Morrey Spaces pp 21-27 | Cite as

# Choquet Integrals

## 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

*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.

*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

*C*were a linear additive measure. Thus in its simplest form, we intend to show: for

*f*,

*g*≥ 0,

*L*

^{ p }(

*C*). And since

*C*can be taken to be \(\tilde{\Lambda }_{0}^{d}\), it follows that

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.

*C*is strongly subadditive and since

*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

*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.

*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

*h*∈ USC

^{+}

*h*∈ USC

^{+}.

*n*= 2, (4.9) is just strong subadditivity. When

*n*= 3, we write

## 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 *H*^{1}-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].

## Theorem 4.2.

*A*depending only on

*d*,

*n*, and

*p*such that

## Proof:

We begin with a lemma:

## Lemma 4.3.

*X*

_{ Q }is the characteristic function of a cube

*Q*(sides parallel to the axes), then

## Proof.

*x*

_{0}be the center of

*Q*, then

*d*∕

*np*< 1.

*f*≥ 0 and

*Q*

_{ j }

^{(k)}, then

*d*∕

*n*<

*p*< 1, then

*p*< 1. Consequently,

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

## 4.3 Notes

### 4.3.1 Further estimates for *M*_{α}*f*

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

## Theorem 4.4.

*d*≤

*n*,

*p*=

*d*∕

*n*,

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

## Theorem 4.5.

*d*≤

*n*, 0 <

*α*<

*n*,

- (a)
*p*≤*q*- (i)\(\dfrac{d} {n} < p < \dfrac{d} {\alpha }\) and
*δ*=*q*(*d*−*α 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) - (ii)\(p = \dfrac{d} {n}\)with \(q = \dfrac{\delta } {(n-\alpha )},\;\delta \geq \dfrac{d} {n}(n-\alpha )\);$$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{q,\infty }(\Lambda ^{\delta })} \leq A_{2}\vert \vert f\vert \vert _{L^{d/n}(\Lambda ^{d})} }$$(4.14)
- (iii)\(p = \dfrac{d} {\alpha }\)Here \(L^{q,p}(\Lambda ^{d})\) is the corresponding Lorentz space with respect to \(\Lambda ^{d}\) (see [St2]).$$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{\infty }} \leq A_{3}\vert \vert f\vert \vert _{L^{d/\alpha }(\Lambda ^{d})}. }$$(4.15)

- (i)
- (b)
*q*<*p*iff$$\displaystyle{ \vert \vert M_{\alpha }f\vert \vert _{L^{q}(\omega \Lambda ^{d-\alpha p})} \leq A_{4}\vert \vert f\vert \vert _{L^{p}(\Lambda ^{d})},\quad \alpha p < d, }$$(4.16)$$\displaystyle{ \vert \vert \omega \vert \vert _{L^{p/(p-q)}(\Lambda ^{d-\alpha p})} < \infty. }$$

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?

*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

*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.

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