Get Fatou Type Theorems: Maximal Functions and Approach Regions PDF

By Fausto Di Biase

A easy precept governing the boundary behaviour of holomorphic func­ tions (and harmonic capabilities) is that this: less than convinced development stipulations, for nearly each element within the boundary of the area, those features advert­ mit a boundary restrict, if we technique the bounda-ry element inside of sure process areas. for instance, for bounded harmonic capabilities within the open unit disc, the average method areas are nontangential triangles with one vertex within the boundary aspect, and completely inside the disc [Fat06]. in truth, those traditional process areas are optimum, within the feel that convergence will fail if we process the boundary within greater areas, having the next order of touch with the boundary. the 1st theorem of this kind is because of J. E. Littlewood [Lit27], who proved that if we exchange a nontangential zone with the rotates of any mounted tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that during Euclidean part­ areas (and the unit disc) there are in influence areas of convergence that aren't nontangential: those greater process areas include tangential sequences (as against tangential curves). The phenomenon chanced on via Nagel and Stein shows that the boundary behaviour of ho)omor­ phic services (and harmonic functions), in theorems of Fatou sort, is regulated through a moment precept, which predicts the life of areas of convergence which are sequentially better than the usual ones.

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Fatou Type Theorems: Maximal Functions and Approach Regions (Progress in Mathematics) by Fausto Di Biase


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