Fatou Type Theorems: Maximal Functions and Approach Regions by Fausto Di Biase (auth.)

By Fausto Di Biase (auth.)

A simple precept governing the boundary behaviour of holomorphic func­ tions (and harmonic capabilities) is that this: less than sure development stipulations, for nearly each aspect within the boundary of the area, those capabilities advert­ mit a boundary restrict, if we process the bounda-ry aspect inside definite technique areas. for instance, for bounded harmonic services within the open unit disc, the average technique areas are nontangential triangles with one vertex within the boundary element, and fully inside the disc [Fat06]. actually, those common process areas are optimum, within the feel that convergence will fail if we strategy the boundary within higher areas, having the next order of touch with the boundary. the 1st theorem of this type 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 impression areas of convergence that aren't nontangential: those greater process areas comprise tangential sequences (as against tangential curves). The phenomenon came upon through Nagel and Stein exhibits that the boundary behaviour of ho)omor­ phic capabilities (and harmonic functions), in theorems of Fatou variety, is regulated by means of a moment precept, which predicts the lifestyles of areas of convergence which are sequentially higher than the normal ones.

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Example text

E. ely often in both directions" [Pom92, p. 127] and [DBFU]; cf. [Bur89]. In particular, the corkscrew at a twist point of the snowflake domain has a twisted shape. 4) define an approach system for (D, bD). Proof. For w E bD and r > 0 let Jffi(W, r) ~ B(w, r) n bD.

For zED and w E W. The shadow nJ.. (z) . (s) E W: SnL(w) # 0} is the shadow of S projected by n. Let n be an approach family for (D, W), u a real-valued function defined on D, >. > 0, and w E W. } R 2sup0 = o. } . zERCw) 30 CHAPTER 2. 3 Let 9 be an approach family for (D, Till). following are equivalent: Then the 1. gi(z) is open for all zED; 2 . the set {supu>>'} 9 is open for each positive function u defined on D and each>' > O. Proof. (1 =? l-(z). (2 =? 1) Fix zED. Let u = 1 at the point z and 0 elsewhere.

3) 2. ]Rn \ D satisfies the corkscrew condition; 3. D satisfies the m-Harnack chain condition: If E > 0 and Zl and Z2 belong to D, d(zj, bD) > E and IZ1 - z21 < C . E, then there exists an m-Harnack chain from Zl to Z2 whose length depends on C but not on E. Observe that, by Harnack's inequality, if u is a positive harmonic function on D and there is an m-Harnack chain from Zl to Z2, then ~ . U(Z2) < u(zd < c· U(Z2), where c depends only on m and on the length of the Harnack chain between Zl and Z2.

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