Morrey Spaces pp 21-27

# Choquet Integrals

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 () (vi). Then the Choquet Integral $$\int \vert f\vert \;dC$$ is sublinear iff C is strongly subadditive (i.e., property () 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 ()(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
$$\displaystyle\begin{array}{rcl} & & \int X_{K_{1}\cup K_{2}}\;dC +\int X_{K_{1}\cap K_{2}}\;dC =\int (X_{K_{1}\cup K_{2}} + X_{K_{1}\cap K_{2}})\;dC {}\\ & =& \int (X_{K_{1}} + X_{K_{2}})\;dC \leq \int X_{K_{1}}\;dC +\int X_{K_{2}}\;dC. {}\\ \end{array}$$
□

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 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
(4.17)
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 ? 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 .

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