Download e-book for kindle: Fatou Type Theorems: Maximal Functions and Approach Regions by Fausto Di Biase

By Fausto Di Biase

A easy precept governing the boundary behaviour of holomorphic func­ tions (and harmonic features) is that this: lower than definite development stipulations, for nearly each aspect within the boundary of the area, those features advert­ mit a boundary restrict, if we procedure the bounda-ry element inside of yes technique areas. for instance, for bounded harmonic features within the open unit disc, the traditional process areas are nontangential triangles with one vertex within the boundary element, and fully inside the disc [Fat06]. in truth, those typical strategy areas are optimum, within the experience that convergence will fail if we method the boundary within higher areas, having a better 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 method areas comprise tangential sequences (as against tangential curves). The phenomenon came upon by means of Nagel and Stein exhibits that the boundary behaviour of ho)omor­ phic capabilities (and harmonic functions), in theorems of Fatou style, is regulated by way of a moment precept, which predicts the lifestyles of areas of convergence which are sequentially better than the ordinary ones.

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

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