Vol. 107, No. 2, 1983

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On the behavior near a torus of functions holomorphic in the ball

Patrick Robert Ahern

Vol. 107 (1983), No. 2, 267–278

If f is bounded and holomorphic in the unit ball in Cn then it has radial limits at almost all points of the boundary of the ball. More is true; for example, f will have limits almost everywhere with respect to arclength on any arc that forms part of the boundary of an anlaytic disc. Motivated by these considerations we consider an n-dimensional torus in the boundary of the ball and ask if there are growth conditions less restrictive than boundedness that imply the existence of radial limits on this torus. We show that the answer is no for some of the standard function classes. For example, we show that there is holomorphic function of bounded mean oscillation in the ball that has a finite radial limit at no point of the torus.

Mathematical Subject Classification 2000
Primary: 32E35
Secondary: 32A35, 32A40
Received: 22 September 1981
Revised: 5 May 1982
Published: 1 August 1983
Patrick Robert Ahern
Department of Mathematics
University of Wisconsin
Madison WI
United States