Decompositions for local Morrey spaces

We develop and apply a decomposition theory for generic local Morrey spaces. Our results are smooth and nonsmooth decompositions, which follows from the fact that local Morrey spaces are isomorphic to local Hardy–Morrey spaces and local Triebel–Lizorkin–Morrey spaces in the generic case. As an application of our results, we consider a bilinear estimate for the fractional integral operators.