Grand complementary Morrey spaces and Hardy operators

We introduce grand complementary Morrey spaces, in a general form, on open sets \(\varOmega\) $\Omega$ in \(\mathbb R^n\) ${\mathbb{R}}^n$ . In case of unbounded sets \(\varOmega\) $\Omega$ , this requires the notion of aggrandizer. We find natural conditions on the aggrandizer under which the grand space is larger than the initial space. We provide a counterexample showing that this embedding is strict in general. In the case \(\varOmega =\mathbb {R}^n\) $\Omega ={\mathbb{R}}^n$ , we study weighted Hardy operators with radial weights and find conditions for their boundedness in grand complementary Morrey spaces. In case of quasi-monotone weights, we also show that the conditions on the weight in terms of Matuszewska–Orlicz indices sufficient for the boundedness of Hardy operators in complementary Morrey spaces also remain to be sufficient for grand complementary Morrey spaces.