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Journal of the London Mathematical Society 2000 62(2):625-639; doi:10.1112/S002461070000123X
© 2000 by London Mathematical Society
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© The London Mathematical Society

On Hardy–Littlewood Inequality for Brownian Motion on Riemannian Manifolds

Alexander Grigor'yan and Mark Kelbert

Department of Mathematics, Imperial College London SW7 2BZ, a.grigoryan{at}ic.ac.uk
European Business Management School, University of Wales Swansea Singleton Park, Swansea SA2 8PP, m.kelbert{at}swansea.ac.uk

Received 12 January 1998. Revision received 16 September 1999.

Let {Xi}i≥1 be a sequence of independent random variables taking the values ±1 with the probability 1/2, and let us set Sn = X1 + X2 +...+ Xn. A classical theorem of Hardy and Littlewood (1914) says that, for any C > 0 and for all n large enough, we have


Formula

with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequality


Formula

for any {varepsilon} > 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).


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