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

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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

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