  We study integrodifferential operators and regularity estimates for solutions
to integrodifferential equations. Our emphasis is on kernels with a critically
low singularity which does not allow for standard scaling. For example, we
treat operators that have a logarithmic order of differentiability. For
corresponding equations we prove a growth lemma and derive a priori estimates.
We derive these estimates by classical methods developed for partial
differential operators. Since the integrodifferential operators under
consideration generate Markov jump processes, we are able to offer an
alternative approach using probabilistic techniques.
