On the Dimensions of Spaces of Harmonic Functions with Polynomial Growth

2019 ◽  
Vol 39 (5) ◽  
pp. 1219-1234
Author(s):  
Xiantao Huang
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth

scholarly journals Polynomial growth harmonic functions on finitely generated abelian groups

2013 ◽  
Vol 44 (4) ◽  
pp. 417-432 ◽  
Author(s):  
Bobo Hua ◽  
Jürgen Jost ◽  
Xianqing Li-Jost
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Abelian Groups ◽  
Finitely Generated

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 Polynomial growth harmonic functions on groups of polynomial volume growth

2015 ◽  
Vol 280 (1-2) ◽  
pp. 551-567 ◽  
Author(s):  
Bobo Hua ◽  
Jürgen Jost
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth

Sobolev Inequalities and Harmonic Functions of Polynomial Growth

1993 ◽  
Vol s2-48 (3) ◽  
pp. 452-464 ◽  
Author(s):  
G. Alexopoulos ◽  
N. Lohoué
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Sobolev Inequalities

scholarly journals Ancient Caloric Functions on Graphs With Unbounded Laplacians

2020 ◽  
Author(s):  
Bobo Hua
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Equations ◽  
Intrinsic Metric ◽  
Ancient Solutions

Abstract We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We extend Colding and Minicozzi’s theorem [12] on manifolds and the result [22] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: for a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor.

scholarly journals On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

2020 ◽  
Vol 2020 (762) ◽  
pp. 281-306 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Asymptotic Behavior ◽  
Harmonic Functions ◽  
Tangent Cone ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Volume Ratio ◽  
Growth Order ◽  
Maximal Volume ◽  
Nonnegative Ricci Curvature ◽  
Asymptotic Volume

AbstractSuppose {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of {h_{d}(M)}. In this paper, we study the behavior of {h_{d}(M)} when d is large. More precisely, suppose {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}}. When {\alpha=\omega_{n}}, i.e., {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of {h_{d}(\mathbb{R}^{n})}.

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