In this survey, we describe joint work in collaboration with A. Stokolos, O. Svensson and T. Weiss. We consider the following question: How sharp is the Stolz approach region condition for the almost everywhere convergence of bounded harmonic functions? The issue was first settled in the rotation invariant case in the unit disc by Littlewood in 1927 and later examined, under less stringent conditions, by Aikawa in 1991. We show that our results are, in a precise sense, sharp. In particular, we show that a problem, first posed by Littlewood in 1927, leads to a statement that is independent of ZFC. This statement is concerned with the boundary behavior of bounded holomorphic functions in the unit disc along curves that end tangentially to the boundary.
On the sharpness of certain approach regions
DI BIASE, FAUSTO
2006-01-01
Abstract
In this survey, we describe joint work in collaboration with A. Stokolos, O. Svensson and T. Weiss. We consider the following question: How sharp is the Stolz approach region condition for the almost everywhere convergence of bounded harmonic functions? The issue was first settled in the rotation invariant case in the unit disc by Littlewood in 1927 and later examined, under less stringent conditions, by Aikawa in 1991. We show that our results are, in a precise sense, sharp. In particular, we show that a problem, first posed by Littlewood in 1927, leads to a statement that is independent of ZFC. This statement is concerned with the boundary behavior of bounded holomorphic functions in the unit disc along curves that end tangentially to the boundary.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
