Density bases associated with Nagel–Stein approach regions
Tom 251 / 2020
Streszczenie
Let $f$ be an integrable function on $\mathbb {R}^n$ and $u(x,y)$ its associated Poisson integral on the upper half-space $\mathbb {R}_+^{n+1}$. A classical result associated to the work of Fatou is that $u(x,y)$ tends to $f(x^0)$ for a.e. $x^0 \in \mathbb {R}^n$ as $(x,y)$ approaches $(x^0,0)$ nontangentially. On the other hand, Littlewood and later Zygmund showed that this limit does not necessarily hold if $(x,y)$ is allowed to approach $(x^0,0)$ with no restrictions on the nature of the convergence. Subsequently, Nagel and Stein proved the existence of approach regions not contained in any cone for which a.e. convergence does hold, in fact providing a characterization of the approach regions for which the associated maximal function is of weak type $(1,1)$. The purpose of this paper is to give a generalization of Nagel and Stein’s result, and to provide a characterization of the approach regions for which the associated maximal function satisfies so-called “Tauberian conditions”, corresponding to the regions for which a.e. convergence holds provided $f$ is the characteristic function of a set. As a consequence of this characterization, we will see that if an approach region enables a.e. convergence when the associated boundary function is the characteristic function of a set, it also enables a.e. convergence when the boundary function is in $L^p(\mathbb {R}^n)$ for $p \geq 1$.
