Sufficient conditions for bounded point evaluations
If \({\mathbb{R}}\) denotes the real line and \({\mathbb{R}}^ 2\) denotes the two- dimensional plane with respective Lebesgue measures m, \(m_ 2\), then the set \(A\subseteq {\mathbb{R}}^ 2\) is said to be a patch if it is of the form \((F\times I_ 1)\cup (I_ 2\times E)\), where \(m(F\cap I_ 2)>0\), \(m(E\cap I_ 1)>0\), and \(I_ 1\), \(I_ 2\) are intervals of \({\mathbb{R}}\). In the main result of this paper, the author shows that if \(E\subseteq {\mathbb{R}}^ 2\), \(m_ 2(E)>0\), then there is a path S such that \(m_ 2(S\setminus E)=0\). If, for \(1\leq r<\infty\), \(P^ r(\mu)\) denotes the closure in \(L^ r(\mu)\) of analytic polynomials, then \(P^ r(\mu)\) is said to have bounded point evaluation at \(\rho\) if the linear functional F defined by \(F(p)=P(\rho)\) is bounded on \(P^ r(\mu)\). The main theorem is applied to show that if \(E\subseteq {\mathbb{R}}^ 2\) is such that \(m(E)>0\), then \(P^ 1(\chi_ Edm_ 2)\) has a bounded point evaluation, and if f is non-negative \(m_ 2\)-a.e., has compact support and is of bounded variation in \({\mathbb{R}}^ 2\), then \(P^ 1(fdm_ 2)\) has a bounded point evaluation. Statements in the introduction of the paper refer to alternative considerations for establishing bounded point evaluation in a paper by J. Thomson to be published.
scientific article; zbMATH DE number 4161438
scientific article