Coercive Inequalities on Metric Measure Spaces

We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincar\'e and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds

Hardy Spaces on Weighted Homogeneous Trees

We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space (V, d, μ) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H1(μ) on (V, d, μ) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H1(μ)

Axial tubule junctions control rapid calcium signaling in atria.

The canonical atrial myocyte (AM) is characterized by sparse transverse tubule (TT) invaginations and slow intracellular Ca2+ propagation but exhibits rapid contractile activation that is susceptible to loss of function during hypertrophic remodeling. Here, we have identified a membrane structure and Ca2+-signaling complex that may enhance the speed of atrial contraction independently of phospholamban regulation. This axial couplon was observed in human and mouse atria and is composed of voluminous axial tubules (ATs) with extensive junctions to the sarcoplasmic reticulum (SR) that include ryanodine receptor 2 (RyR2) clusters. In mouse AM, AT structures triggered Ca2+ release from the SR approximately 2 times faster at the AM center than at the surface. Rapid Ca2+ release correlated with colocalization of highly phosphorylated RyR2 clusters at AT-SR junctions and earlier, more rapid shortening of central sarcomeres. In contrast, mice expressing phosphorylation-incompetent RyR2 displayed depressed AM sarcomere shortening and reduced in vivo atrial contractile function. Moreover, left atrial hypertrophy led to AT proliferation, with a marked increase in the highly phosphorylated RyR2-pS2808 cluster fraction, thereby maintaining cytosolic Ca2+ signaling despite decreases in RyR2 cluster density and RyR2 protein expression. AT couplon "super-hubs" thus underlie faster excitation-contraction coupling in health as well as hypertrophic compensatory adaptation and represent a structural and metabolic mechanism that may contribute to contractile dysfunction and arrhythmias

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Approaching criticality via the zero dissipation limit in the abelian avalanche model

The discrete height abelian sandpile model was introduced by Bak, Tang & Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for $d > 4$.Comment: 46 pages, substantially revised 4th version, title has been changed. The main new material is Section 6 on toppling probabilities and the toppling probability exponen

Potential theory results for a class of PDOs admitting a global fundamental solution

We outline several results of Potential Theory for a class of linear par-tial differential operators L of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for L; under different geometrical assumptions on L (mainly, under global doubling/Poincar\ue9 assumptions), it is described how to obtainan invariant, non-homogeneous Harnack inequality. When L is equipped with a global fundamental solution \u393, further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on L ensuring that such a \u393 exists