The authors study maximal operators related to bases on the infinite-dimensional torus \(\mathbb T^\omega\). A basis \(\mathfrak B\) is a collection of Borel measurable subsets of the infinite-dimensional torus with finite positive measure and the maximal function \(M^\mathfrak B\) is defined as usual by \[ M^\mathfrak B f(x)=\sup_{x\in Q\in \mathfrak B} \frac{1}{|Q|}\int_Q |f(x)|\,dx, \] where \(dx\) denotes the Haar measure on the group \(\mathbb T^\omega\). There are two concrete bases that have already been considered: the so-called Rubio de Francia basis denoted by \(\mathfrak R\), which plays the role of the cubes in \(\mathbb R^n\), and a subclass denoted by \(\mathfrak R_0\), the so-called restricted Rubio de Francia basis, which plays the role of the dyadic cubes in \(\mathbb R^n\). It is known that \(M^\mathfrak R\) is not of weak type \((1,1)\) (see [D. Kosz, Stud. Math. 258, No. 1, 103–119 (2021; Zbl 1466.43004)]) while \(M^\mathfrak {R_0}\) is of weak type \((1,1)\) (see [E. Fernández and L. Roncal, Potential Anal. 53, No. 4, 1449–1465 (2020; Zbl 1443.43007)]).
The authors find a wide class of bases \(\mathfrak R'\) such that \(\mathfrak R_0\subset \mathfrak R'\subset \mathfrak R\) for which the corresponding maximal functions are not even of restricted weak type \((q,q)\). This is done using what they call, for \(0<\varepsilon\le 1/2\) and \(l\in \mathbb N\), \((\varepsilon,l)\)-configurations which are collections \(\{Q^{(1)},\dots, Q^{(l)}\}\subset \mathfrak R\) such that there exists some measurable \(A\subset \mathbb T^\omega\) such that the \(Q^{(k)}\setminus A\) are pairwise disjoint and satisfy that \(|A|=|Q^{(k)}|\) and \(\varepsilon|A|\le |Q^{(k)}\cap A|\le (1-\varepsilon)|A|\) for each \(k\). It is shown that any basis \(\mathfrak {R'}=\mathfrak R_0\cup (\bigcup_j S_j)\) where \(S_j\) are \((\varepsilon_j,l_j)\)-configurations with the condition \(\sup_{j} \varepsilon_j^{q+1}l_j=\infty\) satisfies that \(M^{\mathfrak R'}\) is not of restricted weak type \((q,q)\). Actually, for each \(1\le q_0<\infty\) they manage to find the precise condition in case of bases generated using \((\varepsilon,l)\)-configurations to characterize when \(M^\mathfrak {R'}\) is not of restricted weak type \((q,q)\).
On the other hand, they consider also the weighted setting and introduce the Muckenhoupt class \(A_p^{\mathfrak R}(\mathbb T^\omega)\) and the reverse Hölder class \(RH_r^{\mathfrak R}(\mathbb T^\omega)\) associated to a basis \(\mathfrak R\). They manage to show that if \(w\in A_p^{\mathfrak R}(\mathbb T^\omega)\) for some \(1<p<\infty\), then for each \(1\le q<\infty\) there exists \(\mathfrak R'\) such that \(M^\mathfrak {R'}\) is not of weighted restricted weak type \((q,q)\). In particular if either \(w\in A_p^{\mathfrak R}(\mathbb T^\omega)\) for some \(1<p<\infty\) or \(w\in RH_r^{\mathfrak R}(\mathbb T^\omega)\) for some \(1<r<\infty\), then \(M^\mathfrak {R}\) is not of weighted restricted weak type \((q,q)\) for all \(1\le q <\infty\).