Essential norms of the Neumann operator of the arithmetical mean

Král, Josef, and Medková, Dagmar. "Essential norms of the Neumann operator of the arithmetical mean." Mathematica Bohemica 126.4 (2001): 669-690. <http://eudml.org/doc/248866>.

@article{Král2001,
abstract = {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 \[ W\_K\varphi (z)=\{1\over \sigma \_m\}\underset\{\mathbb \{R\}^m \setminus K\}\{\rightarrow \}\int \mathop \{\mathrm \{g\}rad\}\nolimits \varphi (x)\cdot \{z-x\over |z-x|^m\}\ x \] 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 $K$.},
author = {Král, Josef, Medková, Dagmar},
journal = {Mathematica Bohemica},
keywords = {double layer potential; Neumann’s operator of the arithmetical mean; essential norm; double layer potential; Neumann's operator of the arithmetical mean; essential norm},
language = {eng},
number = {4},
pages = {669-690},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Essential norms of the Neumann operator of the arithmetical mean},
url = {http://eudml.org/doc/248866},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Král, Josef
AU - Medková, Dagmar
TI - Essential norms of the Neumann operator of the arithmetical mean
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 4
SP - 669
EP - 690
AB - 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 \[ W_K\varphi (z)={1\over \sigma _m}\underset{\mathbb {R}^m \setminus K}{\rightarrow }\int \mathop {\mathrm {g}rad}\nolimits \varphi (x)\cdot {z-x\over |z-x|^m}\ x \] 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 $K$.
LA - eng
KW - double layer potential; Neumann’s operator of the arithmetical mean; essential norm; double layer potential; Neumann's operator of the arithmetical mean; essential norm
UR - http://eudml.org/doc/248866
ER -