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MATHEMATICA BOHEMICA, Vol. 123, No. 4, pp. 419-436 (1998)
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Essential norms of a potential theoretic boundary integral operator in $L^1$

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Josef Kral, Dagmar Medkova

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

**Abstract:** Let $G \subset\Bbb R^m$ $(m \ge2)$ be an open set with a compact boundary $B$ and let $\sigma\ge0$ be a finite measure on $B$. Consider the space $L^1(\sigma)$ of all $\sigma$-integrable functions on $B$ and, for each $f \in L^1(\sigma)$, denote by $f \sigma$ the signed measure on $B$ arising by multiplying $\sigma$ by $f$ in the usual way. $\Cal N_{\sigma}f$ denotes the weak normal derivative (w.r. to $G$) of the Newtonian (in case $m >2$) or the logarithmic (in case $n=2$) potential of $f\sigma$, correspondingly. Sharp geometric estimates are obtained for the essential norms of the operator $\Cal N_{\sigma} - \alpha I$ (here $\alpha\in\Bbb R$ and $I$ stands for the identity operator on $L^1(\sigma)$) corresponding to various norms on $L^1(\sigma)$ inducing the topology of standard convergence in the mean w.r. to $\sigma$.

**Keywords:** single layer potential, weak normal derivative, essential norm

**Classification (MSC2000):** 31B20, 31B25, 31A10

**Full text of the article:**

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