Vol 51, No 6 (2016)
- Year: 2016
- Articles: 3
- URL: https://journals.rcsi.science/1068-3623/issue/view/14064
Functional Analysis
Real and Complex Analysis
Fine properties of functions from Hajłasz–Sobolev classes Mαp, p > 0, I. Lebesgue points
Abstract
Let X be a metric measure space satisfying the doubling condition of order γ > 0. For a function f ∈ Llocp(X), p > 0 and a ball B ⊂ X by IB(p)f we denote the best approximation by constants in the space Lp(B). In this paper, for functions f from Hajłasz–Sobolev classes Mαp(X), p > 0, α > 0, we investigate the size of the set E of points for which the limit limr→+0IB(x,r)(p)f = f*(x). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x ∈ E
On unconditional basis property in the space H1(R) of a system of Franklin functions with vanishing means
Abstract
We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H1(R).