  We prove a new type of pointwise estimate of the
Kalamajska-Mazya-Shaposhnikova type, where sparse averaging operators replace
the maximal operator. It allows us to extend the Gagliardo-Nirenberg
interpolation inequality to all rearrangement invariant Banach function spaces
without any assumptions on their upper Boyd index, i.e. omitting problems
caused by unboundedness of maximal operator on spaces close to $L^1$. In
particular, we remove unnecessary assumptions from the Gagliardo-Nirenberg
inequality in the setting of Orlicz and Lorentz spaces. The applied method is
new in this context and may be seen as a kind of sparse domination technique
fitted to the context of rearrangement invariant Banach function spaces.
