Another counterexample to Zygmund’s conjecture

Application of Perron trees to geometric maximal operators

“…It appears that stated in this form, Zygmund's conjecture is false and this was proved by Soria in the simplest case n = 3 and k = 2. More recently, Rey exhibited new class of counter examples to this conjecture in [4]. However, those negative results do not necessarily indicate that the idea behind Zygmund's conjecture is false but rather that it is not correctly formulated.…”

Sharp weak‐type estimates for maximal operators associated to rare bases

“…Geometric maximal operators associated to rare bases occupy a fascinating middle ground between the Hardy–Littlewood and strong maximal operators. Significant mathematical work on the topic on rare bases has been done by, among others, Zygmund [13], Córdoba [1], Soria [8], and Rey [7]. For the purposes of this paper, a rare basis is a translation invariant collection of some (but not necessarily all) intervals in $$\mathbb {R}^n$$.…”

Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases

“…Geometric maximal operators associated to rare bases occupy a fascinating middle ground between the Hardy-Littlewood and strong maximal operators. Significant mathematical work on the topic on rare bases has been done by, among others, Zygmund [14], Córdoba [1], Soria [9], and Rey [8]. For the purposes of this paper, a rare basis is a translation invariant collection of some (but not all) intervals in R n .…”