Operator Theoretic Treatment of Linearabel Integral Equations of First Kind

We consider a linear Abel integral operator A : L 2 (0; 1) ?! L 2 (0; 1) deened by (A y)(t) = 1 ?() Z t 0 (t ? s) ?1 K(t; s)y(s)ds; 0 t 1; 0 < 1: We construct a scale fX g 2R of Hilbert spaces of functions in (0; 1) and relate it with a Hilbert scale of Sobolev spaces. Under suitable assumptions on K, we prove that kA uk L 2 (0;1) gives an equivalent norm in X ?. On the basis of this equivalence, we nd a lower and upper estimate for the singular values of A and, furthermore a HH older estimate for kuk L 2 (0;1) by kA uk L 2 (0;1) provided that kuk X q with q > 0 is uniformly bounded. Finally we discuss convergence rates of regularized solutions obtained by a Tikhonov method. In this paper, we consider a linear Abel integral operator A : L 2 (0; 1) ?! L 2 (0; 1), deened by (1.1) (A y)(t) = 1 ?() Z t 0 (t ? s) ?1 K(t; s)y(s)ds; 0 t 1; 0 < 1: