论文标题
Riesz电位理论中的谐波测度，平衡度量和无穷大的薄度
Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials
论文作者
论文摘要
Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass,建立地图的阴道连续性$ y \ mapsto \ varepsilon_y^a $在内部$α$ - $ a $的内部$α$ irratorgular点以外，并获得了$ \ varepsilon_y^a $的必要条件$ \ varepsilon_y^a（\ mathbb r^n）\ equiv1 $。这些标准是根据$α$ - 细度的新定义的概念和Infination的$α$ - 脱硫的概念给出的，分别概括了Doob和Brelot在Infinity的稀薄概念。此外，通过验证公式$μ^a = \ int \ varepsilon_y^a \，dμ（y）$，我们将其中一些结果中的一些扩展到$μ^a $常规。我们还表明，所有$μ$都有$k_σ$ -SET $ a_0 \ subset a $，因此$μ^a =μ^{a_0} $，并给出此定理的各种应用。特别是，我们证明了内部扫荡的模糊而强的连续性。平衡，衡量$ a $ a nutunary的近似值，从而加强了为$ a $ borel建立的fuglede的结果（Acta Math。，1960年）。即使是$α= 2 $的新事物，获得的结果也呈现出牛顿内部能力和内部牛顿balayage理论的进一步发展，该理论源自卡坦。
Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner capacity) as well as for $\varepsilon_y^A(\mathbb R^n)\equiv1$ to hold. Those criteria are given in terms of the newly defined concepts of $α$-thinness and $α$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to $μ^A$ general by verifying the formula $μ^A=\int\varepsilon_y^A\,dμ(y)$. We also show that there is a $K_σ$-set $A_0\subset A$ such that $μ^A=μ^{A_0}$ for all $μ$, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. equilibrium, measure under the approximation of $A$ arbitrary, thereby strengthening Fuglede's result established for $A$ Borel (Acta Math., 1960). Being new even for $α=2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.
