Choquet Integrals

  • David R. Adams
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

One of the new development in the Theory of Morrey spaces - as far as this author is concerned - is the use of Hausdorff capacity and its corresponding Choquet Integral \((\int f\;d\,\Lambda ^{d})\) in the development of the predual to \(L^{p,\lambda }(\mathbb{R}^{n})\). This we do in the next chapter once we have thoroughly discussed all the tools that are needed to understand and apply these “non-linear” integrals. And it is because \(\Lambda ^{d}\) is not a measure (i.e., not countable additive on disjoint sets) that we must carefully define and expose all of the relevant properties of such an integral.

4.1 Definition and basic properties: sublinear vs. strong subadditivity

One of the new development in the Theory of Morrey spaces - as far as this author is concerned - is the use of Hausdorff capacity and its corresponding Choquet Integral \((\int f\;d\,\Lambda ^{d})\) in the development of the predual to \(L^{p,\lambda }(\mathbb{R}^{n})\). This we do in the next chapter once we have thoroughly discussed all the tools that are needed to understand and apply these “non-linear” integrals. And it is because \(\Lambda ^{d}\) is not a measure (i.e., not countable additive on disjoint sets) that we must carefully define and expose all of the relevant properties of such an integral.

We begin with the definition.

Definition

For \(f \in C_{0}(\mathbb{R}^{n})\; =\) continuous function on \(\mathbb{R}^{n}\) with compact support, set

$$\displaystyle{ \int \vert f\vert ^{p}\;d\Lambda ^{d}\; =\;\int _{ 0}^{\infty }\Lambda ^{d}([\vert f\vert \geq t])\;dt^{p}. }$$
(4.1)
In other words, the Choquet Integral of a real valued function is defined distributionally and extended by functional completion. In (4.1) the set [ | f | ≥ t] is compact and hence the integrals involved are all finite numbers.

But to make use of this functional, we need a metric and in particular a triangle inequality. This was an early key result of Choquet. Our version follows:

Theorem 4.1.

Let C(⋅ ) be a capacity in the sense of N.G. Meyers which also satisfies ( 3.2) (vi). Then the Choquet Integral \(\int \vert f\vert \;dC\) is sublinear iff C is strongly subadditive (i.e., property ( 3.9) or
$$\displaystyle{ C(E_{1} \cup E_{2})\; +\; C(E_{1} \cap E_{2})\; \leq \; C(E_{1})\; +\; C(E_{2}).) }$$
(4.2)
Notice that (4.2) would trivially be true - with equality if C were a linear additive measure. Thus in its simplest form, we intend to show: for f, g ≥ 0,
$$\displaystyle{ \int (f + g)\;dC \leq \int f\;dC\; +\;\int g\;dC }$$
(4.3)
iff (4.2) holds for all \(E_{1},E_{2} \subset \mathbb{R}^{n}\).
Notice that Theorem 4.1 implies that
$$\displaystyle{ \bigg(\int \vert f\vert ^{p}\;dC\bigg)^{1/p},\;1 \leq p < \infty }$$
is a norm by the usual arguments, a norm on what we will eventually call L p (C). And since C can be taken to be \(\tilde{\Lambda }_{0}^{d}\), it follows that
$$\displaystyle{ \bigg\{\int \vert f\vert ^{p}\;d\Lambda ^{d}\bigg\}^{1/p},\;1 \leq p < \infty, }$$
(4.4)
defines a quasi-norm (a norm except that the triangle inequality holds with a fixed constant). And it is these facts that will soon send us on our way to the predual promised in the Introduction.

Theorem 4.1 is due to Choquet [C], but we choose to follow the arguments of Topsøe as given in B. Anger [An]. Also, because of the above, we can characterize \(L^{p}(\Lambda ^{d})\) as consisting of \(\Lambda ^{d}\) - quasi-continuous functions for which (4.4) is finite; i.e., those f such that for any ε > 0 there is a subset \(E \subset \mathbb{R}^{n}\) with \(\Lambda ^{d}(E) <\epsilon\) and f restricted to \(\mathbb{R}^{n}\setminus \;E\) is continuous. Here one proceeds as in measure theory.

Proof.

Proof of Theorem 4.1. First of all, if C is strongly subadditive and since
$$\displaystyle{ C(K)\; =\;\int X_{K}\;dC, }$$
(4.5)
X K being the characteristic function of the compact set K, it follows that if \(K_{1} \subset K_{2} \subset K_{3} \subset \cdots \subset K_{n}\), then
$$\displaystyle{ \int \sum _{i=1}^{n}X_{ K_{i}}\;dC\; =\;\sum _{ i=1}^{n}C(K_{ i}). }$$
(4.6)
The first of these follows from the definition of the Choquet Integral, the second upon breaking up the distribution integral into a sum of integrals from i to i + 1,   i = 0, 1, ⋯ , n and evaluating. This is clearly easy to do here since the sequence {K i } is increasing or at least monotone.
Next observe that (4.4) and (4.5) together with ( 3.2)(vi) implies
$$\displaystyle{ \int \inf \;h_{i}\;dC\; =\;\inf \int h_{i}\,dC, }$$
(4.7)
where the infimum is taken over all downward directed sequences {h i } of h i  ∈ USC+  = non-negative upper semi-continuous functions on \(\mathbb{R}^{n}\). Thus to show (4.3), we may restrict our attention to “compact simple functions,” i.e., functions of the form
$$\displaystyle{ \sum _{i=1}^{n}\alpha _{ i}\;X_{K_{i}},\;\alpha \in \mathbb{R}^{+},\;K_{ i}\mbox{ compact sets in }\mathbb{R}^{n}, }$$
due to (4.7) and the fact that the sum of two compact simple functions is again a compact simple function. Hence, it suffices to prove that strong subadditivity implies
$$\displaystyle{ \int \bigg(\sum _{i=1}^{n}\alpha _{ i}\;X_{k_{i}}\bigg)\;dC \leq \sum _{i=1}^{n}\alpha _{ i}\;C(k_{i}) }$$
(4.8)
From (4.6), we have that if h ∈ USC+
$$\displaystyle{ \int h\;dC\; =\;\inf \bigg\{ \frac{1} {m}\sum _{i=1}^{n}C(K_{ i}): m,n \in \mathbb{N},K_{i} \uparrow \mbox{ and with } \frac{1} {m}\sum _{i=1}^{n}X_{ K_{i}} \geq h\bigg\}. }$$
And so if we set
$$\displaystyle{ \hat{C} (h) =\inf \bigg\{\sum _{ i=1}^{n}\alpha _{ i}C(K_{i}): n \in \mathbb{N},\;\alpha _{i} \in \mathbb{R}^{+},\;\sum _{ i=1}^{n}\alpha _{ i}\;X_{K_{i}} \geq h\bigg\}, }$$
then
$$\displaystyle{ \hat{C} (h) =\inf \bigg\{ \frac{1} {m}\sum _{i=1}^{n}C(K_{ i}): n,m \in \mathbb{N}, \frac{1} {m}\sum _{i=1}^{n}X_{ K_{i}} \geq h\bigg\}, }$$
and hence \(\int h\;dC \geq \hat{C} (h),\) for all h ∈ USC+.
So the final observation will be that strong subadditivity implies \(\int h\;dC \leq \hat{C} (h)\) and then (4.8). To see this last inequality, we show
$$\displaystyle{ \sum _{i=1}^{n}C(K_{ i}) \geq \sum _{i=1}^{n}C(K_{ i}^{{\prime}}) }$$
(4.9)
where \(K_{i}^{{\prime}} =\; \cup \{\cap _{ j\in J}K_{j}: J \subset \{ 1,2,\cdots \,,n\},\;\vert J\vert = n - i + 1\}\), i.e., ordering the sets into an increasing sequence of compact sets \(K_{1}^{{\prime}} \subset,K_{2}^{{\prime}}\subset,\cdots \,,K_{n}^{{\prime}}\). Equation (4.9) is the key to the proof. When n = 2, (4.9) is just strong subadditivity. When n = 3, we write
$$\displaystyle\begin{array}{rcl} C(K_{1}^{{\prime}})& +& C(K_{ 2}^{{\prime}}) + C(K_{ 3}^{{\prime}}) = C(K_{ 1} \cap K_{2} \cap K_{3}) \\ & +& C[(K_{1} \cap K_{2}) \cup (K_{2} \cap K_{3}) \cup (K_{1} \cap K_{3})] + C(K_{1} \cup K_{2} \cup K_{3}) \\ & \leq & C(K_{1} \cap K_{2}) + C(F) + C(K_{1} \cup K_{2} \cup K_{3}), {}\end{array}$$
(4.10)
where \(F = (K_{2} \cap K_{3}) \cup (K_{1} \cup K_{3}) = (K_{1} \cup K_{2}) \cap K_{3}\).
Hence strong subadditivity gives (4.10) as not exceeding
$$\displaystyle{ C(K_{1} \cap K_{2}) + C(K_{1} \cup K_{2}) + C(K_{3}) \leq C(K_{1}) + C(K_{2}) + C(K_{3}). }$$
This then shows how to go from two sets to three sets in the induction process. The rest is similar but messy.
For the converse, we can simply write
$$\displaystyle\begin{array}{rcl} & & \int X_{K_{1}\cup K_{2}}\;dC +\int X_{K_{1}\cap K_{2}}\;dC =\int (X_{K_{1}\cup K_{2}} + X_{K_{1}\cap K_{2}})\;dC {}\\ & =& \int (X_{K_{1}} + X_{K_{2}})\;dC \leq \int X_{K_{1}}\;dC +\int X_{K_{2}}\;dC. {}\\ \end{array}$$
 □ 

4.2 Adams-Orobitg-Verdera Theorem

With these preliminaries out of the way, we are now ready for the real substance - the boundedness of the Hardy-Littlewood maximal function on the spaces \(L^{p}(\Lambda ^{d})\). The following theorem is due to Adams [A5] in the case p=1 and to Orobitg-Verdera for p < 1. For p > 1, the result is of course classical. The case p = 1 was accomplished using H1-BMO duality (see [St2] or [To]), whereas p < 1 was done the old fashioned way, via a covering lemma. For our purposes, we need only a small part of that proved in [OV].

Here, of course

Theorem 4.2.

There is a constant A depending only on d,  n, and p such that
$$\displaystyle{ \int (M_{0}f)^{p}\;d\Lambda ^{d}\; \leq \; A \cdot \int \vert f\vert ^{p}\;d\Lambda ^{d} }$$
(4.11)
for \(0 < d \leq n,\;\;d/n < p < \infty \).

Proof:

We begin with a lemma:

Lemma 4.3.

If X Q is the characteristic function of a cube Q (sides parallel to the axes), then
$$\displaystyle{ \int M_{0}(X_{Q})^{p}\;d\Lambda ^{d}\; \leq C \cdot l(Q)^{d} \mbox{ for }d/n < p. }$$

Proof.

Let x0 be the center of Q, then
$$\displaystyle{ M_{0}(X_{Q})(x) \leq C \cdot \inf \bigg (1, \frac{l(Q)^{n}} {\vert x - x_{0}\vert ^{n}}\bigg), }$$
for \(x \in \mathbb{R}^{n}\). Then
$$\displaystyle\begin{array}{rcl} \int M_{0}(X_{Q})^{p}\;d\Lambda ^{d}& \leq & C \cdot l(Q)^{d} + C\int _{ 0}^{1}l(Q)^{d}t^{-d/np}\;dt {}\\ & =& C^{{\prime}}\cdot l(Q)^{d}, {}\\ \end{array}$$
since dnp < 1.
With this, we proceed as follows: for f ≥ 0 and
$$\displaystyle{ \{x: 2^{k} < f(x) \leq 2^{k+1}\} \subset \cup _{ j}Q_{j}^{(k)} }$$
for some non-overlapping dyadic cubes Q j (k), then
$$\displaystyle{ \sum _{j}l(Q_{j}^{(k)})^{d} \leq 2\;\tilde{\Lambda }^{d}\;(\{x: 2^{k} < f(x) \leq 2^{k+1}\}) }$$
Setting \(g =\sum _{k}2^{(k+1)p}\;X_{A_{k}},\;A_{k} = \cup _{j}Q_{j}^{(k)}\), we have
$$\displaystyle{ f^{p} \leq g. }$$
So now if dn < p < 1, then
$$\displaystyle{ f \leq \sum _{k}2^{k+1}\;X_{ A_{k}}, }$$
and
$$\displaystyle{ (M_{0}f)^{p} \leq \sum _{ k}2^{(k+1)p} \cdot \sum _{ j}M_{0}\;\left (X_{Q_{j}^{(k)}}\right )^{p} }$$
because p < 1. Consequently,
$$\displaystyle\begin{array}{rcl} \int (M_{0}f)^{p}\;d\Lambda ^{d}& \leq & c \cdot \sum _{ k}2^{(k+1)p} \cdot \sum _{ j}l\left (Q_{j}^{(k)}\right )^{d} {}\\ & \leq & c\;\int f^{p}\;d\Lambda ^{d}. {}\\ \end{array}$$
 □ 

For the corresponding “weak-type” estimate at p = dn, the covering lemma mentioned earlier comes into play. See [OV] and below in Notes.

4.3 Notes

4.3.1 Further estimates for Mαf

Using a covering lemma in [OV], the authors were able to prove the following much more difficult result.

Theorem 4.4.

For 0 < d ≤ n,  p = dn,
$$\displaystyle{ \Lambda ^{d}\bigg\{[M_{ 0}f > t]\bigg\} \leq C\;t^{-d/n} \cdot \int \vert f\vert ^{d/n}\;d\Lambda ^{d}. }$$
(4.12)

But then the present author responded by proving the following extension of Theorems 4.2 and 4.4:

Theorem 4.5.

For 0 < d ≤ n,  0 < α < n,
  1. (a)
    p ≤ q
    1. (i)
      \(\dfrac{d} {n} < p < \dfrac{d} {\alpha }\) and δ = q(dα p)∕p, then
      $$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{q,p}(\Lambda ^{\delta })} \leq A_{1}\vert \vert f\vert \vert _{L^{p}(\Lambda ^{d})}; }$$
      (4.13)
       
    2. (ii)
      \(p = \dfrac{d} {n}\)
      $$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{q,\infty }(\Lambda ^{\delta })} \leq A_{2}\vert \vert f\vert \vert _{L^{d/n}(\Lambda ^{d})} }$$
      (4.14)
      with \(q = \dfrac{\delta } {(n-\alpha )},\;\delta \geq \dfrac{d} {n}(n-\alpha )\);
       
    3. (iii)
      \(p = \dfrac{d} {\alpha }\)
      $$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{\infty }} \leq A_{3}\vert \vert f\vert \vert _{L^{d/\alpha }(\Lambda ^{d})}. }$$
      (4.15)
      Here \(L^{q,p}(\Lambda ^{d})\) is the corresponding Lorentz space with respect to \(\Lambda ^{d}\) (see [St2]).
       
     
  2. (b)
    q < p
    $$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{q}(\omega \Lambda ^{d-\alpha p})} \leq A_{4}\vert \vert f\vert \vert _{L^{p}(\Lambda ^{d})},\quad \alpha p < d, }$$
    (4.16)
    iff
    $$\displaystyle{ \vert \vert \omega \vert \vert _{L^{p/(p-q)}(\Lambda ^{d-\alpha p})} < \infty. }$$
     

The reader is refereed to [A7] for more details.

4.3.2 Speculations on weighted Hausdorff Capacity

In [A6], the author asked the question: Is weighted capacity related to the Choquet Integral of the weight with respect to the unweighted capacity?

We may have more to say on this point in Chapter  9 when various L p - capacities are examined. But for now we might just ask this question for the weighted Hausdorff capacities of [Ni], i.e., suppose ω is an \(A_{\infty }\)- weight in the sense of Muckenhoupt [St2] and set One can define the usual dyadic version of this weighted set function, so the question becomes: do such capacities satisfy all the properties discussed in Chapter  3? Perhaps of more interest is: when is the following true?
$$\displaystyle{ \int f^{p}\;d\Lambda ^{d/\omega } \sim \int f^{p}\,\omega \,d\Lambda ^{d}, }$$
(4.18)
f ≥ 0? And what about an analogue of Theorem 4.5 for the fractional maximal operator M α , with \(\Lambda ^{d}\) replaced by \(\Lambda ^{d/\omega }\)? One can think of (4.18) as a sort of Radon-Nikodym Theorem for weighted Hausdorff Capacities. More on this on Chapter  9.

Bibliography

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    Choquet, G., Theory of capacities, Ann. Inst. Fourier Grenoble, 5(1953), 131–295.CrossRefMathSciNetGoogle Scholar
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    Nieminen, E., Hausdorff measures, capacities, and Sobolev spaces with weights, Ann. Acad. Sci. Finland, Math. Dissertations, Helsinki 1991.Google Scholar
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    Orobitg, J. Verdera, J., Choquet integrals, Hausdorff content and the Hardy-Littlewood maximal operator, Bull. London Math. Society, 30(1998), 145–150.CrossRefMathSciNetGoogle Scholar
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    , Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton U. Press 1993.Google Scholar
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • David R. Adams
    • 1
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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