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Balayage of measures with respect to classes of all subharmonic or harmonicfunctions on an open set of a plane or finite-dimensional Euclidean space isone of the main objects of potential theory and its applications to the complexanalysis. For a class $H$ of functions on $O$, a measure $\omega$ on $O$ is abalayage of a measure $\delta$ on $O$ with respect to this class $H$ if $\int_Oh\, d \delta\leq \int_O h\, d\omega$ for each $h\in H$. In our previous workswe used this concept to study envelopes relative to classes of subharmonic andharmonic functions and apply them to describe zero sets of holomorphicfunctions on $O$ with growth restrictions near the boundary of $O$. In thisarticle, we consider the complex plane $\mathbb C$ as $O$, and instead of theclasses of all (sub)harmonic functions on $\mathbb C$, we use only the classesof harmonic polynomials of degree at most $p$, often together with thelogarithmic functions-kernels $z\mapsto \ln |w-z|$, $w\in \mathbb C$. Ourresearch has show that this case has both many similarities and featurescompared to previous situations. The following issues are considered: thesensitivity of balayage of measures to polar sets; the duality between balayageof measures and their logarithmic potentials, together with a complete internaldescription of such potentials; extension/prolongation of balayage with respectto polynomials and logarithmic kernels to balayage with respect to subharmonicfunctions of finite order $p$. The planned applications of these results to thetheory of entire and meromorphic functions of finite order are not discussedhere and will be presented later.

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