Gagliardo-Nirenberg inequality via a new pointwise estimate

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.