Some regularity estimates for convolution semigroups on a group of polynomial growth

2004 ◽  
Vol 77 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Nick Dungey
Keyword(s):  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Kernels ◽  
High Order ◽  
Convolution Semigroup ◽  
Geometric Condition ◽  
Convolution Semigroups ◽  
Regularity Estimates ◽  
Convolution Powers ◽  
Gaussian Estimates

AbstractWe study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

scholarly journals Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015 ◽  
Vol 07 (03) ◽  
pp. 407-451 ◽  
Author(s):  
Urs Frauenfelder ◽  
Clémence Labrousse ◽  
Felix Schlenk
Keyword(s):  
Lower Bound ◽  
Polynomial Growth ◽  
Polynomial Complexity ◽  
Cohomology Ring ◽  
Volume Growth ◽  
Universal Cover ◽  
Geodesic Flows ◽  
Dynamical Complexity ◽  
Reeb Flows ◽  
Rank One

We give a uniform lower bound for the polynomial complexity of Reeb flows on the spherization (S*M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our lower bound is in terms of the polynomial growth of the homology of the based loop space of M. As an application, we extend the Bott–Samelson theorem from geodesic flows to Reeb flows: If (S*M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.

scholarly journals A Liouville theorem for Lévy generators

Positivity ◽  
2020 ◽  
Author(s):  
Franziska Kühn
Keyword(s):  
Weak Solution ◽  
Laplace Equation ◽  
Lévy Process ◽  
Lévy Processes ◽  
Infinitesimal Generator ◽  
Polynomial Growth ◽  
Liouville Theorem ◽  
Levy Processes ◽  
Levy Process ◽  
Regularity Estimates

AbstractUnder mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

scholarly journals Dimensional Bounds for Ancient Caloric Functions on Graphs

2019 ◽  
Author(s):  
Bobo Hua
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Equations ◽  
Ancient Solutions

Abstract We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi’s theorem [9] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.

scholarly journals Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature

2015 ◽  
Vol 2015 (700) ◽  
pp. 1-36 ◽  
Author(s):  
Bobo Hua ◽  
Jürgen Jost ◽  
Shiping Liu
Keyword(s):  
Riemannian Manifolds ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Geometric Analysis ◽  
Nonnegative Curvature ◽  
Doubling Property ◽  
Alexandrov Geometry ◽  
Combinatorial Curvature ◽  
Volume Doubling

AbstractWe apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. Finally, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.

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