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Essential norms of the Neumann operator of the arithmetical mean. (English) Zbl 0998.31003
Summary: Let $$K\subset \mathbb R^m$$ ($$m\geq 2$$) be a compact set; assume that each ball centered on the boundary $$B$$ of $$K$$ meets $$K$$ in a set of positive Lebesgue measure. Let $${\mathcal C}_0^{(1)}$$ be the class of all continuously differentiable real-valued functions with compact support in $$\mathbb R^m$$ and denote by $$\sigma _m$$ the area of the unit sphere in $$\mathbb R^m$$. With each $$\varphi \in {\mathcal C}_0^{(1)}$$ we associate the function $W_K\varphi (z)={1\over \sigma _m}\int _{\mathbb R^m \setminus K}\text{grad }\varphi (x)\cdot {z-x\over |z-x|^m} dx$ of the variable $$z\in K$$ (which is continuous in $$K$$ and harmonic in $$K\setminus B$$). $$W_K\varphi$$ depends only on the restriction $$\varphi |_B$$ of $$\varphi$$ to the boundary $$B$$ of $$K$$. This gives rise to a linear operator $$W_K$$ acting from the space $${\mathcal C}^{(1)}(B)=\{ \varphi |_B$$; $$\varphi \in {\mathcal C}_0^{(1)}\}$$ to the space $${\mathcal C}(B)$$ of all continuous functions on $$B$$. The operator $${\mathcal T}_K$$ sending each $$f\in {\mathcal C}^{(1)}(B)$$ to $${\mathcal T}_Kf=2W_Kf-f \in {\mathcal C}(B)$$ is called the Neumann operator of the arithmetical mean; it plays a significant role in connection with boundary value problems for harmonic functions. If $$p$$ is a norm on $${\mathcal C}(B)\supset {\mathcal C}^{(1)}(B)$$ inducing the topology of uniform convergence and $$\mathcal G$$ is the space of all compact linear operators acting on $${\mathcal C}(B)$$, then the associated $$p$$-essential norm of $${\mathcal T}_K$$ is given by $\omega _p {\mathcal T}_K=\inf_ {Q\in {\mathcal G}} \sup \bigl \{ p[({\mathcal T}_K-Q)f]; \;f\in {\mathcal C}^{(1)}(B), \;p(f)\leq 1\bigr \} .$ In the present paper estimates (from above and from below) of $$\omega _p {\mathcal T}_K$$ are obtained resulting in precise evaluation of $$\omega _p {\mathcal T}_K$$ in geometric terms connected only with $$K$$.
##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 45P05 Integral operators 47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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