On the definition of $B$-points of a~Borel charge on the real line

P. A. Mozolyako

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 191-205

Résumé

Let $\mu$ be a Borel charge (i.e., a real Borel measure) on $\mathbb R$, and let $P_{(y)}(t)=\frac y{\pi(y^2+t^2)}$, $y>0$, $t\in\mathbb R$, denote the Poisson kernel. Bourgain proved in [1,2] that for a nonnegative $\mu$ and for numerous $x\in\mathbb R$ the variation of the function $y\mapsto(\mu*P_{(y)})(x)$ on $(0,1]$ is finite. This is true in particular for the so-called $B$-points $x$ (see e.g., [4]). In the present article new descriptions of $B$-points are given adjusted to some applications of this notion.