Operator theoretic treatment of linear Abel integral equations of first kind

Abstract

We consider a linear Abel integral operatorA _α: L²(0, 1) → L²(0, 1) defined by

$$\left( {A_\alpha y} \right)\left( t \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_0^t {\left( {t - s} \right)^{\alpha - 1} K\left( {t - s} \right)y\left( s \right)ds, 0 \leqslant t \leqslant 1, 0 \leqslant \alpha \leqslant 1} $$

. We construct a scale {X_β}_β∈ℝ of Hilbert spaces of functions in (0, 1) and relate it with a Hilbert scale of Sobolev spaces. Under suitable assumptions onK, we prove that$\left\| {A_\alpha u} \right\|_{L^2 \left( {0,1} \right)}$ gives an equivalent norm inX _α. On the basis of this equivalence, we find a lower and upper estimate for the singular values ofA _α and, furthermore a Hölder estimate for$\left\| u \right\|_{L^2 \left( {0,1} \right)}$ by$\left\| {A_\alpha u} \right\|_{L^2 \left( {0,1} \right)}$ provided that$\left\| u \right\|_{Xq}$, withq > 0 is uniformly bounded. Finally we discuss convergence rates of regularized solutions obtained by a Tikhonov method.