Essential norms of the Neumann operator of the arithmetical mean
Josef Král, Dagmar Medková · Mathematica Bohemica · 2001
Let $K\subset \mathbb{R}^m$ ($m\ge 2$) 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 ${C}_0^{(1)}$ 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 $\varphi \in {C}_0^{(1)}$ we associate the function \[ W_K\varphi (z)={1\over \sigma _m}\underset{\mathbb{R}^m \setminus K}{\rightarrow }\int \mathop {\mathrm grad} olimits \varphi (x)\cdot {z-x\over |z-x|^m}\ x \] 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 ${C}^{(1)}(B)=\lbrace \varphi |_B; \varphi \in {C}_0^{(1)}\rbrace $ to the space ${C}(B)$ of all continuous functions on $B$. The operator ${T}_K$ sending each $f\in {C}^{(1)}(B)$ to ${T}_Kf=2W_Kf-f \in {C}(B)$ 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 ${C}(B)\supset {C}^{(1)}(B)$ inducing the topology of uniform convergence and $G$ is the space of all compact linear operators acting on ${C}(B)$, then the associated $p$-essential norm of ${T}_K$ is given by \[ \omega _p {T}_K=\underset{Q\in {G}}{\rightarrow }\inf \sup \bigl \lbrace p[({T}_K-Q)f]; \ f\in {C}^{(1)}(B), \ p(f)\le 1\bigr \rbrace . \] In the present paper estimates (from above and from below) of $\omega _p {T}_K$ are obtained resulting in precise evaluation of $\omega _p {T}_K$ in geometric terms connected only with $K$.