New bounds on Cantor maximal operators


  • Pablo Shmerkin Department of Mathematics, the University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
  • Ville Suomala Research Unit of Mathematical Sciences, P.O. Box 8000, FI-90014, University of Oulu, Finland



 We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors-regular Salem Cantor sets of any dimension $> 1/2$ such that the associated maximal operator is bounded on $L^2(\mathbb{R})$. We follow the overall scheme of Łaba-Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures.


Download data is not yet available.


J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 0874045.

K. Falconer, Fractal Geometry, third edition, John Wiley & Sons, Chichester, 2014. MR 3236784.

M. Hochman, Some problems on the boundary of fractal geometry and additive combinatorics, in Recent Developments in Fractals and Related Fields, 129–174, Trends Math, Birkhäuser/Springer, Cham, 2017. MR 3775463.

S. Janson, Large deviations for sums of partly dependent random variables, Random Structures Algorithms 24 (2004), no. 3, 234–248. MR 2068873.

I. Łaba, Maximal operators and decoupling for $Lambda(p)$ Cantor measures, Ann. Fenn. Math. 46 (2021), no. 1, 163–186. MR 4277805.

I. Łaba and M. Pramanik, Maximal operators and differentiation theorems for sparse sets, Duke Math. J. 158 (2011), no. 3, 347–411. MR 2805064.

J. L. Rubio de Francia, Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), no. 2, 395–404. MR 0850542.

P. Shmerkin and V. Suomala, Spatially independent martingales, intersections, and applications, Mem. Amer. Math. Soc. 251 (2018), no. 1195. MR 3756896.

P. Shmerkin and V. Suomala, Patterns in random fractals, Amer. J. Math. 142 (2020), no. 3, 683–749. MR 4101330.

E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 0420116.