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Essential norms of a potential theoretic boundary integral operator in $$L^1$$. (English) Zbl 0936.31007
Summary: Let $$G\subset \mathbb R^m$$ $$(m\geq 2)$$ be an open set with a compact boundary $$B$$ and let $$\sigma \geq 0$$ 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 $$f\sigma$$, respectively. 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 $$\sigma$$.
##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 31A10 Integral representations, integral operators, integral equations methods in two dimensions 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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