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

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.