We prove a homogenization formula for a sub-Laplacian are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1to weak-L1.

Let µ be a probability measure with finite support on a discrete group of polynomial volume growth. The main purpose of this paper is to study the asymptotic behavior of the convolution powers µ * n of µ. If µ is centered, then we prove upper and lower Gaussian estimates. We prove a central limit theorem and we give a generalization of the Berry-Esseen theorem. These results also extend to noncentered probability measures. We study the associated Riesz transform operators. The main tool is a parabolic Harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of Krylov and Safonov. This inequality implies that the positive µ-harmonic functions are constant. Finally we give a characterization of the µ-harmonic functions which grow polynomially.

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Background-Dyspnea and fatigue are the main causes of exercise limitation in chronic heart failure (CHF) patients, whose peak inspiratory (Pi max ) and expiratory pressures (Pe max ) are often reduced. The aim of this study was to examine the relationship between respiratory muscle performance and oxygen kinetics. Methods and Results-A total of 55 patients (NYHA class I to III) and 11 healthy subjects underwent cardiopulmonary exercise tests (CPET) on a treadmill. In 45 of the 55 patients (group I) and in healthy subjects (group II), pulmonary function tests, Pi max , and Pe max were measured before and 10 minutes after exercise, and oxygen kinetics were monitored throughout and during early recovery from CPET. The first degree slope of oxygen consumption (V O 2 ) decline during early recovery (V O 2 /t-slope) and V O 2 half-time (T 1/2 ) were calculated. In 10 of the 55 CHF patients (group III), the measurements of Pi max were repeated 2, 5, and 10 minutes after CPET. A Ͼ10% reduction in Pi max after CPET (subgroup IA) was measured in 11 of 45 patients. In contrast, 34 of 45 CHF patients (subgroup IB) and all control subjects (group II) had Pi max Ͼ90% of baseline value after CPET. Subgroup IA patients had significantly lower peak V O 2 (13.5Ϯ2.1 versus 17.8Ϯ5.6 mL ⅐ kg Ϫ1 ⅐ min Ϫ1 ; PϽ0.001), lower anaerobic thresholds (10.1Ϯ2.4 versus 13.6Ϯ4.6 mL ⅐ kg Ϫ1 ⅐ min

Abstract. Let L be a left invariant sub-Laplacian on a connected Lie group G of polynomial volume growth, and let {Ex, A > 0} be the spectral resolution of L and m a bounded Borel measurable function on [0, oo). In this article we give a sufficient condition on m for the operator m{L) = /0°° m{k) dE^ to extend to an operator bounded on LP{G), 1 < p < co , and also from L1{G) to weak-L'(G).

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