  The (prefix-free) Kolmogorov complexity of a finite binary string is the
length of the shortest description of the string. This gives rise to some
`standard' lowness notions for reals: A is K-trivial if its initial segments
have the lowest possible complexity and A is low for K if using A as an oracle
does not decrease the complexity of strings by more than a constant factor. We
weaken these notions by requiring the defining inequalities to hold up only up
to all $\Delta^0_2$ orders, and call the new notions $\Delta^0_2$-bounded
K-trivial and $\Delta^0_2$-bounded low for K. Several of the `nice' properties
of K-triviality are lost with this weakening. For instance, the new weaker
definitions both give uncountable set of reals. In this paper we show that the
weaker definitions are no longer equivalent, and that the $\Delta^0_2$-bounded
K-trivials are cofinal in the Turing degrees. We then compare them to other
previously studied weakenings, namely infinitely-often K-triviality and weak
lowness for K (in each, the defining inequality must hold up to a constant, but
only for infinitely many inputs). We show that $\Delta^0_2$-bounded K-trivial
implies infinitely-often K-trivial, but no implication holds between
$\Delta^0_2$-bounded low for K and weakly low for K.
