The paper deals with the theory of potentials with respect to the α-Riesz kernel |x − y|^α−n of order α ∈ (0,2] on \(\mathbb R^{n}\), \(n\geqslant 3\). Focusing first on the inner α-harmonic measure \({\varepsilon _{y}^{A}}\) (ε_y being the unit Dirac measure at \(y\in \mathbb R^{n}\), 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 \({\varepsilon _{y}^{A}}(\mathbb R^{n})\equiv 1\) to hold. Those criteria are given in terms of newly defined concepts of inner α-thinness and inner α-ultrathinness of A at infinity that for α = 2 and A Borel coincide with the concepts of outer 2-thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to μ^A general by verifying the integral representation formula \(\mu ^{A}={\int \limits \varepsilon _{y}^{A}} d\mu (y)\). We also show that for every \(A\subset \mathbb R^{n}\), there exists a K_σ-set A₀ ⊂ A such that \(\mu ^A=\mu ^{A_0}\) for all μ, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. inner equilibrium, measure under an 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.

本文讨论了有关α- Riesz核的势能理论| x − y | ^{α - Ñ}顺序的α＆Element;（0,2]上\（\ mathbb R 2 {N} \） ，\（N \ geqslant 3 \）首先着眼于内。α K谐波测量\（{\ varepsilon _ { Y} ^ {A}} \） （ε _ÿ处于所述单元狄拉克量度\（Y \在\ mathbb R 2 {N} \） ，和μ^阿内α -Riesz一个Radon测度的balayage μ至\（ A \ subset \ mathbb R ^ {n} \）任意），我们描述了它的欧几里得支持度，提供了一个评估其总质量的公式，建立了内部α-外的图\（y {\ mapsto \ varepsilon _ {y} ^ {A}} \）的模糊连续性- A的不规则点，并获得必要的充分条件，使\（{\ varepsilon _ {y} ^ {A}} \）具有有限能量（更普遍地，对于\（{\ varepsilon _ {y} ^ {A }} \）在内部容量方面是绝对连续的，也要保持\（{\ varepsilon _ {y} ^ {A}}（\ mathbb R ^ {n}）\ equiv 1 \）保持不变。这些标准是根据新定义的A的内部α-厚度和内部α-尿素概念给出的在无穷大处，α = 2和A Borel分别与Doob和Brelot提出的在无穷远处的外部2稀薄度概念一致。此外，我们扩展一些这些结果到μ^阿通过验证积分表示式通用\（\亩^ {A} = {\ INT \限制\ varepsilon _ {Y} ^ {A}} d \亩（Y）\ ）。我们还表明，对于每\（A \子集\ mathbb R 2 {N} \） ，存在ķ _σ -set甲₀ ⊂甲使得\（\亩^ A = \亩^ {A_0} \）为全部μ，并给出该定理的各种应用。特别是，我们证明了内部扫描的模糊性和强连续性。内平衡，近似所测甲任意的，从而加强了对建立Fuglede的结果甲波雷尔（学报数学。，1960）。即使对于α = 2，它也是新的，所获得的结果也提出了由Cartan提出的内部牛顿容量和内部牛顿平衡理论的进一步发展。