Maximal operators on Lorentz spaces in non-doubling setting

We study mapping properties of the centered Hardy–Littlewood maximal operator $$\mathcal {M}$$ acting on Lorentz spaces $$L^{p,q}({\mathfrak {X}})$$ in context certain non-doubling metric measure $${\mathfrak {X}}$$ . The special class for which these are very peculiar is introduced and many examples given. In particular, each $$p_0, q_0, r_0 \in (1, \infty )$$ with $$r_0 \ge q_0$$ we construct a space associated bounded from $$L^{p_0,q_0}({\mathfrak to $$L^{p_0,r}({\mathfrak if only $$r r_0$$