An analogue for harmonic functions of Kolmogorov's law of the iterated logarithm
type: Item
instance of: scholarly article
title: An analogue for harmonic functions of Kolmogorov's law of the iterated logarithm
zbMATH Open document ID: 0666.31002
DOI: 10.1215/S0012-7094-88-05702-X
author: Rodrigo Bañuelos
author: Charles N. Moore
author: Ivo Klemeš
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The authors prove the ``law'' in the title in its analysis setting. This result is in turn a consequence of the following ``restricted version''. Let \(L(t)=(2t^ 2\log \log t)^{1/2}.\) In the upper half-plane \(D=\{(x,y):\quad y>0\},\) let \(\Gamma^ 1_{\alpha}(x)\) be the non- tangential cone of vertex (x,0), of half-aperture \(\tan^{-1}\alpha\) and truncated at \(y=1\). For any f continuous in D, let \(A^ 1_{\beta}(f)(x)\) be the square root of the integral in \(\Gamma^ 1_{\beta}(x)\) of \((\partial f/\partial \xi)^ 2\); further, let \(A^ 1_{\beta}(f)(x,\epsilon)\) denote the same integral on the same cone but translated to vertex (x,\(\epsilon)\) and truncated at \(y=1\). Then, given \(0<\alpha <\beta\), \(0<\gamma <1\), there exists a constant C such that, for every u harmonic in D satisfying \(\int^{\infty}_{-\infty}| u(x,y)| dx<\infty\) \((y>0)\), we have that, for almost every x, the condition \(A^ 1_{\beta}(u)(x)<\infty\) implies sup\(\{\) \(| u(\xi,\eta)|:\) \((\xi,\eta)\in \Gamma^ 1_{\alpha}(x)\}<\infty\), while \(A^ 1_{\beta}(u)(x)=\infty\) implies \[ \limsup | u(\xi,\eta)| /L(A^ 1_{\beta}(u)(\xi,\gamma \eta))<C(1+\alpha^{- 3/2})\quad as\quad (\xi,\eta)\to (x,0) \] from inside \(\Gamma^ 1_{\alpha}(x)\).
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 4089874
zbMATH Keywords: iterated logarithm
zbMATH Keywords: upper half-plane
zbMATH Keywords: harmonic
MaRDI profile type: Publication
cites work: Probabilistic Behaviour of Functions in the Zygmund Spaces ∧ * and λ *
cites work: Q3025975
cites work: Q3833343
cites work: Sharp Estimates for the Nontangential Maximal Function and the Lusin Area Function in Lipschitz Domains
cites work: Variants on the Law of the Iterated Logarithm
cites work: Parabolic maximal functions associated with a distribution
cites work: Parabolic maximal functions associated with a distribution. II
cites work: Some weighted norm inequalities concerning the Schrödinger operators
cites work: On the Maximum Partial Sums of Sequences of Independent Random Variables
cites work: Q4114551
cites work: Q3217388
cites work: Q3229521
cites work: Q3922907
cites work: The Martingale Version of a Theorem of Marcinkiewicz and Zygmund
cites work: The density of the area integral in \({\mathbb{R}}_+^{n+1}\)
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Weighted H p Spaces on Lipschitz Domains
cites work: An iterated logarithm law for local time
cites work: On the Distortion of Boundary Sets Under Conformal Mappings
cites work: Good λ inequalities for the area integral and the nontangential maximal function
cites work: Almost sure invariance principles for partial sums of weakly dependent random variables
cites work: Invariance principles for martingales and sums of independent random variables
cites work: On the law of iterated logarithm for Bloch functions
cites work: Q5798186
cites work: Q5607484
cites work: Q5611461
cites work: An invariance principle for the law of the iterated logarithm
cites work: Almost sure invariance principles for lacunary trigonometric series
cites work: The Law of the Iterated Logarithm for Lacunary Trigonometric Series
full work available at URL: s0012-7094-88-05702-x
OpenAlex ID: W1557845660
description: scientific article; zbMATH DE number 4089874