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 $ω$ on $O$ is a balayage of a measure $δ$ on $O$ with respect to this class $H$ if $\int_O h\, d δ\leq \int_O h\, dω$ 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.
Geometry of logarithmic forms and deformations of complex structures
利用Kahler流形上电流的调和积分理论，给出了求解某些对数微分形式的$\bar{\partial}$ -方程的新方法.这个结果可以看作是对数形式的一个$\bar{\partial}$ -引理.作为应用，我们推广了Deligne关于对数形式的闭性的结果，给出了对数Hodge to de Rham谱序列的Deligne退化定理在$E_1$ -水平上的几何证明和简单证明，以及紧致Kahler流形上的某些内射性定理.此外，对于Kahler流形上一类复结构的对数变形，我们构造了中心纤维上任意对数$(n,q)$ -形式的扩张，从而通过将迭代法推广到对数形式，得到了log Calabi-Yau结构的局部稳定性.最后用微分几何方法证明了对数Calabi-Yau对和Calabi-Yau流形上的对的变形的无障碍性。
