Riesz transforms on graphs for $1 \leq p \leq 2$

Authors
Publisher
MATHEMATICA SCANDINAVICA
Publication Date

Abstract

omslaga 1..6 {orders}ms/000562/russ.3d -3.10.00 - 08:31 RIESZ TRANSFORMS ON GRAPHS FOR 1 � p � 2 EMMANUEL RUSS Abstract We prove, for 1 < p � 2, the Lp-boundedness of Riesz transforms on graphs satisfying the dou- bling property and a on-diagonal estimate of the Markov kernel. In [6], Coulhon and Duong proved the analogous result on Riemannian manifolds. We follow closely Coulhon and Duong’s work. However, the discrete setting creates difficulties which do not appear in [6]. 1. Introduction This paper deals with Riesz transforms on graphs endowed with suitable Markov kernels. In this setting, one may define a discrete gradient r and a ‘‘Laplace operator’’ �. The issue is to know whether krf kp and k I ÿ P ÿ12f kp are comparable uniformly in f . It is clear when p 2. The question arises when p 6 2 and is equivalent to the Lp-continuity of the op- erator r�ÿ12, which is called the Riesz transform. Let ÿ be a infinite graph, endowed with a measure m satisfying 8x 2 ÿ; m x > 0: 1 We assume that ÿ is connected and locally uniformly finite, which means that sup x2ÿ N x <1 where, for x 2 ÿ , N x is the number of neighbours of x. We also assume that ÿ is endowed with its natural distance d. Denote by B x; r the closed ball of center x and of radius r, and by V x; r its volume. We assume that ÿ has the doubling property, i. e. there exists C > 0 such that V x; 2r � CV x; r ; 8x 2 ÿ; r > 0: 2 That property implies that there exists D > 0 such that V x; �r � C�DV x; r ; 8x 2 ÿ; r > 0; � > 1: 3 MATH. SCAND. 87 (2000), 133^160 Received November 24, 1999. {orders}ms/000562/russ.3d -3.10.00 - 08:32 Let p be a Markov kernel on ÿ , i. e. a non-negative map defined on ÿ � ÿ such that X y2ÿ p x; y 1; 8x 2 ÿ: Assume that p is reversible with respect to m, which means that m x p x; y m y p y; x for all x; y 2 ÿ: 4 We also assume that there exists r0 > 0 such that p x; y 0 whenever d x; y � r0 5 and that inf d x;y �1 p x; y > 0: 6 The iter

Seen <100 times

More articles like this

Jan 10, 2014

Jan 01, 2004

Jun 13, 2012

7Li, 31P and 1H Static and MAS NMR investigations...

Jan 01, 2004
More articles like this..