Morrey Spaces pp 21-27 | Cite as

# Choquet Integrals

Chapter

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## 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.

## Bibliography

- [A5]The existence of capacitary strong-type estimates in \(\mathbb{R}^{n}\), Ark. Math. 14(1976), 125–140.Google Scholar
- [A6]
- [A7]A note on Choquet integrals with respect to Hausdorff capacity, Function Spaces and Appl., Proc. Lund 1986, Lecture Notes in Math., Springer 1988.Google Scholar
- [An]Anger, B., Representation of capacities, Math. Ann. 229(1997), 245–258.CrossRefMathSciNetGoogle Scholar
- [C]Choquet, G., Theory of capacities, Ann. Inst. Fourier Grenoble, 5(1953), 131–295.CrossRefMathSciNetGoogle Scholar
- [Ni]Nieminen, E., Hausdorff measures, capacities, and Sobolev spaces with weights, Ann. Acad. Sci. Finland, Math. Dissertations, Helsinki 1991.Google Scholar
- [OV]Orobitg, J. Verdera, J., Choquet integrals, Hausdorff content and the Hardy-Littlewood maximal operator, Bull. London Math. Society, 30(1998), 145–150.CrossRefMathSciNetGoogle Scholar
- [St2], Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton U. Press 1993.Google Scholar
- [To]Torchinsky, A., Real-variable methods in harmonic analysis, 123 Pure & Appl. Math. series, Academic Press 1986.Google Scholar

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