ON FUNCTIONS OF BOUNDED β -DIMENSIONAL MEAN OSCILLATION

A BSTRACT . In this paper, we deﬁne a notion of β -dimensional mean oscillation of functions u : Q 0 ⊂ R d → R which are integrable on β -dimensional subsets of the cube Q 0 :


Introduction
A fundamental result of F. John and L. Nirenberg [17] asserts that if Q 0 is a cube in ℝ d and u ∈ L 1 (Q 0 ) satisfies sup then the integrand in (1.1) enjoys an exponential decay estimate of its level sets: There exist constants c, C > 0 such that for all finite Q ⊂ Q 0 .Here we use u Q to denote the average value of u over Q and all Q are assumed to be finite and parallel to Q 0 .That (1.2) implies (1.1) is an easy consequence of expressing the Lebesgue integral as integration over the level sets, so that somewhat surprisingly (1.1) and (1.2) are equivalent in characterizing what was subsequently termed the space of functions of bounded mean oscillation in Q 0 : Typical examples of functions of bounded mean oscillation include bounded functions, f * log | ⋅ | for any f ∈ L 1 (ℝ d ), and functions in critical Sobolev spaces, though the sense in which any of these examples is generic is itself an interesting question.Clearly bounded functions admit a significant improvement to (1.2), while the latter examples have special properties of their Fefferman-Stein decomposition (originally established in [10,11], with a constructive proof given in [38], special properties demonstrated in [5,12,13], and discussed further in [36]) as well as the well-definedness of restrictions on lower dimensional sets of arbitrarily small dimension with exponential decay of the Hausdorff content of their level sets [1,8,18,20].
The starting point of this paper is the consideration of the exponential decay of the Hausdorff content established in [20] alongside the John-Nirenberg result on the equivalence of (1.1) and (1.2).To illustrate the idea let us consider a simple example, that of W it is a consequence of the results in [20] that such u admits an exponential decay of the β-dimensional spherical Hausdorff content H β ∞ of its level sets for any β ∈ (0, d]: for some c β , C β > 0 and all t > 0, where we recall that and ) is a normalization constant.One is tempted to compare inequalities (1.2) and (1.3), though as only the former is scale invariant it is more appropriate to find a replacement for the latter which satisfies this property, which for the moment we refer to as (1.2   ).In the identification of the existence of such an inequality the idea of the paper emerges: The equivalence of (1.1) and (1.2) and the parallel structure of (1.2) and this (1.2   ) prompts one to wonder whether there is a counterpart of (1.1) involving the β-dimensional Hausdorff content which is equivalent to (1.2   ).The main contribution of this paper is the pursuit of these ideas, and in particular, to give a definition of this counterpart (1.1   ), the β-dimensional mean oscillation, such that functions for which this quantity is bounded admit suitable analogues to properties known in the classical theory of functions of bounded mean oscillation.The relationships we identify can be represented schematically as while our results comprise the discovery of the existence and equivalence of the bottom portion of the diagram.
A posteriori one finds that (1.2  ) is given by for some c  β , C  β > 0, all t > 0, all Q ⊂ Q 0 parallel to Q 0 , and suitable c Q ∈ ℝ, while to define (1.1  ), we first recall a few relevant notions related to β-dimensional integrability of functions f : Q 0 ⊂ ℝ d → ℝ.We say that a function f is H which enables us to define, for a signed With this preparation we now introduce the vector space It can be shown that (1.5) is a quasi-norm, that there exists a norm equivalent¹ to (1.5) such that L1 (Q 0 ; H β ∞ ) equipped with this equivalent norm is a Banach space, that the space of continuous and bounded functions in Q 0 for which this equivalent norm (or alternatively (1.5)) is finite is dense in this Banach space, and that the topological dual of this space can be identified with the Morrey space where M loc (Q 0 ) is the set of locally finite Radon measures in Q 0 (and extended by zero outside Q 0 ), see Section 2 below.For the present we only mention one further consequence of these facts, which follows from this duality, the validity of the formula where the supremum is taken over all finite subcubes Q ⊂ Q 0 parallel to Q 0 .We then define the space of functions of bounded β-dimensional mean oscillation in Q 0 : As an almost immediate consequences of the definition, let us record two theorems of interest.First, the space BMO β (Q 0 ) is closed under composition with Lipschitz functions, a result we record in: Second, by the functional property (1.6), for β = k ∈ ℕ, such functions restrict to k-dimensional hyperplanes and are of bounded mean oscillation on these subspaces, a result we record in: Theorem 1.2 and the classical argument of John and Nirenberg together imply that u ∈ BMO k (Q 0 ) satisfies (1.2) on k-hyperplanes.Our main result is that for any β ∈ (0, d], and for any β-dimensional set, these spaces satisfy an analogue of (1.2): for every t > 0, u ∈ BMO β (Q 0 ), all finite subcubes Q ⊂ Q 0 parallel to Q 0 , and where c Q ∈ ℝ is given by equation Our proof of Theorem 1.3 follows the argument of John and Nirenberg, with suitable modifications necessitated by the replacement of the Lebesgue measure with the Hausdorff content, which is not a measure but only an outer measure with some additional properties.More precisely, the actual arguments in our proof are via a more convenient, though equivalent (see Section 2), quantity, the dyadic Hausdorff content adapted to any given cube Q ⊂ Q 0 .This requires several results for this object that may be of independent interest, including a weak form of the Lebesgue differentiation theorem we prove in Corollary 3.3 and a Calderón-Zygmund decomposition we establish in Theorem 3.4, see Section 3 for further details.These results parallel the standard ones for measures and make use of the fact that the Hausdorff content is in a certain sense doubling.There are some difficulties, however, due to the nonlinearity of the Choquet integral, which we handle through the use of an additional covering lemma, see Proposition 2.11 in Section 2 below.Theorem 1.3 has a number of interesting consequences.First, a standard application of inequality (1.7) is an exponential integrability result.In our setting, we have the following corollary which asserts that functions in BMO β (Q 0 ) are exponentially integrable on sets of dimension β.
for every u ∈ BMO β (Q 0 ) with ‖u‖ BMO β (Q 0 ) ≤ 1 and for all finite subcubes Q ⊂ Q 0 parallel to Q, and where c Q ∈ ℝ is given by equation (4.3) in Lemma 4.2.
Second, a classical result for the space of functions of bounded mean oscillation is the equivalence of a family of semi-norms indexed by p ∈ [1, ∞).A similar result for the space of functions of bounded β-dimensional mean oscillation follows easily from Theorem 1.3 or Corollary 1.4.That is, if we introduce 1 p , we will show: for all u ∈ BMO β (Q 0 ).
Third, from Theorem 1.3 one obtains a nesting of the spaces, a result we record in: Remark 1.7.The preceding results imply that functions in BMO α (Q 0 ) are exponentially integrable on sets of every dimension β ∈ [α, d] and that for k ∈ ℕ, k ≥ β, such functions restrict to k-dimensional hyperplanes and are of bounded mean oscillation in these hyperplanes with respect to the natural measure.This gives another perspective of the question of restriction of functions of bounded mean oscillation addressed in [19,23].
Finally, we return to the impetus for the initial consideration of these quantities, the question of improvements to critical Sobolev embeddings studied in [20].An interesting case which was not treated there is the Riesz potential I α acting on the Morrey space M d−α (ℝ d ).This space contains L d/α,∞ (ℝ d ) (the weak L d/α (ℝ d ) space), whose image under the Riesz potential was shown to exhibit linear exponential decay of the Hausdorff content of its level sets.Theorem 1.3 and the following refinement of D. R. Adams' BMO estimate [2] together imply that one has the same linear exponential decay of Hausdorff content of the level sets of I α μ for μ ∈ M d−α (ℝ d ), provided the dimension of the content is strictly larger than d − α.
The local capacitary integrability assumption ) for all finite cubes Q ⊂ ℝ d is analogous to D. R. Adams' local Lebesgue integrability assumption [2, Proposition 3.3 on p. 770], without which it is clear that one cannot obtain ).However, one wonders whether it is necessary to assume this or if it can be obtained from the other hypotheses, which we here ask as: The answer is no when ϵ = 0, as we demonstrate in the following examples.First, in the simpler case α = k ∈ ℕ, with the choice μ = H d−k | S for a portion of a (d − k)-dimensional hypersurface S, one finds that |I k μ| behaves locally like |log(d(x, S))| where d(x, S) is the distance to the surface (see [35] for the case α = 1 and [29] for the case α = d − 1, analogous estimates can be made for the other cases).In these cases |I k μ| = +∞ on the (d − k)-dimensional hypersurface S and therefore for any Q containing a portion of S. When α is not an integer, the argument is slightly more involved.In particular, fixing d − α ∉ ℕ one can apply a construction of J. Hutchinson [16] to generate a d − α regular measure μ supported in a cube Q 0 , i.e. μ which satisfies . Indeed, if this were the case, one would have However, this would imply that the exists and is finite for μ almost every x ∈ ℝ d , which along with the upper and lower density conditions (1.8), would yield by [21,Theorem 4.5] that d − α is an integer, a contradiction.Therefore This does not rule out the possibility that μ which happen to satisfy this property might enjoy a similar conclusion to that of Theorem 1.8, which is our: Before discussing the plan of the paper, a few remarks are in order on the study of these spaces BMO β (Q 0 ) and their relationship with the results of Jean Van Schaftingen [40].The observation that there are spaces of interest between critical Sobolev spaces and BMO is the subject of Jean Van Schaftingen's paper [40].In particular, he there utilizes endpoint L 1  inequalities pioneered by J. Bourgain and H. Brezis in [6,7], and to which he contributed [39][40][41][42], as a starting place to define such spaces.The spaces he studies are the dual of closed k-forms with integrable coefficients for k = 1, . . ., d − 1, a discrete family of spaces for which he establishes various properties in a similar spirit to what we record here, including nesting in his Theorem 3.1, extension in his Theorem 3.2, restriction in his Theorem 3.4, and exponential integrability in his Proposition 5.6.For Van Schaftingen's spaces, there are still a number of open questions, including whether they are closed under composition with Lipschitz functions and admit restriction/exponential integrability up to the endpoint (see [40, end of Section 5 on p. 18 and Section 6]), results which we can settle for the spaces defined in this paper.It seems an interesting problem to compare the spaces he considers and the ones we define here for suitable choices of k and β.
The plan of the paper is as follows.In Section 2, we recall some necessary preliminaries related to the Hausdorff content and Choquet integrals.In Section 3, we establish results for the dyadic Hausdorff content adapted to an arbitrary cube Q which are analogous to results known for the Lebesgue measure.This includes a weak-type estimate for a dyadic Hausdorff content maximal function in Theorem 3.1, a weak form of the Lebesgue differentiation theorem in Corollary 3.3, and a Calderón-Zygmund decomposition in Theorem 3.4.In Section 4, we use the results up to this point in the paper to prove Theorem 1.3 and then from this theorem we deduce Corollaries 1.4, 1.5, and 1.6.Finally, in Section 5 we prove the remainder of the results asserted in the introduction, Theorems 1.1, 1.2, and 1.8.

Preliminaries
The purpose of this section is to collect some necessary preliminaries concerning the dyadic Hausdorff contents adapted to cubes, the Choquet integrals defined in terms of these contents, and the function spaces generated by these integrals.We rely on results established in [3, 4, 24-26, 32, 33], see also [14,15,27,28,30,31,34] for other results concerning Choquet integrals.The subtle point is that these objects are not measures, so that integration with respect to them is a delicate matter.We begin with some general definitions concerning set functions, and to this end we let P(ℝ d ) denote the class of all subsets ℝ d .
Definition 2.1.We say that a set function Remark 2.2.In the work of D. R. Adams [3,4], outer measures are also referred to as capacities in the sense of N. Meyers.
For such set functions, the monotonicity property allows us to define a notion of integration: Definition 2.3.Given an outer measure H, we define the Choquet integral of a non-negative function f over a cube Q by where the right-hand side is understood as the Lebesgue integral of the function The Choquet integral is not linear, and not even sublinear, unless one assumes additional properties of H.We here introduce two assumptions that have been used to establish sublinearity.
Definition 2.4.We say that a set function for all E, F ⊂ ℝ d .
Definition 2.5.We say that a set function for every increasing sequence of arbitrary sets If H is a set function which satisfies the preceding assumptions, the Choquet integral of such a set function is sublinear and satisfies a monotone convergence theorem (see [33, Proposition 3.2]): Proposition 2.6.Suppose H is an outer measure which is strongly subadditive and continuous from below.Then the Choquet integral with respect to H satisfies ( (3) For non-negative functions f n , Property (3) asserts that under such assumptions one has a sublinear integral.When restricted to a suitable class of functions, this will allow us to define a norm.To this end, we next introduce the notion of H-quasicontinuity.We are now prepared to introduce a norm on the space of H-quasicontinuous functions.For a signed H-quasicontinuous function f we define and we let 3) The fact that this object is a norm is one part of the following theorem on the properties of this space which are derived from the assumptions on H. Theorem 2.8.Suppose H is an outer measure which is strongly subadditive and continuous from below, Q 0 ⊂ ℝ d is a cube, and let L 1 (Q 0 ; H) be as defined in (2.3).Then: (3) The space of continuous and bounded functions in Q Proof.It follows from Proposition 2.6 above ([33, Proposition 3.2]) that under these assumptions the integral is sublinear, so that the rest of the assertions follow from [24, Proposition 2.2 and 2.3].
With these results collected, we next introduce the outer measures of relevance for our consideration.First, we again recall the definition of the spherical Hausdorff content where we recall the definition of the dyadic Hausdorff content subordinate to Q: Here D(Q) denotes the set of all dyadic cubes generated by the cube Q, where we follow the convention of [32,33] with cubes taken as half-open.For example, in the case Q = [0, 1) d , this is the usual dyadic lattice and in this case H Hβ ∞ , the usual dyadic Hausdorff content.While the spherical Hausdorff content and dyadic Hausdorff content adapted to a cube are both outer measures, the spherical Hausdorff content may fail to be strongly subadditive.This is not the case for the dyadic content, which satisfies the necessary properties to generate a well-behaved Choquet integral.In particular, a suitable modification of [33, Proposition 3.5 and 3.6] to the dyadic lattice generated by Q yields: Proposition 2.9.The map H β,Q ∞ is strongly subadditive and continuous from below.
Therefore as a consequence of Theorem 2.8, the function space of H β,Q ∞ -quasicontinuous functions for which norm (2.3) is finite is complete and enjoys approximation by bounded continuous functions.The motivation for introducing the spherical content here is that it is a useful tool to move between the various dyadic contents, as one has their equivalence as set functions.The following result asserting this is proved in [43,Proposition 2.3] in the usual dyadic case, though the argument extends to a general cube by rotation and dilation.

Proposition 2.10. There exist a universal constant C
for all E ⊂ ℝ d .
The preceding proposition shows that all of the dyadic Hausdorff contents are equivalent with a constant independent of the cube, so that if one works in a fixed cube Q 0 , the functional spaces ∞ ) are all equivalent (though with different norms).Therefore if we introduce we have finally collected all of relevant references of several facts claimed in the introduction -that (2.4) is a norm which is equivalent to the quasi-norm (1.5), that ) is a Banach space, and that the space of functions in The last of the claims to address is the duality asserted in the introduction, that the dual of This is a result of D. R. Adams from [3, Proposition 1 on p. 118] (see also the detailed argument presented by H. Saito and H. Tanaka in [32]).From this duality the formula (1.6) follows from the Hahn-Banach theorem.This implies an important consequence we appealed to in the introduction, that for functions f ∈ L 1 (Q 0 ; H β,Q 0 ∞ ) one has an upper bound for the integral of |f| with respect to any β-dimensional growth measure ν: There exists a constant Another important consequence of the equivalence of the spherical Hausdorff content with any of the dyadic Hausdorff contents is that one can replace the semi-norm on BMO β (Q 0 ) with a quantity which simplifies the proofs, namely ‖u‖ (2.5) Indeed, we utilize this semi-norm in the sequel in the argument of Lemma 4.2, which is slightly more general than Theorem 1.3 in that it can be applied to infinite cubes.This is in the spirit of the program instituted in John and Nirenberg [17], analogous to their Lemma 1'.The semi-norm (2.5) also has the advantage that some of the quantities become more transparent, e.g. that in Theorem 1.3 and Corollary 1.4 one can take and that this infimum is attained.Our choice to give definitions and the theorems stated in the introduction in terms of the spherical Hausdorff content stems from our impression that this object is more common in the literature and therefore more accessible to the reader.
Finally, we record a small modification of a lemma from [26] which is attributed there to Mel'nikov [22].In particular, an inspection of the proof in [26] yields the following statement.

Proposition 2.11. Suppose {Q j } is a family of non-overlapping dyadic cubes subordinate to some cube lattice D(Q).
There exists a subfamily {Q j k } and a family of non-overlapping ancestors Qk such that: (1) We have (2) We have ∑ Condition ( 2) is sometimes called a packing condition, an important consequence of which is the existence of a constant C  > 0 such that for every non-negative

Capacitary differentiation theorems
The purpose of this section is to establish for the dyadic Hausdorff content adapted to an arbitrary cube some analogues of classical results known for the Lebesgue measure.
The first result of this section is an analogue of D. R. Adams' result [3] on the boundedness of the Hardy-Littlewood maximal function on the space for the dyadic β-Hausdorff maximal function associated to H Note that the equivalence of all of the Hausdorff contents implies that In particular, we prove: for every family of non-overlapping dyadic cubes {Q k } subordinate to Q which satisfy The argument follows the standard approach for maximal functions, along with some modifications typical to replacing the Lebesgue measure with the Hausdorff content, see e.g.[26,33].We include the proof for the convenience of the reader.
Proof.Without loss of generality we suppose f is non-negative.By the definition of the dyadic maximal function Let {Q j } be the maximal collection of these dyadic cubes, each Q j of which satisfies (3.1) and let {Q j k } and { Qk } be the families of cubes obtained by an application of Proposition 2.11 to this family.By only selecting necessary cubes, Q j k ̸ ⊆ Qm for some Q m in the original family with Qm selected as a covering cube, we have However, since Q j k satisfies (3.1) we find that Finally, the packing condition satisfied by which gives the desired conclusion.
We next establish a precise estimate for upper density of the dyadic Hausdorff content.Our argument follows that of Evans and Gariepy [9, p. 74], where we observe that the use of the dyadic content yields a slightly better result than the upper and lower bounds one has for the upper density of the spherical content.
Lemma 3.2.Let E ⊂ ℝ n .Then lim sup for H β -a.e.x ∈ E, where the limit is taken over all dyadic cubes Q  ∈ D(Q) containing x as l(Q  ) tends to 0.
Proof.As density is a local result, by taking E ∩ Q for a sufficiently large cube Q, we may assume E is bounded and therefore H β,Q ∞ (E) < +∞.For every δ > 0 and 0 < τ < 1, we define We claim that H β (E(δ, τ)) = 0 for every δ > 0 and 0 < τ < 1.To this end, we introduce and observe that H β,Q δ is an outer measure.Thus for every collection of non-overlapping dyadic cubes and l(Q i ) ≤ δ, we have by countable subadditivity, monotonicity, and the fact that Thus, taking the infimum over all such families of cubes in the preceding inequality we deduce that E(δ, τ)) = 0 and therefore H β (E(δ, τ)) = 0 (which follows from [9, Lemma 1 on p. 64] and the equivalence of the dyadic and spherical contents, for example), which proves the claim.Now suppose that x ∈ E and lim sup Then using the fact that the inequality is strict and the definition of the limit supremum, we can find a constant Using the fact that for any k ∈ ℕ such that 1/k < δ x .In particular, we have the set inclusion {x ∈ E : lim sup As the sets on the right-hand side have H β measure zero, we deduce and the proof is complete.
Next, we establish a weak Lebesgue differentiation theorem for the Hausdorff content: ∞ ) be non-negative.Then for H β -a.e.x ∈ ℝ d , we have lim sup Proof.We first prove that for H β -a.e.x ∈ ℝ d , we have lim sup For ϵ > 0, we define Note that inequality (3.3) will be verified once we show that H In particular, we have by subadditivity of the integral Since f k is continuous for each k, we have lim sup It follows that for every ϵ > 0, we have the set inclusion Finally, an application of Theorem 3.1 to A As the Hausdorff content and Hausdorff measure have the same null sets [9, Lemma 1 on p. 64], this completes the proof of (3.3).We next show that for H β -a.e.x ∈ ℝ n , we have lim sup (3.4) > t} for every t > 0. Then for every x ∈ L t , we have lim sup An application of Lemma 3.2 yields lim sup That is, for every t > 0, we have In particular, we observe that {x ∈ ℝ d : lim sup where ℚ + denotes the set of all positive rational numbers.Using the subadditivity property of H β , we conclude that As we show, this one-sided Lebesgue differentiation theorem is sufficient to obtain a Calderón-Zygmund decomposition of these spaces: there exists a countable collection of non-overlapping dyadic cubes {Q k } ⊂ Q subordinate to Q such that: Proof.Let {Q i } be the collection of all the dyadic cubes contained in Q which satisfies which is the first claimed property.Toward the second claimed property, for In particular, by Theorem 3.3, we obtain that |f(x

Main result and several consequences
The goal of this section is to prove Theorem 1.3 and then use it to deduce the several corollaries from the introduction.We begin with an exponential decay lemma which is a slight variant of a result from the paper of John and Nirenberg [17].
Iteration of the choice s = C  e in inequality (4.1) a total of m times yields In particular, using the monotonicity properties of the exponential and c β > 1 we find for t ≥ c β , so that the choice gives the desired conclusion for t ≥ c β .Theorem 1.3 is an almost immediate consequence of the following analogue of [17, Lemma 1']: Lemma 4.2.There exist constants c, C > 0 such that Proof.We assume without loss of generality that c Q = 0 and ‖u‖ . Let F(t) be the smallest function for which it only remains to show that , from which the desired inequality follows by an application of Lemma 4.1.The reduction that c Q = 0 and ‖u‖ In particular, if c −1 β t ≥ s ≥ 1, λ = s is an admissible height for an application of Theorem 3.4 to u to obtain a countable collection of non- An application of Proposition 2.11 to this family yields a subfamily {Q j k } and non-overlapping ancestors { Qk } which satisfy: (1) We have (2) We have ∑ , for each dyadic cube Q.
(3) For each Qk , l( Qk ) β ≤ ∑ As before, we assume we avoid redundancy in this covering so that if Q j k ⊂ Qm for some k, m we do not utilize the cube Q j k .The fact that t ≥ c β s > s where N is a null set with respect to H β (and therefore also H As the inclusion is preserved if we intersect with {x ∈ Q : |u(x)| > t}, we have so that by subadditivity of the Hausdorff content we can estimate Next, for We have, by subadditivity of the dyadic Hausdorff content and using s ≥ 1, where we utilize that there is no redundancy in cubes in the collections.Next, we recall that {Q j k } is a subcollection of cubes from the Calderón-Zygmund decomposition and therefore each The use of this inequality in the preceding chain of inequalities leads to the estimate This completes the proof of the claim and therefore the theorem.
Theorem 1.3 is an immediate consequence of Lemma 4.2, when one takes into account the constants computed in Lemma 4.1: Proof.From the proof of Lemma 4.2 we have are the same constants as above derived in Lemma 4.1.It only remains to replace the dyadic Hausdorff content adapted to Q with the Hausdorff content, which by the equivalence recalled in Section 2 yields the desired estimate for all t > 0 with the constants This completes the proof of Theorem 1.3.

Consequences of the John-Nirenberg inequality
We next give the proof of Corollary 1.4.
Proof.From Theorem 1.3 and the fact that ‖u‖ BMO β (Q 0 ) ≤ 1, we have the estimate Thus, for c  > 0 we find In particular, the claimed estimate holds for any c  < c and with constant We next give the proof of Corollary 1.5.
Proof.One direction is straightforward: The pseudo-Hölder inequality satisfied by the Hausdorff content [15, (C7) , from which the desired inequality follows from taking the infimum in c and then the supremum over Q ⊂ Q 0 .For the other direction, Corollary 1.4 asserts is admissible for application of this corollary, it follows from the Taylor expansion of the exponential that, for any The result then follows dividing by l(Q) β , raising both sides to the 1 p power, and the observation that We next prove Corollary 1.6.
β α and the exponential decay proved in Theorem 1.3 we have )) As the infimum over c is smaller than the left-hand side, the result follows from dividing by l(Q) β and taking the supremum over finite subcubes Q ⊂ Q 0 parallel to Q 0 .

Composition, restriction, and a Sobolev embedding
We first prove Theorem 1.1.
Proof.The fact that ϕ(0) = 0 implies which along with the stability of quasicontinuity under composition with Lipschitz functions implies that ϕ But this shows inf We next prove Theorem 1.2.
Proof.Let u ∈ BMO k (Q 0 ) and let H k denote be k-dimensional hyperplane for which , which in turn by the functional property (1.6) implies where we use L k to denote the k-dimensional Lebesgue measure in the k-dimensional space Q 0 ∩ H k .In fact, when applied to sets within H k , the spherical Hausdorff content H k ∞ is precisely the k-dimensional Lebesgue measure L k .Therefore Let u Q∩H k to denote the average of u| Q 0 ∩H k over the set Q ∩ H k with respect to L k .Then by the usual trick for BMO, adding and subtracting c, we have k  , so that desired inclusion BMO k (Q 0 ) ⊂ BMO(Q 0 ∩ H k ) then follows from the combination of the two preceding inequalities and taking the supremum over finite subcubes Q ⊂ Q 0 which are parallel to Q.
In the case k = d, the preceding argument again yields the inclusion BMO d (Q 0 ) ⊂ BMO(Q 0 ), where one observes that there is no need to even define the restriction.Moreover, as H d ∞ = L d , the other direction follows easily in the same spirit: from which the inclusion BMO(Q 0 ) ⊂ BMO d (Q 0 ) follows from taking the supremum over finite subcubes Q ⊂ Q 0 which are parallel to Q.
We conclude the paper with a proof of Theorem 1.8.
Proof.We now show that I α μ ∈ BMO d−α+ϵ (ℝ d ) for ϵ ∈ (0, α).To this end, let Q ⊂ ℝ d be any finite cube and for μ ∈ M d−α (ℝ d ) we let 2Q be the cube with the same center and twice the side length and define Then for any ν ∈ M d−α+ϵ (ℝ d ) with ‖ν‖ M d−α+ϵ ≤ 1 and any c ∈ ℝ we have Concerning I, we have the bound However, a dyadic splitting yields  As the infimum over c is smaller than the left-hand side, the claim then follows by taking the supremum over cubes Q.This completes the proof.

β∞
-quasicontinuous if for every ϵ > 0, there exists an open set O such that H β ∞ (O) < ϵ and f| O c is continuous.For a non-negative H β ∞ -quasicontinuous function f , we define the Choquet integral of f in Q 0 via the formula