Real-valued valuations on Sobolev spaces

Ma, Dan

doi:10.1007/s11425-015-5101-6

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2016

2024

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Cited by 24 publications

(16 citation statements)

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“…Valuations on the space of functions of bounded variations and on Sobolev spaces have been recently studied by Wang and Ma respectively, in [21], [20], [14] and [13].…”

Section: Introductionmentioning

confidence: 99%

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Valuations on the Space of Quasi-Concave Functions

Colesanti¹,

Lombardi²

2017

Lecture Notes in Mathematics

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“…Valuations on the space of functions of bounded variations and on Sobolev spaces have been recently studied by Wang and Ma respectively, in [21], [20], [14] and [13].…”

Section: Introductionmentioning

confidence: 99%

“…, N}. Then φ has finite integral with respect to the measure S k (f ; ·) for every f ∈ C N if and only if φ verifies(14).…”

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confidence: 99%

Valuations on the Space of Quasi-Concave Functions

Colesanti¹,

Lombardi²

2017

Lecture Notes in Mathematics

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“…See, for instance, [1], [3], [12], [13], [15], [16]. See also [18], [14], [19] for recent developments related to our paper. References in [15], [16] provide a broad vision of the field.…”

Section: Introductionmentioning

confidence: 95%

Radial continuous rotation invariant valuations on star bodies

Villanueva

2016

Advances in Mathematics

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We characterize the positive radial continuous and rotation invariant valuations $V$ defined on the star bodies of $\mathbb R^n$ as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is, $$V(K)=\int_{S^{n-1}}\theta(\rho_K)dm,$$ where $\theta$ is a positive continuous function, $\rho_K$ is the radial function associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.Comment: Two minor gaps and several typos corrected thanks to the refere

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“…The results achieved so far embrace various types of spaces, such as Lebesgue spaces, Sobolev spaces, functions of bounded variations, convex functions, etc. For more details the reader is referred to the papers: [[2]- [4], [9]- [16], [18], [21]- [25]].…”

Section: Introductionmentioning

confidence: 99%