Essential norms of a potential theoretic boundary integral operator in $L^1$

Král, Josef, and Medková, Dagmar. "Essential norms of a potential theoretic boundary integral operator in $L^1$." Mathematica Bohemica 123.4 (1998): 419-436. <http://eudml.org/doc/248312>.

@article{Král1998,
abstract = {Let $G \subset \mathbb \{R\}^m$$(m \ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ 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. $\mathcal \{N\}_\{\sigma \}f$ 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 $\mathcal \{N\}_\{\sigma \} - \alpha I$ (here $\alpha \in \mathbb \{R\}$ 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 $.},
author = {Král, Josef, Medková, Dagmar},
journal = {Mathematica Bohemica},
keywords = {single layer potential; weak normal derivative; essential norm; single layer potential; weak normal derivative; essential norm},
language = {eng},
number = {4},
pages = {419-436},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Essential norms of a potential theoretic boundary integral operator in $L^1$},
url = {http://eudml.org/doc/248312},
volume = {123},
year = {1998},
}

TY - JOUR
AU - Král, Josef
AU - Medková, Dagmar
TI - Essential norms of a potential theoretic boundary integral operator in $L^1$
JO - Mathematica Bohemica
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 123
IS - 4
SP - 419
EP - 436
AB - Let $G \subset \mathbb {R}^m$$(m \ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ 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. $\mathcal {N}_{\sigma }f$ 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 $\mathcal {N}_{\sigma } - \alpha I$ (here $\alpha \in \mathbb {R}$ 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 $.
LA - eng
KW - single layer potential; weak normal derivative; essential norm; single layer potential; weak normal derivative; essential norm
UR - http://eudml.org/doc/248312
ER -