Abstract dyadic cubes, maximal operators and Hausdorff content

Dual of the Choquet spaces with general Hausdorff content

“…In what follows, we summarize the elementary properties of the general Hausdorff content H λ and the Choquet integrals with respect to H λ . All the relevant proofs are found in [5].…”

“…In the next section, we summarize the elementary properties of H λ and the Choquet integral with respect to H λ . We emphasize that to ensure the left continuity of H λ in Proposition 2.2, we cannot drop the continuity condition (1.3): see[5, Example 3.7].…”

A Note on Embedding Inequalities for Weighted Sobolev and Besov Spaces

“…In this section, following [7,8], we first summarize some important properties of the weighted Hausdorff capacity and the dyadic Hausdorff capacity. These two types of weighted Hausdorff capacity are equivalent each other.…”

Some Remarks on Capacitary Integrals and Measure Theory

“…These inequalities in turn serve as a basis for the study of a family of Banach spaces of functions L p (H), those suitably regular functions whose pth power has finite Choquet integral. Here typical questions have concerned the boundedness of maximal functions [10,47,49], characterizations of the topological duals [10,48], and interpolation theory [25][26][27]. The assumption one has a capacity in the sense of Choquet ensures that even without the full strength of results from Measure Theory one has a number of useful tools, e.g.…”