On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions

For the one-sided Hardy–Littlewood maximal operator $M_d^{+}$ on ${\mathbb{R}}^d$, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak $(p,p)$ inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint $p=1$ in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman–Stein-type question and the related weighted weak-type $(1,1)$ question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every $d\ge 3$ and $p>1$. More precisely, for every $1<p<\mathrm{\infty }$ and every $d\ge 3$, there exist weights $w$ and $v$ such that $$A_{p,d}^{+}(w,v)<\mathrm{\infty },\mathrm{\parallel }{M}_d^{+}{\mathrm{\parallel }}_{L^p(v)\to L^{p,\mathrm{\infty }}(w)}=\mathrm{\infty }\mathrm{.}$$ The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by $c_p(\log N{)}^{1\mathrm{/}{p}^{\mathrm{\prime }}}$. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension $d>3$.