The Journal of Geometric Analysis

, Volume 25, Issue 3, pp 1459–1475 | Cite as

On Generalizations of Fatou’s Theorem for the Integrals with General Kernels

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Abstract

We define \(\lambda (r)\)-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson–Stieltjes integrals for general kernels \(\{\varphi _r\}\), forming an approximation of identity. We prove that the bound
$$\begin{aligned} \limsup _{r\rightarrow 1}\lambda (r) \Vert \varphi _r\Vert _\infty <\infty \end{aligned}$$
is necessary and sufficient for almost everywhere \(\lambda (r)\)-convergence of the integrals
$$\begin{aligned} \int _\mathbb T\varphi _r(t-x)d\mu (t). \end{aligned}$$

Keywords

Fatou theorem Littlewood theorem Harmonic functions 

Mathematics Subject Classification

Primary 42B25 Secondary 32A40 

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Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Institute of Mathematics of Armenian National Academy of SciencesYerevanArmenia
  2. 2.Yerevan State University YerevanArmenia

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