Balayage of Measures on the Complex Plane with respect to Harmonic Polynomials and Logarithmic Kernels

Abstract: Balayage of measures with respect to classes of all subharmonic or harmonic functions on an open set of a plane or finite-dimensional Euclidean space is one of the main objects of potential theory and its applications to the complex analysis. For a class $H$ of functions on $O$, a measure $\omega$ on $O$ is a balayage of a measure $\delta$ on $O$ with respect to this class $H$ if $\int_O h\, d \delta\leq \int_O h\, d\omega$ for each $h\in H$. In our previous works we used this concept to study envelopes relative to classes of subharmonic and harmonic functions and apply them to describe zero sets of holomorphic functions on $O$ with growth restrictions near the boundary of $O$. In this article, we consider the complex plane $\mathbb C$ as $O$, and instead of the classes of all (sub)harmonic functions on $\mathbb C$, we use only the classes of harmonic polynomials of degree at most $p$, often together with the logarithmic functions-kernels $z\mapsto \ln |w-z|$, $w\in \mathbb C$. Our research has show that this case has both many similarities and features compared to previous situations. The following issues are considered: the sensitivity of balayage of measures to polar sets; the duality between balayage of measures and their logarithmic potentials, together with a complete internal description of such potentials; extension/prolongation of balayage with respect to polynomials and logarithmic kernels to balayage with respect to subharmonic functions of finite order $p$. The planned applications of these results to the theory of entire and meromorphic functions of finite order are not discussed here and will be presented later.