the context of coorbit space theory, sooner or later convolution estimates of Young type have to be employed, which yield bounded mappings between $L_p$-spaces with *different* integrability exponents for domain and codomain if the convolution kernel is not in $L_1$. Concerning the atomic decomposition part, the situation is even more involved. It turns out that for any element in the coorbit space a suitable approximation by linear combinations of the atoms can be derived, but at the price that the weighted sequence norms of the expansion coefficients cannot be uniformly bounded by the coorbit norm. These results will be stated and proved in Sect. \[sec:discretization\], see in particular Theorem \[maintheorem\].

Looking at these results, the inclined reader might have the impression that the authors were simply unable to prove sharper results, whereas such results might still be true, and provable with a more refined analysis. This might be true, but only partially. Indeed, in Sect. \[sec:Obstructions\] we prove an additional result which shows that, under some very natural conditions, [*uniform*]{} bounds can only be obtained if the kernel operator acts as a bounded operator on the weighted $L_p$-spaces, that is, this additional assumption is necessary for obtaining uniform bounds. These 