On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric
Let \(D\) be a domain of a space of homogeneous type \(\mathbb{X}\) and \(\text{BMO}(\mathbb{X})\) be the space of bounded mean oscillation function on \(\mathbb{X}\). The authors study the problem of existence of the continuous linear operator \(\text{ext}:\text{BMO}(D)\to\text{BMO}(\mathbb{X})\) such that every function \(\varphi\) in \(\text{BMO}(D)\) is the restriction of some function \(\widetilde\varphi=\text{ext}(\varphi)\) on \(D\). One of the main results of the article reads that the operator ext exists if and only if \(D\) is a uniform domain, i.e. there are constants \(a\) and \(b\) such that any \(x_1\) and \(x_2\) in \(D\) can be connected by the rectifiable curve \(\gamma\) in \(D\) with conditions \(\iota(\gamma)\leq a\cdot d(x_1,x_2)\) and \(\min\iota(\gamma(x_j,x))\leq b\cdot d(x,\partial D)\) \((x\in\gamma)\), where \(\iota(\gamma)\) denotes the length of \(\gamma\), \(\gamma(x_j,x)\) is the arc of \(\gamma\) with endpoints \(x_j\) and \(x\), \(d(x,\partial D)\) is the distant between \(x\) and \(\partial D\). The authors investigate conditions for a domain \(D\) to be uniform and in particular to obtain the following result: a ball in the Carnot-Carathéodory metric on a Heisenberg group is a uniform domain.
scientific article; zbMATH DE number 923558
scientific article