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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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