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On Hardy–Littlewood Inequality for Brownian Motion on Riemannian Manifolds
Author(s) -
Grigor'yan Alexander,
Kelbert Mark
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070000123x
Subject(s) - mathematics , brownian motion , sequence (biology) , motion (physics) , combinatorics , pure mathematics , mathematical analysis , physics , statistics , classical mechanics , genetics , biology
Let { X i } i ⩾1 be a sequence of independent random variables taking the values ±1 with the probability ½, and let us set S n = X 1 + X 2 +…+ X n . A classical theorem of Hardy and Littlewood (1914) says that, for any C > 0 and for all n large enough, we haveS n ⩽ C n log n ,( 1 )with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequalityS n ⩽( 2 + ɛ ) n log log n ,( 2 )for any ɛ > 0. In view of Khinchin's result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this paper, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2).