Hausdorff measure, BMO, and analytic functions

Hausdorff-analytic Functions of Matrices1

Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property (M̃) and Property (P ) of X , prove that Property (M̃) implies Property (P ) and give some equivalent characterizations of Property (M̃) and Property (P ). If X has Property (P ), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the se...

Hausdorff measure

Here, instead of taking a σ-algebra as given and then defining a measure on this σ-algebra (namely, on the measurable sets), we take an outer measure as given and then define measurable sets using this outer measure. Carathéodory’s theorem states that the collection M of ν-measurable sets is a σ-algebra and that the restriction of ν to M is a complete measure. Suppose that (X, ρ) is a metric sp...

The Bloch Space and Bmo Analytic Functions in the Tube over the Spherical Cone

We prove that the Bloch space coincides with the space BMOA in the tube over the spherical cone of R.3; this extends a well-known onedimensional result. Introduction. Let Q be a symmetric Siegel domain of type II contained in Cn. Let V denote the Lebesgue measure in fi and H{Q) the space of holomorphic (or analytic) functions in fi. When n = 1 and fi = tt+ = {z £ C: Imz > 0}, a Bloch function i...

Quasi-orthogonal expansions for functions in BMO

For {φ_n(x)}, x ε [0,1] an orthonormalsystem of uniformly bounded functions, ||φ_n||_{∞}≤ M