**
MATHEMATICA BOHEMICA, Vol. 126, No. 4, pp. 669-690 (2001)
**

# Essential norms of the Neumann operator

of the arithmetical mean

## Josef Kral, Dagmar Medkova

* Josef Kral*, Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Praha 1, Czech Republic

* Dagmar Medkova*, Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail: ` medkova @math.cas.cz`

**Abstract:**
{Let $K\subset \Bbb R^m$ ($m\ge 2$) 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 ${\Cal C}_0^{(1)}$ be the class of all continuously differentiable real-valued functions with compact support in $\Bbb R^m$ and denote by $\sigma _m$ the area of the unit sphere in $\Bbb R^m$. With each $\varphi \in {\Cal C}_0^{(1)}$ we associate the function $$ W_K\varphi (z)={1\over \sigma _m}\underset {\Bbb R^m \setminus K} \to \int \grad \varphi (x)\cdot {z-x\over |z-x|^m} \dd x $$ 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 ${\Cal C}^{(1)}(B)=\{ \varphi |_B; \varphi \in {\Cal C}_0^{(1)}\} $ to the space ${\Cal C}(B)$ of all continuous functions on $B$. The operator ${\Cal T}_K$ sending each $f\in {\Cal C}^{(1)}(B)$ to ${\Cal T}_Kf=2W_Kf-f \in {\Cal C}(B)$ 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 ${\Cal C}(B)\supset {\Cal C}^{(1)}(B)$ inducing the topology of uniform convergence and $\Cal G$ is the space of all compact linear operators acting on ${\Cal C}(B)$, then the associated $p$-essential norm of ${\Cal T}_K$ is given by $$ \omega _p {\Cal T}_K=\underset {Q\in {\Cal G}} \to \inf \sup \bigl \{ p[({\Cal T}_K-Q)f]; f\in {\Cal C}^{(1)}(B), \^^Mp(f)\le 1\bigr \} . $$ In the present paper estimates (from above and from below) of $\omega _p {\Cal T}_K$ are obtained resulting in precise evaluation of $\omega _p {\Cal T}_K$ in geometric terms connected only with $K$.}

**Keywords:** double layer potential, Neumann's operator of the arithmetical mean, essential norm

**Classification (MSC2000):** 31B10, 45P05, 47A30

**Full text of the article:**

[Next Article] [Contents of this Number]

*
© 2005 ELibM and
FIZ Karlsruhe / Zentralblatt MATH
for the EMIS Electronic Edition
*