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On Tangential and Projectively Adjacent Approach Regions

Fausto Di Biase, Haguma Gratien, Olof Svensson

TL;DR

The paper investigates the boundary behavior of bounded holomorphic functions on the unit disk, focusing on almost-everywhere convergence along various approach regions. It introduces projectively adjacent approach regions, a new class that includes both curvilinear and sequential tangential ends, and proves a sharp negative Fatou-type result for this class: there exists $h\, extin H^ty() such that for a.e. boundary point $oldsymbol{w}$, the limit along any projectively adjacent region fails to exist. This result extends classical Littlewood-type theorems and unifies prior negative results with a broader, countably infinite family of regions, while preserving a connection to seminal works of Fatou, Rudin, and Nagel–Stein. The analysis leverages Carleson tents, Poisson integrals, and a refined shadow/pointwise-conditions framework, and discusses possible higher-dimensional extensions and open problems, including regularity conditions and harmonic analogues. Overall, the work significantly advances understanding of when boundary limits fail for Fatou-type questions and provides a robust toolkit for analyzing tangential and projectively adjacent approach regions.

Abstract

In 1906 Fatou proved that bounded holomorphic functions on the unit disc converge a.e. on the boundary along nontangential approach regions. In 1927 Littlewood proved a negative result, i.e., that a.e. convergence fails for certain approach regions: More precisely, it fails for the rotationally invariant families of tangential approach regions that end curvilinearly at the boundary. The fact that tangential approach regions which end sequentially at the boundary may instead be very well conducive to a.e. convergence was understood more recently by W. Rudin (in 1979) and A. Nagel and E.M. Stein (in 1984), in contributions that provided additional and much needed insight, and that prompted the question of giving an a priori description of those families of tangential approach regions which end sequentially at the boundary and for which a.e. convergence fails. Our main result is the first one of this kind. Indeed, we prove the failure of a.e. convergence for a class of approach regions, introduced in this work, which we call projectively adjacent, that contains curvilinear approach regions as well as sequential ones. Our result recaptures the aforementioned theorem of Littlewood as well as some of the other theorems of that type, but its novelty lies in the fact that, while all previous negative results for tangential approach (Littlewood, 1927; Lohwater and Piranian, 1957; Aikawa, 1990-1991, and others) dealt with approach regions that either are curves or share with curves a certain topological property that excludes the possibility that they could end sequentially at the boundary, our negative result deals with a class of approach regions that contains both the curvilinear and the sequential ones. Hence we present a significant extension of the class of those families of approach regions for which a.e. convergence fails.

On Tangential and Projectively Adjacent Approach Regions

TL;DR

The paper investigates the boundary behavior of bounded holomorphic functions on the unit disk, focusing on almost-everywhere convergence along various approach regions. It introduces projectively adjacent approach regions, a new class that includes both curvilinear and sequential tangential ends, and proves a sharp negative Fatou-type result for this class: there exists oldsymbol{w}$, the limit along any projectively adjacent region fails to exist. This result extends classical Littlewood-type theorems and unifies prior negative results with a broader, countably infinite family of regions, while preserving a connection to seminal works of Fatou, Rudin, and Nagel–Stein. The analysis leverages Carleson tents, Poisson integrals, and a refined shadow/pointwise-conditions framework, and discusses possible higher-dimensional extensions and open problems, including regularity conditions and harmonic analogues. Overall, the work significantly advances understanding of when boundary limits fail for Fatou-type questions and provides a robust toolkit for analyzing tangential and projectively adjacent approach regions.

Abstract

In 1906 Fatou proved that bounded holomorphic functions on the unit disc converge a.e. on the boundary along nontangential approach regions. In 1927 Littlewood proved a negative result, i.e., that a.e. convergence fails for certain approach regions: More precisely, it fails for the rotationally invariant families of tangential approach regions that end curvilinearly at the boundary. The fact that tangential approach regions which end sequentially at the boundary may instead be very well conducive to a.e. convergence was understood more recently by W. Rudin (in 1979) and A. Nagel and E.M. Stein (in 1984), in contributions that provided additional and much needed insight, and that prompted the question of giving an a priori description of those families of tangential approach regions which end sequentially at the boundary and for which a.e. convergence fails. Our main result is the first one of this kind. Indeed, we prove the failure of a.e. convergence for a class of approach regions, introduced in this work, which we call projectively adjacent, that contains curvilinear approach regions as well as sequential ones. Our result recaptures the aforementioned theorem of Littlewood as well as some of the other theorems of that type, but its novelty lies in the fact that, while all previous negative results for tangential approach (Littlewood, 1927; Lohwater and Piranian, 1957; Aikawa, 1990-1991, and others) dealt with approach regions that either are curves or share with curves a certain topological property that excludes the possibility that they could end sequentially at the boundary, our negative result deals with a class of approach regions that contains both the curvilinear and the sequential ones. Hence we present a significant extension of the class of those families of approach regions for which a.e. convergence fails.

Paper Structure

This paper contains 96 sections, 23 theorems, 205 equations, 3 figures.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Four notable classes of approach regions
  4. Notation
  5. Approach regions and limiting values
  6. ($\boldsymbol{\mathsf{t}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$)
  7. ($\boldsymbol{\mathsf{c}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$)
  8. ($\boldsymbol{\mathsf{s}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$)
  9. ($\boldsymbol{\mathsf{a}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$)
  10. Projectively adjacent approach regions
  11. ($\boldsymbol{\mathsf{p}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$)
  12. Further background:
  13. Our main result, its significance, with a look at open problems
  14. The significance of our main result
  15. Comment on our proof
  16. ...and 81 more sections

Key Result

Lemma 1.1

$\boldsymbol{\mathsf{c}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]\subsetneq\boldsymbol{\mathsf{a}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]$ and $\boldsymbol{\mathsf{a}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]\cap\boldsymbol{\mathsf{s}}[\mathop{\mathrm{\mathtt{w}}}\nolimits]=\emptyset$.

Figures (3)

  • Figure 1: The cross-section condition fails in (T) but holds in (S) and (O).
  • Figure 2: A diagram with Littlewood, or not Littlewood, subsets of ${ { \boldsymbol{[\![} \mathop{\mathrm{\partial\mathop{\mathrm{\mathbb{D} }}\nolimits }}\nolimits \mathrel{ \mkern2mu \clipbox{{.5} 0 0 0}{$$}} \mathop{\mathrm{\mathcal{P}}}\nolimits{}{(\mathop{\mathrm{\mathbb{D} }}\nolimits)} \boldsymbol{]\!]} } }_{\boldsymbol{\ell}}$.
  • Figure 3: A larger Littlewood set.

Theorems & Definitions (57)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • Example 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 47 more