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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 \(K\).
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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