < s < 1, we shall repeat the main points of the argument here. Traces of Function Spaces Ë 113

P Z First, to obtain the convergence of the series n≥0 In u, x ϵ ∈ (0, s) such that r := Q/(Q + ϵ) < p. For a xed ball B ⊂ Z with radius 1, we may use the doubling property (2.1) as in the proof of [2, Lemma 2.3] to obtain 1/r X X X r |IZ u|dµ µB(e)|u(e)| µB(e)2|e|ϵ|u(e)| , ˆ n . . n≥0 B |e|≥0 |e|≥0 B(e)∩B=∅̸ B(e)∩B=∅̸ ϵ where the implied constant in the last inequality depends on B. The latter quantity is essentially a local Ir,r norm of u. Since we are only dealing with vertices e with |e| ≥ 0 and ϵ < s, we may easily estimate this quantity s s from above by a local Ir,q or Jr,q norm (depending on which space u belongs to), and since r < p, we obtain a nite upper bound by using Hölder’s inequality. As the ball B with radius 1 was arbitrary, we have the desired P Z convergence of n≥0 In u. s For the part corresponding to negative indices, we shall consider the case u ∈ Jp,q(E) with admissible s parameters; the case with u ∈ Ip,q(E) can be handled in a similar manner. We shall examine the convergence −k of this part in B := B(ξ0, 2 ), where k < 0 is an arbitrary integer. For all n ≤ k, the Lipschitz continuity of the Z functions ψx yields Z Z n−k X In u(ξ) − In u(ξ0) dµ(ξ) µ(B)2 |u(e)|, ˆ . B |e|=n B(e)∩B=∅̸ so that 1/q X Z Z X |e| X |e|s q In u(ξ) − In u(ξ0) dµ(ξ) 2 |u(e)| 2 |u(e)| ˆ . . n≤k B |e|≤k |e|≤k B(e)∩B=∅̸ B(e)∩B=∅̸ (obvious modication for q = ∞); here we used the knowledge that #{e ∈ EZ : |e| = n and B(e) ∩ B ≠ ∅} is bounded uniformly in n ≤ k and the assumption that either s < 1 or q ≤ 1. Writing tB(e) := tB(e−) ∪ tB(e+) for a suitable t > 1, we further get

1/q X Z Z X |e|s q In u(ξ) − In u(ξ0) dµ(ξ) . inf 2 |u(e)| χtB(e)(ξ) . ˆ ξ∈B n≤k B |e|≤k

The latter inmum is nite, since by Remark 2.2 (vii), the function in question belongs to Lp(Z). Z (ii) There is nothing to prove if diam (Z) < ∞, because then TM u is simply a constant function for M < 0 with large enough absolute value. We can thus assume that diam (Z) = ∞ in the remainder of this proof.

Write kduk for kduk s or kduk s , whichever is nite. With ξ and η xed, we can for all M < 0 Ip,q(E) Jp,q(E) Z with suciently large absolute value nd a vertex xM ∈ X such that B(x) contains both ξ and η. By the −M doubling property, µ B(xM) is comparable to µ B(η, 2 ) uniformly in M. Using the Lipschitz continuity Z of the functions ψx , we obtain

Z Z X Z Z M X TM u(ξ) − TM u(η) ≤ u(x) − u(xM)| ψx (ξ) − ψx (η) . 2 |(du)(e)| Z e : x ∈{e ,e } x∈XM M − + M(1−s) −1/p M(1−s) −M −1/p . 2 µ B(xM) kduk ≈ 2 µ B(η, 2 ) kduk.

Since either s < 1 or µ(Z) = ∞, the latter quantity tends to zero as M → −∞.

A.3 Proof of Lemma 5.3

Finally we present the proof of the auxiliary result that was used in Section 5 when handling the cases with p < 1.

Proof of Lemma 5.3. (i) Note that

p p p p k k 1 ≤ k − ( )k + ( ) | ( )| . (A.2) f L (B(x)) f Pf x L1(B(x)) µ B x Pf x 114 Ë Eero Saksman and Tomás Soto

For the latter term, we have the following simple estimates:

|Pf(x)|p = − |Pf (x)|p dµ(ξ) ˆB(x) Z p Z p ≤ − |Pf (x) − T0 (Pf)(ξ)| dµ(ξ) + − |T0 (Pf)(ξ)| dµ(ξ) ˆB(x) ˆB(x) −1 X p ( ) | ( )( )| + | Z( )( )|p ( ) , . µ B x d Pf e χB(e) Lp(σB(x)) T0 Pf ξ dµ ξ ˆ ( ) |e|=0 B x and multiplying both sides by µ(B(x))p gives a good enough estimate for the latter term in (A.2). The necessary estimate for the term k − ( )kp in (A.2) is basically contained in the proof of Lemma f Pf x L1(B(x)) A.1 above, so we only give a rough outline here that takes into account the multiplicative constants depending Z 1 on µ(B(x)). Using again the fact that Tk (Pf) → f in Lloc(Z) as k → ∞, the assumption that p = 1 − (ϵ/Q)p and the doubling property of µ, we get

X p k − ( )kp ( ) | ( )( )|p f Pf x L1(B(x)) . µ B e d Pf e |e|≥0 B(e)∩B(x)=∅̸ −(ϵ/Q)p X |e|ϵp p . µ B(x) 2 µ B(e) |d(Pf )(e)| |e|≥0 B(e)∩B(x)=∅̸ p−1 X X p ( ) 2kϵp | ( )( )| , . µ B x d Pf e χB(e) Lp(σB(x)) k≥0 |e|=k which nishes the proof of part (i). Z (ii) This follows from part (i) by covering B with an optimal collection of balls B(xi), xi ∈ X0 , and noting p−1 Q−Qp p−1 that the doubling property yields µ(B(xi)) . r µ(B) for all xi.

Acknowledgement: E.S. was supported by the Finnish CoE in Analysis and Dynamics Research, and by the Academy of Finland, projects 113826 and 118765. T.S. was supported by the Finnish CoE in Analysis and Dy- namics Research, and by the Väisälä foundation.

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