LAUSR

We prove theorems on the boundedness of commutators $$[a,H_w^\alpha ]$$[a,Hwα] of the weighted multidimensional Hardy operator $$H^\alpha _w:= w H^\alpha \frac{1}{w}$$Hwα:=wHα1w from a generalized local Morrey space $$\mathcal {L}^{p,\varphi ;0}({\mathbb {R}^n})$$Lp,φ;0(Rn) to local or global space $$\mathcal {L}^{q,\psi }({\mathbb {R}^n})$$Lq,ψ(Rn). The main impacts of these theorems are1.the use of CMO$$_s$$s-class of coefficients a for the commutators;2.the general setting when the function $$\varphi $$φ defining the Morrey space and the weight w are independent of one another and the weight w is not assumed to be in $$A_p$$Ap;3.recovering the Sobolev–Adams exponent q instead of Sobolev–Spanne type exponent in the case of classical Morrey spaces4.boundedness from local to global Morrey spaces;5.the obtained estimates contain the parameter $$s > 1$$s>1 which may be arbitrarily chosen. Its choice regulates in fact an equilibrium between assumptions on the coefficient a and the characteristics of the space. The obtained results are new also in non-weighted case.