Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 253-263

COMPLEX GEODESICS, THEIR BOUNDARY REGULARITY, AND A HARDY-LITTLEWOOD-TYPE LEMMA

Gautam Bharali

Indian Institute of Science, Department of Mathematics
Bangalore 560012, India; bharali 'at' math.iisc.ernet.in

Abstract. We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to ∂D. This example suggests that continuity at the boundary of the complex geodesics of a convex domain Ω ⊂ Cⁿ, n ≥ 2, is affected by the extent to which ∂Ω curves or bends at each boundary point. We provide a sufficient condition to this effect (on C¹-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy-Littlewood-type lemma that might be of independent interest.

2010 Mathematics Subject Classification: Primary 30H05, 32H40; Secondary 32F45.

Key words: Boundary regularity, complex geodesics, Hardy-Littlewood lemma.

Reference to this article: G. Bharali: Complex geodesics, their boundary regularity, and a Hardy-Littlewood-type lemma. Ann. Acad. Sci. Fenn. Math. 41 (2016), 253-263.

Full document as PDF file

doi:10.5186/aasfm.2016.4116