Positive definite temperature functions on the Heisenberg group

2006 ◽  
Vol 85 (8) ◽  
pp. 987-1000 ◽  
Author(s):  
Jinman Kim ◽  
M. W. Wong
Keyword(s):  
Heisenberg Group ◽  
Positive Definite ◽  
Definite Temperature

Positive definite temperature functions and a correspondence to positive temperature functions

1997 ◽  
Vol 127 (5) ◽  
pp. 947-961 ◽  
Author(s):  
Soon-Yeong Chung
Keyword(s):  
Positive Measure ◽  
Positive Definite ◽  
Positive Temperature ◽  
Stieltjes Transform ◽  
Schwartz Distributions ◽  
Image Position ◽  
Definite Temperature

SynopsisPositive definite temperature functionsu(x, t) in ℝn+1= {(x, t)|x∈ ℝn,t> 0} are characterised bywhere μ is a positive measure satisfying that for every ℰ > 0,is finite. A transformis introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions inThen correspondence given bygeneralises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.

INTEGRAL REPRESENTATIONS FOR LOCALLY DEFINED POSITIVE DEFINITE FUNCTIONS ON LIE GROUPS

1991 ◽  
Vol 02 (03) ◽  
pp. 257-286 ◽  
Author(s):  
PALLE E. T. JORGENSEN
Keyword(s):  
Hilbert Space ◽  
Spectral Analysis ◽  
Heisenberg Group ◽  
Lie Groups ◽  
Positive Definite ◽  
Integral Representations ◽  
Positive Definite Functions ◽  
Additional Symmetry ◽  
Noncommuting Operators ◽  
Unitary Dilation

It is well known that locally defined positive definite functions on Lie groups G generally do not extend to positive definite functions which are defined on the whole group. We introduce two stronger positivity concepts for locally defined functions, and show that they are equivalent to extendability. We then apply this to the case when G is the Heisenberg group. When an additional symmetry is imposed, we obtain a complete spectral analysis of the (locally defined) positive definite functions. The methods of proof are based on unitary dilation techniques (i.e., carefully chosen extensions of some underlying Hilbert space associated to the problem), and on spectral theory for noncommuting operators.

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators
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