Green's theorem and balayage
Let Q denote a cube in \({\mathbb{R}}^ d\) with sides parallel to the coordinate axes, let \(\ell (Q)\) denote its sidelength, and let \(\hat Q=Q\times (0,\ell (Q))\). A function \(\phi \in L^ 1_{loc}({\mathbb{R}}^ d)\) is said to be in BMO if \[ \| \phi \|_*\equiv \sup_{Q}[\ell (Q)]^{-d}\int_{Q}| \phi -\phi_ Q| dx<\infty, \] where \(\phi_ Q\) denotes the mean of \(\phi\) over Q. A Borel measure \(\mu\) on \({\mathbb{R}}^ d\times (0,\infty)\) is called a Carleson measure if \(\| \mu \|_ C\equiv \sup_{Q}[\ell (Q)]^{-d}| \mu | (\hat Q)<\infty.\) Further, let \[ S_{\mu}(x)=\int_{{\mathbb{R}}^ d\times (0,\infty)}y| (x,0)-(t,y)|^{-d-1} d\mu (t,y). \] It is a fact that, if \(\mu\) is a Carleson measure, then \(S_{\mu}\in BMO\) and \(\| S_{\mu}\|_*\leq C(d)\| \mu \|_ C.\) Conversely: Theorem. Let \(\phi\in BMO\) have support in \([-1,1]^ d\). There exist \(g\in L^{\infty}({\mathbb{R}}^ d)\) and a finite Carleson measure \(\mu\) such that \(\phi =g+S_{\mu}\) with \(\| g\|_{\infty}+\| \mu \|_ c\leq C(d)\| \phi \|_*\) and \(\sup p g\subseteq [-2,2]^ d.\) Apart from the conclusion that g has compact support, this theorem (in some extra generality) is due to \textit{L. Carleson} [Adv. Math. 22, 269- 277 (1976; Zbl 0357.46058)]. However, the author presents a new, more direct proof using Green's theorem and the semigroup property of the Poisson kernel.
scientific article; zbMATH DE number 4084039
scientific article