On the sharpness of the stolz approach

We study the sharpness of the Stolz approach for the a.e. convergence of functions in the Hardy spaces in the unit disc, first settled in the rotation invariant case by J. E. Littlewood in 1927 and later examined, under less stringent, quantitative hypothesis, by H. Aikawa in 1991. We introduce a new regularity condition, of a qualitative type, under which we prove a version of Littlewood's theorem for tangential approach whose shape may vary from point to point. Our regularity condition can be extended in those contexts where no group is involved, such as NTA domains in Rn. We show exactly in what sense our regularity condition is sharp.