Hausdorff measures and functions of bounded quadratic variation

Abstract

Abstract. To each function f of bounded quadratic variation we associate a Hausdorff measure μ f . We show that the map f → μ f is locally Lipschitz and onto the positive cone of M[0, 1]. We use the measures {μ f : f ∈ V 2 } to determine the structure of the subspaces of V 0 2 which either contain c 0 or the square stopping time space S 2 .

On Interval Based Generalizations of Absolute Continuity for Functions on R_n

“…Thus all squares Q (m+1)j are contained in the union of of squares A 1 B 1 C 1 D 1 and A 3 B 3 C 3 D 3 , whose union has measure equal to 1 8 of the measure of square Q mk , so that (5.11) is satisfied.…”

“…. , r m , the disjoint intervals T mk ⊂ Q mk are 1 2 -regular, and by (5.11), for every δ > 0, there exists m 0 ∈ N so that…”

“…There are several natural ways of generalizing the definition of absolute continuity for functions of several variables (cf. [12,20,27,33,1]).…”

“…On the other hand, the Banach space of generalized absolutely continuous functions on [0, 1] is closely related to the famous James space, and it is an example of a separable space not containing ℓ 1 but with a non-separable dual [22,21]. Moreover it has a very rich subspace structure [1,2], but several questions about the Banach space structure of this space remain open, see [1].…”

Variants of the James Tree space

“…The authors modestly gave their example the name James function space and denoted it by JF . A detailed study of the subspaces of JT and JF can be found in the papers [4], [5] of D. Apatsidis, the first author, and V. Kanellopoulos and in the paper [10] of the first two authors and M. Petrakis where the James function space is defined and studied with domain subsets of R n .…”