  We study the application of Tikhonov regularization to ill-posed nonlinear
operator equations. The objective of this work is to prove low order
convergence rates for the discrepancy principle under low order source
conditions of logarithmic type. We work within the framework of Hilbert scales
and extend existing studies on this subject to the oversmoothing case. The
latter means that the exact solution of the treated operator equation does not
belong to the domain of definition of the penalty term. As a consequence, the
Tikhonov functional fails to have a finite value.
