On pointwise convergence of singular integrals of Christ-Journé type

Abstract

The primary objective of this paper is to establish the pointwise almost everywhere convergence of the singular integrals of Christ-Journé type. To this end, we prove the weak type $(1,1)$ estimate for the corresponding maximal operator defined as \[ T^*[a] f(x):= \sup_{r>0}\left|\int_{|x-y| > r} K(x-y) \int_0^1 a(tx + (1-t)y)\, dt f(y)\, dy\right|, \] where $K$ is a Calderón--Zygmund kernel and $a\in L^\infty$. This result resolves a problem posed by Seeger (Rev. Mat. Iberoam., 2014).