Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm

Citation:
Karlovich, Alexei Yu. "Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm." Operators and Matrices. 1.3 (2007): 427-444.

Abstract:

Suppose \(\Gamma\) is a Carleson Jordan curve with logarithmic whirl points, \(\varrho\) is a Khvedelidze weight, \(p:\Gamma\to(1,\infty)\) is a continuous function satisfying \(|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|\) for \(|\tau-t|\le 1/2\), and \(L^{p(\cdot)}(\Gamma,\varrho)\) is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with \(N\times N\) matrix piecewise continuous coefficients are Fredholm on \(L_N^{p(\cdot)}(\Gamma,\varrho)\).

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Preprint in arXiv:

http://arxiv.org/abs/0704.3973