Convolution-type operators in grand Lorentz spaces

We introduce and study a novel grand Lorentz space—that we believe is appropriate for critical cases—that lies “between” the Lorentz–Karamata space and the recently defined grand Lorentz space from Ahmed et al. (Mediterr J Math 17:57, 2020). We prove both Young’s and O’Neil’s inequalities in the newly introduced grand Lorentz spaces, which allows us to derive a Hardy–Littlewood–Sobolev-type inequality. We also discuss Köthe duality for grand Lorentz spaces, from which we obtain a new Köthe dual space theorem in grand Lebesgue spaces.