Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces Tsp,q(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{p,q}_s(X)$$\end{document}, called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.

considered as forming a scale of function spaces in their own right until the work of Hofmann et al. [15, §8.3], in which factorisation and complex interpolation theorems are obtained for these spaces.
In this article we further explore the weighted tent space scale. In the interests of generality, we consider weighted tent spaces T p,q s (X ) associated with a metric measure space X , although our theorems are new even in the classical case where X = R n equipped with the Lebesgue measure. Under sufficient geometric assumptions on X (ranging from the doubling condition to the assumption that X = R n ), we uncover two previously unknown novelties of the weighted tent space scale.
First, we identify some real interpolation spaces between T p 0 ,q s 0 and T p 1 ,q s 1 whenever s 0 = s 1 . In Theorem 2.4 we prove that for appropriately defined parameters, where the scale of 'Z -spaces' is defined in Definition 2.3. We require p 0 , p 1 , q > 1 in this identification, but in Theorem 2.9 we show that in the Euclidean setting the result holds for all p 0 , p 1 > 0 and q ≥ 1. In the Euclidean setting, Z -spaces have appeared previously in the work of Barton and Mayboroda [4]. In their notation we have Z p,q s (R n ) = L( p, ns + 1, q). Barton and Mayboroda show that these function spaces are useful in the study of elliptic boundary value problems with boundary data in Besov spaces. The connection with weighted tent spaces shown here is new.
Second, we have continuous embeddings whenever the parameters satisfy the relation This is Theorem 2.19. Thus a kind of Hardy-Littlewood-Sobolev embedding theorem holds for the weighted tent space scale, and by analogy we are justified in referring to the parameter s in T p,q s as a regularity parameter. We also identify complex interpolation spaces between weighted tent spaces in the Banach range. This result is already well-known in the Euclidean setting, and its proof does not involve any fundamentally new arguments, but we include it here for completeness.
These results in this paper play a crucial role in recent work of the author and Pascal Auscher [2], in which we use weighted tent spaces and Z -spaces to construct abstract homogeneous Hardy-Sobolev and Besov spaces associated with elliptic differential operators with rough coefficients, extending the abstract Hardy space techniques initiated independently by Auscher et al. [3] and Hofmann and Mayboroda [14]. Notation Given a measure space (X, μ), we write L 0 (X ) for the set of μ-measurable functions with values in the extended complex numbers C ∪ {±∞, ±i∞}. As usual, by a 'measurable function', we actually mean an equivalence class of measurable functions which are equal except possibly on a set of measure zero. We will say that a function f ∈ L 0 (X ) is essentially supported in a subset E ⊂ X if μ{x ∈ X \ E : f (x) = 0} = 0.
A quasi-Banach space is a complete quasi-normed vector space; see for example [18, §2] for further information. If B is a quasi-Banach space, we will write the quasinorm of B as either ||·|| B or ||· | B||, according to typographical needs.
As is now standard in harmonic analysis, we write a b to mean that a ≤ Cb for some unimportant constant C ≥ 1 which will generally change from line to line. We also write a c 1 ,c 2 ,... b to mean that a ≤ C(c 1 , c 2 , . . .)b.

Metric Measure Spaces
A metric measure space is a triple (X, d, μ), where (X, d) is a nonempty metric space and μ is a Borel measure on X . For every x ∈ X and r > 0, we write B(x, r ) := {y ∈ X : d(x, y) < r } for the ball of radius r , and we also write V (x, r ) := μ(B(x, r )) for the volume of this ball. The generalised half-space associated with X is the set X + := X × R + , equipped with the product topology and the product measure dμ(y) dt/t.
This immediately implies that the measure space (X, μ) is σ -finite, as X may be written as an increasing sequence of balls for any point x 0 ∈ X . Nondegeneracy also implies that the metric space (X, d) is separable [7, Proposition 1.6]. To rule out pathological behaviour (which is not particularly interesting from the viewpoint of tent spaces), we will always assume nondegeneracy. Generally we will need to make further geometric assumptions on (X, d, μ). In this article, the following two conditions will be used at various points. We say that A consequence of the doubling condition is that there exists a minimal number n ≥ 0, called the doubling dimension of X , and a constant C ≥ 1 such that for all x ∈ X and 0 < r ≤ R < ∞.
For n > 0, we say that (X, d, μ) is AD-regular of dimension n if there exists a constant C ≥ 1 such that for all x ∈ X and all r < diam(X ). One can show that AD-regularity (of some dimension) implies doubling. Note that if X is unbounded and AD-regular of dimension n, then (5) holds for all x ∈ X and all r > 0.

Unweighted Tent Spaces
Throughout this section we suppose that (X, d, μ) is a nondegenerate metric measure space. We will not assume any further geometric conditions on X without explicit mention. All of the results here are known, at least in some form. We provide statements for ease of reference and some proofs for completeness. For x ∈ X we define the cone with vertex x by and for each ball B ⊂ X we define the tent with base B by Equivalently, T (B) is the set of points (y, t) ∈ X + such that B(y, t) ⊂ B. From this characterisation it is clear that if (y, t) ∈ T (B), then t ≤ r B , where we define Note that it is possible to have r B(y,t) > t.
Fix q ∈ (0, ∞) and α ∈ R. For f ∈ L 0 (X + ), define functions A q f and C q α f on X by and for all x ∈ X , where the supremum in (7) is taken over all balls B ⊂ X containing x. We abbreviate C q := C q 0 . Note that the integrals above are always defined (though possibly infinite) as the integrands are non-negative, and so we need not assume any local q-integrability of f . We also define and Geometrically (x) + s is a 'vertically translated' cone, and (x) + s ⊃ (x) + r for all r < s. The triangle inequality implies that Since the sequence of sets O ∩ ( (x) + n −1 ) is increasing in n, and since O has positive measure, we find that there exists n ∈ N such that O ∩ ( (x) + n −1 ) has positive measure. Thus for all x ∈ B(x, n −1 ), has positive measure, and so (A ∞ f )(x ) > λ. Therefore A ∞ f is lower semicontinuous.
The argument for C ∞ α is simpler. We have (C ∞ α f )(x) > λ if and only if there exists a ball B x such that This immediately yields (C ∞ α f )(x ) > λ for all x ∈ B, and so C ∞ α f is lower semicontinuous.
For the sake of notational clarity, we will write T p,q rather than T p,q (X ) unless we wish to emphasise a particular choice of X . Although we will always refer to tent space 'quasi-norms', these are norms when p, q ≥ 1.

Remark 1.3
Our definition of A ∞ f gives a function which is less than or equal to the corresponding function defined by Coifman, Meyer, and Stein [11], which uses suprema instead of essential suprema. We also do not impose any continuity conditions in our definition of T p,∞ . Therefore our space T p,∞ (R n ) is strictly larger than the Coifman-Meyer-Stein version.
By a cylinder we mean a subset C ⊂ X + of the form C = B(x, r ) × (a, b) for some (x, r ) ∈ X + and 0 < a < b < ∞. We say that a function f ∈ L 0 (X + ) is cylindrically supported if it is essentially supported in a cylinder. In general cylinders may not be precompact, and so the notion of cylindrical support is more general than that of compact support. For all p, q ∈ (0, ∞] we define A straightforward application of the Fubini-Tonelli theorem shows that for all q ∈ (0, ∞) and for all f ∈ L 0 (X + ), and so T q,q = L q (X + ). When q = ∞ this is true by definition. Proposition 1.4 For all p, q ∈ (0, ∞), the subspace T p,q;c ⊂ T p,q is dense in T p,q . Furthermore, if X is doubling, then for all p, q ∈ (0, ∞], T p,q is complete, and when p, q = ∞, L q c (X + ) is densely contained in T p,q .
Proof The second statement has already been proven in [1, Proposition 3.5], 1 so we need only prove the first statement. Suppose f ∈ T p,q and fix a point x 0 ∈ X . For each k ∈ N, define Then each f k is cylindrically supported. We have All interchanges of limits and integrals follow by dominated convergence. Hence we have f = lim k→∞ f k , which completes the proof.

Proposition 1.5
Suppose that X is doubling, p ∈ [1, ∞), and q ∈ (1, ∞). Then the L 2 (X + ) inner product identifies the dual of T p,q with T p ,q .
Suppose that p ∈ (0, 1], q ∈ [p, ∞], and B ⊂ X is a ball. We say that a function a ∈ L 0 (X + ) is a T p,q atom (associated with B) if a is essentially supported in T (B) and if the size estimate ||a|| T q,q ≤ μ(B) δ p,q holds (recall that δ p,q := q −1 − p −1 ). A short argument shows that if a is a T p,q -atom, then ||a|| T p,q ≤ 1.
Theorem 1.6 (Atomic decomposition) Suppose that X is doubling. Let p ∈ (0, 1] and q ∈ [p, ∞]. Then a function f ∈ L 0 (X + ) is in T p,q if and only if there exists a sequence (a k ) k∈N of T p,q -atoms and a sequence ( with convergence in T p,q . Furthermore, we have where the infimum is taken over all decompositions of the form (10). This is proven by Russ when q = 2 [22], and the same proof works for general q ∈ [p, ∞). For q = ∞ we need to combine the original argument of Coifman, Meyer, and Stein [11,Proposition 2] with that of Russ. We defer this to Sect. 3.2.

Weighted Tent Spaces: Definitions, Duality, and Atoms
We continue to suppose that (X, d, μ) is a nondegenerate metric measure space, and again we make no further assumptions without explicit mention.
For each s ∈ R, we can define an operator V s on L 0 (X + ) by for all (x, t) ∈ X + . Note that for r, s ∈ R the equality V r V s = V r +s holds, and also that V 0 is the identity operator. Using these operators we define modified tent spaces, which we call weighted tent spaces, as follows.
For q = ∞, and with an additional parameter α ∈ R, we define T where the supremum is taken over all balls B ⊂ X . Note that T ∞,q Remark 1.8 The weighted tent space quasi-norms of Hofmann et al. [15, §8.3] (with p = ∞) and Huang [16] (including p = ∞ with α = 0) are given by which are equivalent to those of our spaces T p,q s/n (R n ). In general, when X is unbounded and AD-regular of dimension n, the quasi-norm in (11) (with X replacing R n ) is equivalent to that of our T p,q s/n . We have chosen the convention of weighting with ball volumes, rather than with the variable t, because this leads to more geometrically intrinsic function spaces and supports embedding theorems under weaker assumptions.
For all r, s ∈ R, the operator V r is an isometry from T p,q s to T p,q s+r . The operator V −r is also an isometry, now from T p,q s+r to T p,q s , and so for fixed p and q the weighted tent spaces T p,q s are isometrically isomorphic for all s ∈ R. Thus by Proposition 1.4, when X is doubling, the spaces T p,q s are all complete. Recall the L 2 (X + ) inner product (9), which induces a duality pairing between T p,q and T p ,q for appropriate p and q when X is doubling. For all s ∈ R and all f, g ∈ L 2 (X + ) we have the equality which yields the following duality result.
Then the L 2 (X + ) inner product (9) identifies the dual of T and V s g ∈ T p ,q , so by Proposition 1.5 and (12) we have determines a bounded linear functional on T p,q with norm dominated by ||ϕ||. Hence by Proposition 1.5 there exists a functiong ∈ T p ,q with ||g|| T p ,q ||ϕ|| such that we are done.
There is also a duality result for p < 1 which incorporates the spaces T ∞,q s;α with α > 0. Before we can prove it, we need to discuss atomic decompositions.
Suppose that p ∈ (0, 1], q ∈ [p, ∞], s ∈ R, and B ⊂ X is a ball. We say that a function a This is equivalent to demanding that a is essentially supported in T (B) and that The atomic decomposition theorem for unweighted tent spaces (Theorem 1.6) immediately implies its weighted counterpart.

Proposition 1.10 (Atomic decomposition for weighted tent spaces)
with convergence in T p,q s . Furthermore, we have where the infimum is taken over all decompositions of the form (13).
Using this, we can prove the following duality result for p < 1.

Theorem 1.11
Suppose that X is doubling, p ∈ (0, 1), q ∈ [1, ∞), and s ∈ R. Then the L 2 (X + ) inner product (9) identifies the dual of T we write f as a sum of T p,q s -atoms as in (13) and get Taking the infimum over all atomic decompositions completes the argument.
Conversely, suppose that ϕ ∈ (T p,q s ) . Exactly as in the classical duality proof (see [1, Proof of Proposition 3.10]), using the doubling assumption, there exists a function for each ball B ⊂ X by duality: Hölder's inequality implies that when f is essentially supported in T (B), so we have and therefore which completes the proof.

Remark 1.12
Note that q = 1 is included here, and excluded in the other duality results of this article. Generally the spaces T p,q with p ≤ q are easier to handle than those with p > q.
We end this section by detailing a technique, usually referred to as 'convex reduction', which is very useful in relating tent spaces to each other. Suppose f ∈ L 0 (X + ) and M > 0. We define a function f M ∈ L 0 (X + ) by and for α ∈ R we also have and likewise for p = ∞ and q < ∞ we have The case p = q = ∞ behaves in the same way: , and vice versa. We will use them frequently.

Interpolation and Embeddings
As always, we assume that (X, d, μ) is a nondegenerate metric measure space. We will freely use notation and terminology regarding interpolation theory; the uninitiated reader may refer to Bergh and Löfström [5].

Complex Interpolation
In this section we will make the following identification of the complex interpolants of weighted tent spaces in the Banach range of exponents.

Remark 2.2
In the case where X = R n with the Euclidean distance and Lebesgue measure, this result (with p 0 , p 1 < 1 permitted) is due to Hofmann et al. [15,Lemma 8.23]. A more general result, still with X = R n , is proven by Huang [16,Theorem 4.3] with q 0 , q 1 = ∞ also permitted, and with Whitney averages incorporated. Both of these results are proven by means of factorisation theorems for weighted tent spaces (with Whitney averages in the second case), and by invoking an extension of Calderón's product formula to quasi-Banach spaces due to Kalton and Mitrea [19,Theorem 3.4].
We have chosen to stay in the Banach range with 1 < q 0 , q 1 < ∞ for now, as establishing a general factorisation result would take us too far afield.
Note that if p 0 = ∞ (say) then we are implicitly considering T ∞,q 0 s 0 ;α with α = 0; interpolation of spaces with α = 0 is not covered by this theorem. This is because the method of proof uses duality, and to realise T ∞,q 0 s 0 ;α with α = 0 as a dual space we would need to deal with complex interpolation of quasi-Banach spaces, which adds difficulties that we have chosen to avoid.
Before moving on to the proof of Theorem 2.1, we must fix some notation. For q ∈ (1, ∞) and s ∈ R, write (this notation is consistent with viewing the function V −qs−1 as a weight on the product measure dμ dt/t). An important observation, originating from Harboure, Torrea, and Viviani [13], is that for all p ∈ [1, ∞), q ∈ (1, ∞) and s ∈ R, one can write Hence H is an isometry from T p,q s to L p (X : L q s (X + )). Because of the restriction on q, the theory of Lebesgue spaces (more precisely, Bochner spaces) with values in reflexive Banach spaces is then available to us.
This proof follows previous arguments of the author [1], which are based on the ideas of Harboure et al. [13] and of Bernal [6], with only small modifications to incorporate additional parameters. We include it to show where these modifications occur: in the use of duality, and in the convex reduction.
Proof of Theorem 2.1 First we will prove the result for p 0 , p 1 ∈ (1, ∞). Since H is an isometry from T p j ,q j s j to L p j (X : L q j s j (X + )) for j = 0, 1, the interpolation property implies that H is bounded (with norm ≤ 1 due to exactness of the complex interpolation functor) Here we have used the standard identification of complex interpolants of Banachvalued Lebesgue spaces [5,Theorem 5.1.2]. The standard identification of complex interpolants of weighted Lebesgue spaces [5,Theorem 5.5.3] gives and we conclude that To obtain the reverse inclusion, we use the duality theorem for complex interpolation [5, Theorem 4.5.1 and Corollary 4.5.2]. Since X is doubling, and by our restrictions on the exponents, the spaces T p 0 ,q 0 s 0 and T p 1 ,q 1 s 1 are reflexive (by Proposition 1.9) with intersection dense in both spaces (as it contains the dense subspace L max(q 0 ,q 1 ) c (X + ) by Proposition 1.4). Therefore the assumptions of the duality theorem for complex interpolation are satisfied, and we have where the first two lines follow from Proposition (1.9) and (15), and the third line uses the duality theorem for complex interpolation combined with Proposition 1.9.
We can extend this result to p 0 , p 1 ∈ [1, ∞] using the technique of [1, Proposition 3.18]. The argument is essentially identical, so we will not include the details here.

Real Interpolation: The Reflexive Range
In order to discuss real interpolation of weighted tent spaces, we need to introduce a new scale of function spaces, which we denote by Z and for q ∈ (0, ∞), f ∈ L 0 (X + ), and (x, t) ∈ X + we define the L q -Whitney average ∞), and f ∈ L 0 (X + ), we then define the quasi-norm In this section we will prove the following theorem, which identifies real interpolants of weighted tent spaces in the reflexive range. We will extend this to the full range of exponents in the Euclidean case in the next section.

Theorem 2.4
Suppose that X is AD-regular and unbounded, p 0 , p 1 , q ∈ (1, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1). Then for any c 0 ∈ (0, ∞) and As a corollary, in the case when X is AD-regular and unbounded, and when p, q > 1, the spaces Z p,q s (X ; c 0 , c 1 ) are independent of the parameters (c 0 , c 1 ) with equivalent norms, and we can denote them all simply by Z p,q s . 3 We remark that most of the proof does not require AD-regularity, but in its absence we obtain identifications of the real interpolants which are less convenient.
The proof relies on the following identification of real interpolants of weighted L q spaces, with fixed q and distinct weights, due to Gilbert [12,Theorem 3.7]. The cases p ≤ 1 and q < 1 are not considered there, but the proof still works without any modifications in these cases. Note that the original statement of this theorem contains a sign error in the expression corresponding to (17). (M, μ) is a σ -finite measure space and let w be a weight on (M, μ). Let p, q ∈ (0, ∞) and θ ∈ (0, 1). For all r ∈ (1, ∞), and for f ∈ L 0 (M), the expressions

Theorem 2.5 (Gilbert) Suppose
and s −θ 1 x:w(x)>1/s f L q (M) L p (R + ,ds/s) (19) define equivalent norms on the real interpolation space The first step in the proof of Theorem 2.4 is a preliminary identification of the real interpolation norm.

Proposition 2.6 Let all numerical parameters be as in the statement of Theorem 2.4.
Then for all f ∈ L 0 (X + ) we have the equivalence Proof We use the notation of the previous section. We have already noted that the map The Lions-Peetre result on real interpolation of Banach-valued Lebesgue spaces (see for example [21,Remark 7]) then implies that .
Having proven Proposition 2.6, we can use Theorem 2.5 to provide some useful characterisations of the real interpolation norm. For f ∈ L 0 (X + ) and a, b ∈ [0, ∞], we define the truncation Note that in this theorem we allow for p 0 , p 1 ≤ 1; we will use this range of exponents in the next section.
Theorem 2.7 Suppose p 0 , p 1 , q ∈ (0, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1), and suppose that X is AD-regular of dimension n and unbounded. Let r ∈ (1, ∞). Then for f ∈ L 0 (X + ) we have norm equivalences . (23) Proof First assume that s 1 > s 0 . Let μ q s 0 be the measure on X + given by Since X is AD-regular of dimension n and unbounded, we have that || f || L q (μ q s 0 ) || f || L q s 0 (X + ) . Also define the weight w(y, t) := t −(s 1 −s 0 )n , so that w q μ q s 0 = μ q s 1 . We will obtain the norm equivalence (23). For 1 < r < ∞ and k ∈ Z, we have r −k < w(y, t) ≤ r −k+1 if and only if t ∈ [r (k−1)/n(s 1 −s 0 ) , r k/n(s 1 −s 0 ) ) (here we use s 1 > s 0 ). Using the characterisation (17) of Theorem 2.5, and replacing r with r n(s 1 −s 0 ) , for f ∈ L 0 (X + ) we have This proves the norm equivalence (23) for all f ∈ L 0 (X + ) when s 1 > s 0 . If s 1 < s 0 , one simply uses the identification (L The equivalences (21) and (22) follow from the characterisations (18) and (19) of Theorem 2.5 in the same way, with integrals replacing sums throughout. We omit the details here.
Finally we can prove the main theorem: the identification of the real interpolants of weighted tent spaces as Z -spaces.
Proof of Theorem 2.4. Suppose f ∈ L 0 (X + ). Using the characterisation (23) in Theorem 2.7 with r = c 1 > 1, and using aperture c 0 /c 1 for the tent space (making use of the change of aperture theorem [1, Proposition 3.21]), we have . To prove the reverse estimate we use the same argument, this time using that for r, t ∈ (2 k−1 , 2 k ) we have c 0 ,c 1 (x, t) ⊂ B(x, 2c 0 t)×(c −1 1 2 k−1 , c 1 2 k ). Using aperture 2c 0 for the tent space, we can then conclude that This completes the proof of Theorem 2.4.

Remark 2.8 Note that this argument shows that
whenever X is AD-regular of dimension n and unbounded, for all p 0 , p 1 ∈ (0, ∞), c 0 ∈ (0, ∞), and c 1 ∈ (1, ∞). Therefore, since Theorem 2.7 also holds for this range of exponents, to establish the identification (24) for p 0 , p 1 ∈ (0, ∞) it suffices to extend Proposition 2.6 to p 0 , p 1 ∈ (0, ∞). We will do this in the next section in the Euclidean case.

Real Interpolation: The Non-reflexive Range
In this section we prove the following extension of Theorem 2.4. In what follows, we always consider R n as a metric measure space with the Euclidean distance and Lebesgue measure. Throughout this section we use the real interpolation method for quasi-normed Abelian groups, as described in [5, §3.11].

Theorem 2.9
Suppose that p 0 , p 1 ∈ (0, ∞), q ∈ [1, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1). Then for any c 0 ∈ (0, ∞) and c 1 ∈ (1, ∞) we have the identification The main difficulty here is that vector-valued Bochner space techniques are not available to us, as we would need to use quasi-Banach valued L p spaces with p < 1, and such a theory is not well-developed. Furthermore, although the weighted tent spaces T p,q s embed isometrically into L p (X : L q s (X + )) in this range of exponents, their image may not be complemented, and so we cannot easily identify interpolants of their images. 4 We must argue directly.

Lemma 2.11
Suppose x ∈ X, α ∈ (0, ∞), and let all other numerical parameters be as in the statement of Theorem 2.9. Then for all cylindrically supported f ∈ L 0 (X + ) we have and where the infima are taken over all decompositions f = ϕ 0 + ϕ 1 in L 0 (X + ) with ϕ 0 , ϕ 1 cylindrically supported.
Proof We will only prove the equality (25), as the proof of (26) is essentially the same.
Given a decomposition f = ϕ 0 + ϕ 1 in L 0 (X + ), we have a corresponding decom- and likewise for ϕ 1 . This shows that For the reverse inequality, suppose that 1 (x) f = ϕ 0 + ϕ 1 in L 0 (X + ), and suppose f is essentially supported in a cylinder C. Multiplication by the characteristic function 1 (x)∩C does not increase the quasi-norms of ϕ 0 and ϕ 1 in L q s 0 (X + ) and L q s 1 (X + ) respectively, so without loss of generality we can assume that ϕ 0 and ϕ 1 are cylindrically supported in (x). Now let f = ψ 0 + ψ 1 be an arbitrary decomposition in L 0 (X + ), and define Then f = ψ 0 + ψ 1 in L 0 (X + ), and we have and likewise for ψ 1 . The conclusion follows from the definition of the K -functional.

Lemma 2.12
Suppose f ∈ L q c (X + ). Then A q f is continuous.
Proof Let f be essentially supported in the cylinder C := B(c, r ) × (κ 0 , κ 1 ). First, for all x ∈ X we estimate using the estimate (40) from the proof of Lemma 3.1.
For all x ∈ X we thus have Therefore A q f is continuous.
Having established these lemmas, we can prove the following (half-)extension of Proposition 2.6.

Proposition 2.13
Let all numerical parameters be as in the statement of Theorem 2. 9.
Then for all f ∈ L q c (X + ) the function is measurable on X (using the discrete characterisation of the real interpolation quasinorm), and we have We denote the quantity on the right hand side of (20) by f | I p θ ,q Proof First we take care of measurability. Using Lemma 2.11, for x ∈ X we write where the infima are taken over all decompositions |t −ns f (y, t)| q dy dt t n+1 It has already been shown by Barton and Mayboroda that the Z -spaces form a real interpolation scale [4,Theorem 4.13], in the following sense. We will stop referring to the parameters c 0 and c 1 , as Proposition 2.15 implies that the associated quasi-norms are equivalent.

Proposition 2.17
Suppose that all numerical parameters are as in the statement of Theorem 2.9. Then we have the identification Now we know enough to complete the proof of Theorem 2.9.
Proof of Theorem 2.9. First suppose that p 0 , p 1 ∈ (0, 2]. By Corollary 2.16 we have for j = 0, 1, and so Proof Suppose that the atom a is associated with the ball B ⊂ X . When p = ∞, using the fact that B(x, t) ⊂ B whenever (x, t) ∈ T (B) and that −δ 1, p > 0, we have where we used Hölder's inequality with exponent q/ p ≥ 1 in the third line.
When p = q = ∞ the argument is simpler: we have using the same arguments as before (without needing Hölder's inequality). Now we will prove the embedding theorem. Here is a quick outline of the proof. First we establish the first statement for p 0 = 1 and 1 < p 1 ≤ q by using part (1) of Lemma 2.20. A convexity argument extends this to 0 < p 0 < p 1 ≤ q, with q > 1. Duality then gives the case 1 < q ≤ p 0 < p 1 ≤ ∞, including when p 1 = ∞ and α = 0. A composition argument completes the proof with q > 1. Finally, we use another convexity argument to allow for q ∈ (0, 1] (with p 1 < ∞). To handle the second statement, we argue by duality again.

Proof of Theorem 2.19
The proof is split into six steps, corresponding to those of the outline above.
We remark that this technique also yields the embedding T ∞,q

Remark 2.21
The embeddings of Theorems 2.19, at least for p, q ∈ (1, ∞), also hold with Z p,q s replacing T p,q s on either side (or both sides) of the embedding. This can be proven by writing Z p,q s as a real interpolation space between tent spaces Tp ,q s withp near p ands near s, applying the tent space embedding theorems, and then interpolating again. These embeddings can also be proven 'by hand', even for p, q ≤ 1. We leave the details to any curious readers.

T p,∞ -L ∞ Estimates for Cylindrically Supported Functions
The following lemma, which extends [1, Lemma 3.3] to the case q = ∞, is used in the proof that T p,∞ is complete (see Proposition 1.4).

Lemma 3.1
Suppose that X is doubling and let K ⊂ X + be cylindrical. Then for all p ∈ (0, ∞] and all f ∈ L 0 (X + ), Proof When p = ∞ this reduces to which is immediate. Thus it suffices to prove the result for p < ∞. Write K ⊂ B K × (κ 0 , κ 1 ) for some ball B K = B(c K , r K ) ⊂ X and 0 < κ 0 < κ 1 < ∞.

T p,∞ Atomic Decomposition
As stated above, the atomic decomposition theorem for T p,∞ can be proven by combining the arguments of Coifman-Meyer-Stein (who prove the result in the Euclidean case) and Russ (who proves the atomic decomposition of T p,2 (X ) for 0 < p ≤ 1 when X is doubling). First we recall a classical lemma (see for example [22,Lemma 2.2]), which combines a Vitali-type covering lemma with a partition of unity. This is proven by combining the Vitali-type covering of Coifmann-Weiss [ • if i, j ∈ I are not equal, then B(x i , r (x i )/4) and B(x j , r (x j )/4) are disjoint, and • for all i ∈ I , there exist at most M indices j ∈ I such that B(x j , 5r (x j )) meets B(x i , 5r (x i )).
Moreover, there exist a collection of measurable functions {ϕ i : X → [0, 1]} i∈I such that is finite due to the third condition above).
we shall pause to prove the second inequality. Since X + is a Lindelöf space, there exists a countable set J such that