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...

In this paper, we illustrate certain criteria which are sufficient for a henselian valued field to admit non-isomorphic maximal purely wild extensions.

We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity any $\alpha\in\nobreak (0,1)$ of minimizers the functional $$ v\ \mapsto\ \int\_\Omega \phi(x,|Dv|),dx, where $\phi$ satisfies a $(p,q)$-growth condition. Establishing such regularity theory with sharp, general conditions has been an open problem since 1980s. In contrast to previous results, we fo...