CAPACITARY FUNCTION SPACES

. These notes are devoted to the analysis on a capacity space, with capacities as substitutes of measures in the study of function spaces. The goal is to extend to the associated function lattices some aspects of the theory of Banach function spaces, to show how the general theory can be applied to classical function spaces such as Lorentz spaces


Introducction
The purpose of this paper is to present some basic developments connected with properties of function spaces defined on capacity spaces, instead of measure spaces.It is our feeling that these developments, because of their relations with important aspects of mathematical analysis on one hand and their simple and basic character on the other, deserve to be widely known.
The emphasis of our exposition is placed upon the study of the essential functional analytic elements such that a satisfactory theory can be developed in the context of quasi-Banach spaces.One of the main problems is that we are forced to work with a non-additive integral, the Choquet integral, so that the dual spaces are not easily identifiable and some basic properties, such as the dominated convergence theorem, are not longer available.
In the literature, a capacity on a space Ω is usually supposed to be an increasing set function C : Σ → [0, ∞], with Σ a family of subsets in Ω, with different properties depending on the context, and the Choquet integral is defined as if f ≥ 0 is a measurable function in the sense that {f > t} ∈ Σ for every t > 0.
In many important examples of capacities the domain Σ of C is a σalgebra.This is the case of the variational capacities, and of the Fuglede [18] and Meyers [21] capacities of nonlinear potential theory.They are countably subadditive set functions on all subsets of R n which include the Riesz and the Bessel capacities.Although they are not Caratheodory metric outer measures, they satisfy a Fatou type condition and, by a general theorem due to G. Choquet (cf.[16, Chapter VI]), every Borel set B ⊂ R n is capacitable, this meaning that sup{C(K); K ⊂ B, K compact} = C(B) = inf{C(G); G ⊃ B, G open}.
Then the class of all Borel sets turns out to be a convenient domain for all of them.We refer to [2] and [20] for an extended overview of these capacities.
Another well known class of capacities are the Hausdorff contents.If h is a continuous increasing function on [0, ∞) vanishing only at 0, which is called a measure function in [11], denote µ h the corresponding Hausdorff measure on R n , and let I or I k represent a general cube in R n with its sides parallel to the axes.The use of the corresponding Hausdorff capacity or Hausdorff content, is often more convenient than µ h , and E h (A) = 0 if and only if µ h (A) = 0.If h(t) = t α (α > 0), it is customary to write H ∞ α instead of E h , and this capacity is called the α-dimensional Hausdorff content.The case h(x) := x log(1/x) on [0, 1/e] corresponds to the Shannon entropy considered in [17].
New examples appear when studying interpolation properties of function spaces as in [15].If E is a quasi-Banach function space on the measure space (Ω, Σ, µ), then defines a capacity and, as in the case of Hausdorff capacities, there is a measure µ such that C E (A) = 0 if and only if µ(A) = 0.
The goal of these notes is to clearly set the basic properties of the capacity spaces (Ω, Σ, C) and their associated Lebesgue spaces L p (C) and L p,q (C), to show how the general theory can be applied to function spaces such as classical Lorentz spaces, and to complete the real interpolation theory for these spaces started in [15] and [14].
Further applications of the use of these capacities will appear in forthcoming work.In [3] it will be shown how they are a useful tool to extend the Riesz-Herz estimates concerning the Hardy-Littlewood operator.
The notation A B means that A ≤ γB for some absolute constant γ ≥ 1, and A B if A B A. We refer to [7] for general facts concerning function spaces.

Capacitary function spaces
Let (Ω, Σ) be a measurable space.Sets will always be assumed to be in the σ-algebra Σ and functions will be real mesurable functions on (Ω, Σ).
From now on, by a capacity C we mean a set function defined on Σ satisfying at least the following properties: we say that the capacity is subadditive.
If C is a capacity on Σ, we will say that (Ω, Σ, C) is a capacity space.It will play the role of a measure space (Ω, Σ, µ) in the theory of Banach function spaces.We are going to check which of the properties for measure spaces are still satisfied by capacity spaces.
The distribution function C f and the nonincreasing rearrangement f C are defined as in the case of measures by and Many of the basic properties remain true in this capacitary setting.The following ones are easily proved: Indeed, let In particular, (f A property is said to hold quasi-everywhere (C-q.e. for short) if the exceptional set has zero capacity.
Pointwise convergence satisfies f dC = 0 if and only if f = 0 C-q.e. and it is positive-homogeneous, Moreover, by Fubini's theorem, The relation {f + g > t} ⊂ {f > t/2} ∪ {g > t/2} shows that this integral, defined on nonnegative functions, is quasi-subadditive with constant 2c, Observe that, if f = g C-q.e. and C is subadditive, then This will be also true if C(A n ) → C(A) whenever A n ↑ A. In this case we say that C has the Fatou property (or that it is a Fatou capacity).
If C is a Fatou capacity, the countable unions of C−null sets are also We consider equivalent two functions, f and g, if they are equal C-q.e.In this case |f | dC = |g| dC, since C{|f | > t} = C{|g| > t} for every t ≥ 0. Thus, |f | dC = 0 if and only if f = 0 C-q.e.
Note that if a Fatou capacity is subadditive, then it is σ-subadditive.
The Fatou property can be presented in several equivalent ways: Theorem 1.The following properties are equivalent: Proof.(c) follows from (b) and (3), and (a) follows from (c) by taking and, by (a), and (b) follows: Moreover, (d) implies (a) by taking f n = χ An and f = χ A .Finally, suppose now that C satisfies (a) and that 0 Hence, in the subadditive case (in the general case the proof is the same but the constant 2c from (4) has to be included), we have The Minkowski inequality (6) follows from (5) in the usual way.
One could wonder if these estimates are always true with constant 1.We will see in Section 4 that subadditivity holds only if C is concave.It is easily checked that Hölder's inequality is always true for sets, since but the following example shows that it is not longer true for functions: Example 1.Consider the "Lorentz-type" capacity C(A) := |A| 0 w(t) dt on (0, 1) with w(t) = tχ (0,1) (t), and the functions (on (0, 1)). Then Just note that Hence, there is no hope to obtain the Hölder and Minkowski inequalities with constant 1 in the general case.We do not know whether the subadditivity of the Choquet integral is a necessary condition to get Hölder's estimate with constant 1.

Lebesgue capacitary spaces
From now on, C will represent a Fatou capacity on (Ω, Σ) and c ≥ 1 its subadditivity constant.
In this section we study the completeness of the spaces L p,q (C) (p, q > 0) defined by the condition Observe that f L p,q (C) = 0 if and only if f = 0 C-q.e. and equivalent functions (in the sense of C-q.e. identity) have the same We write As for function spaces, there are several descriptions of these "norms": and the results follow from the estimates for and f + g ≤ (2c) 1/p 2 (1−p)/p ( f + g ).Now, recall that if • is a quasi-seminorm with constant c ≥ 1 and (2c) = 2 then, by Aoki's theorem (cf.Section 3.10 of [8]), and it follows that In the special case f i = χ A i and p = 1 we obtain We say that Note that if the sequence {f n } converges in capacity, then it is a Cauchy sequence in capacity, that is, for every > 0, C{|f p − f q | > } → 0 as p, q → ∞.The converse is also true: Theorem 5. A sequence {f n } is convergent in capacity to a function f if and only if it is a Cauchy sequence in capacity.In this case, the sequence has a subsequence which is C-q.e. convergent to f .Proof.If {f n } is a Cauchy sequence in capacity, then there exists . By (8) and the Fatou property, By the Fatou property Since {f n } is a Cauchy sequence in capacity which has a subsequence which is convergent in capacity to f , {f n } converges also to f in capacity.
The topology and the uniform structure of L p (C) are given by the metric d(f, g) := f − g * , where • * is associated to • L p (C) as in (7).
Proof.We follow some usual arguments of measure theory combined with (9): Therefore, there exists and C(A) = 0 since, by (9), for n large enough.
The proof of completeness of L p (C) can be easily adapted to show that all L p,q (C)-spaces are also complete.
Remark 1.The absence of additivity for the Choquet integral makes it difficult to give a description of the dual of L p (C). See for instance [1, Section 4], where duality in the case of Hausdorff and Bessel capacities is studied.
If p is the conjugate exponent of p ∈ [1, ∞], Hölder's inequality shows that every g ∈ L p (C) + defines a functional u g (f ) := f g dC which is homogeneous and bounded on L p (C) + , , but in general u g is not additive.

Subadditivity
The Choquet integral is subadditive on sets, Then the Choquet integral is also subadditive on nonnegative simple functions.These facts were proved by Choquet in [16] (see also [15] or [14] for a direct elementary proof).
In this case C is said to be strongly subadditive or concave.
Variational capacities and those of Fuglede and Meyers are examples of concave capacities.Shannon entropy is concave if n = 1, but not if n > 1 (see [17]).In the case of the entropies C E associated to Banach function spaces, examples and counterexamples of concave capacities are given in [15].
Concave capacities give rise to normed L p -spaces, since the Minkowski inequality holds with constant 1, and a natural question is to determine when, for a non-concave capacity C, L p (C) is normable, this meaning that there exists in L p (C) a norm which is equivalent to As for usual Lorentz spaces, one could try to substitute f C by and the theorem follows.
We do not have a satisfactory sufficient normability condition.Let us see a restrictive one, which extends a known result for classical Lorentz spaces.
In the rest of the section µ represents a measure on (Ω, Σ) such that µ(Σ) = [0, µ(Ω)] ⊂ [0, ∞], and we will suppose that C is µ-invariant, this meaning that A capacity C on (Ω, Σ) will be said to be quasi-concave with respect to µ if there exists a constant γ ≥ 1 such that, whenever µ(A) ≤ µ(B), the following two conditions are satisfied: Example 2. If J : [0, µ(Ω)] → R is an increasing function such that J(t)/t is decreasing, then it is readily seen that C(A) := J(µ(A)) defines a µ-invariant and quasi-concave capacity with respect to µ.For instance, C(A) := ϕ X (µ(A)) when ϕ X is the fundamental function of an r.i.space.Note that ϕ X is a quasi-concave function.
Theorem 8.If the capacity C is µ-invariant and quasi-concave with respect to µ, then Proof.It is clear that C(A) ≥ 0 and it is readily seen that C is increasing.Let us show that Obviously C(A) ≤ C(A).On the other hand, if ε > 0 is given, we can find and ( 10) follows.
We claim that, if x, y ≥ t > 0, then and the concavity of C follows by taking t = µ(A ∩ B), x = µ(A) and y = µ(B), since then To prove the claim, we may assume that 0 < t < x ≤ y and write Finally, by addition we obtain (12).Since C is concave, it is also subadditive.Let > 0 and A, B ∈ Σ.There exists {A n } with A n ↑ A such that lim n→∞ C(A n ) ≤ C(A) + .Since C C then there exists c > 0 such that for all C(A) ≤ c C(A) for every set A; and hence, , since C has the Fatou property.We get the equivalence of the capacities with the equivalence of C to C. Moreover there exist increasing sequences {A n } and {B n } such that Assume that C(A) + C(B) < ∞.By the concavity of C, and then, from the definition of C, since A n ∪B n ↑ A∪B and A n ∩B n ↑ A∩B, we obtain and the concavity of C follows.
To prove that C has the Fatou property, if {A n } with A n ↑ A, we need to prove C(A) ≤ lim n→∞ C(A n ) if this limit is finite.For each n, there exist Indeed, in this case C(A) := χ A * is a µ-invariant capacity, C(A) C(A) and C is concave: We can suppose 0 ≤ µ(A ∩ B) < µ(A) ≤ µ(B) and define ϕ(µ(A)) := C(A).
Let t = µ(A ∩ B), x = µ(A) and y = µ(B), so that 0 < t < x ≤ y and ϕ(x + y − t) + ϕ(t) ≤ ϕ(x) + ϕ(y).In particular, if m ∈ N and r > 0, then ϕ(mr) ≤ mϕ(r).Moreover, if a ≤ b, then there exists m ≥ 2 such that (m − 1)a ≤ b ≤ ma and But x ≤ y and there is some m ∈ N such that which means that C is quasi-concave respect to the measure µ, with constant γ = 2. Since C C, C is also quasi-concave with respect to µ.

Interpolation
If Ā = (A 0 , A 1 ) is a couple of quasi-Banach spaces, 0 < θ < 1 and 0 < q ≤ ∞, the interpolation space Āθ,q is the quasi-Banach space of all We refer to [8] and [10] for general facts concerning interpolation theory.
Here we are wishing to extend the results on real interpolation of capacitary L p (C)-spaces included in [15] and [14], where the capacities were supposed to be concave and p ≥ 1, since only Banach couples were allowed.
The capacities will be still supposed to be Fatou but the Choquet integral will not be necessarily subadditive anymore, and 0 < p < 1 is also allowed.
For the sake of completeness we include the details of the proof of the estimates of K(t, f ) = K(t, f ; L p (C), L ∞ (C)) similar to those of [15] and [14].
and consider Hence, For the reverse estimate we use that there exists Ω f (t) ⊂ Ω such that Since f 0 , f 1 are disjointly supported, χ {f >y} = χ {f 0 >y} + χ {f 1 >y} , and To prove that also K(t, f ) Conversely, consider f = g + h with g ∈ L p (C) and h ∈ L ∞ (C).Then and then Once we have the description of K(t, f ), real interpolation follows easily as in [8], Theorem 5.2.1: Proof.We know from Theorem 9 that ∞ 0 y p−1 min(C{f > y}, t p )dy t p 0 f C (y) p dy so that, using the integral Minkowski inequality (q/p 0 ≥ 1), Conversely, where we have used that f C is decreasing.
We also want to consider interpolation with change of capacities.Let (C 0 , C 1 ) be a couple of capacities on the same measurable space with the same null sets.Examples are given by Hausdorff capacities and the corresponding Hausdorff measures, or by couples of capacities associated to quasi-Banach spaces, as in (1), on the same measure space.
We will denote which is a quasi-subadditive capacity.
In the proof of the following theorem we use the reiteration properties for the triple (L r (C 0 ), L r (C 1 ), L ∞ ) that we describe in the appendix.
Denote V (A) = A v(t) dt.Then we can write It follows from our results that Λ p µ (v) is a normed space when C is concave, which means that V is concave.But such a remark can be also applied to new Lorentz spaces obtained from some other well known symmetrization method of analysis: • Spherical symmetrization: , where E(s) is the ssection {y ∈ R; (s, y) ∈ E}.Then s * 2 is defined for a simple function s, and finally f * • Discrete rearrangements on trees as in [19].
In [9], Boza and J. Soria consider increasing transformations A → R(A) on measure spaces with the Fatou property, A n ↑ A ⇒ R(A n ) ↑ R(A), that allow to define the corresponding rearrangements of functions that allow to unify various Lorentz spaces found in the literature, included all the mentioned above: and our results on capacities apply to this special case.
As a final example, let us show how interpolation of capacitary Lebesgue spaces can be used in interpolation of Lorentz spaces, (Λ p 0 (v 0 ), Λ p 1 (v 1 )) η,p .

Appendix: interpolation of triples
Let C 0 and C 1 be a couple of Fatou capacities on (Ω, Σ) having the same null sets and X = (L r (C 0 ), L r (C 1 ), L ∞ ) with 0 < r < ∞, a triple of quasi-Banach spaces.Here L ∞ = L ∞ (C 0 ) = L ∞ (C 1 ) and we write X i (i = 0, 1, 2) for the components of X, which are continuously contained in the sum space Σ( X) endowed with If f Σ = 0, it is readily seen that f = 0 C-q.e. and • Σ is a quasi-norm.
In our situation, the fundamental lemma with S ρ holds for X: Lemma 1.Every f ∈ σ ( X) admits a representation as a sum in Σ( X), where k∈Z 2 f k ρ Σ( X) < ∞ and Proof.Exactly as in [4], one can find a family of sets B(k) (k = (k 1 , k 2 ) ∈ Z 2 ) such that the functions g k := |f |χ B(k) satisfy 2 k j g k j (S ρ K(•, f ))(2 k ) (j = 0, 1, 2 with k 0 := 0) capacity.Both C and C are equivalent to C.
).Proof.We write |f g| = (|f | p ) 1/p (|g| p ) 1/p .Since the inequality a ≤ b θ c 1−θ holds if and only if a ds, but unfortunately this average function is subadditive precisely when L p (C) (p ≥ 1) are normed spaces: Theorem 7. f is subadditive with respect to f if and only if C is concave.
Proof.It is clear that C t (A) := min(C(A), t) is a Fatou capacity.For a fixed t > 0, f (t) is subadditive in f if and only if C t is concave, since