The Cauchy-Ahlfors operator, an invariant differential operator for vector fields

1983 ◽  
pp. 391-401
Author(s):  
H. M. Reimann
Keyword(s):  
Differential Operator ◽  
Vector Fields ◽  
Invariant Differential Operator ◽  
Invariant Differential

scholarly journals On an analytic description of the α-cosine transform on real Grassmannians

2016 ◽  
Vol 18 (02) ◽  
pp. 1550025 ◽  
Author(s):  
Semyon Alesker ◽  
Dmitry Gourevitch ◽  
Siddhartha Sahi
Keyword(s):  
Differential Operator ◽  
Radon Transform ◽  
Open Problem ◽  
Analytic Description ◽  
Radon Transforms ◽  
Invariant Differential Operator ◽  
Cosine Transform ◽  
Invariant Differential

The goal of this paper is to describe the [Formula: see text]-cosine transform on functions on real Grassmannian [Formula: see text] in analytic terms as explicitly as possible. We show that for all but finitely many complex [Formula: see text] the [Formula: see text]-cosine transform is a composition of the [Formula: see text]-cosine transform with an explicitly written (though complicated) [Formula: see text]-invariant differential operator. For all exceptional values of [Formula: see text] except one, we interpret the [Formula: see text]-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value [Formula: see text], which is [Formula: see text], is still an open problem.

scholarly journals Asymptotics for the Radon transform on hyperbolic spaces

2019 ◽  
pp. 241-251
Author(s):  
Nils Byrial ANDERSEN ◽  
Mogens FLENSTED-JENSEN
Keyword(s):  
Differential Operator ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Hyperbolic Spaces ◽  
Schwartz Function ◽  
Invariant Differential Operator ◽  
Abel Transform ◽  
Invariant Functions ◽  
Invariant Differential

Let G/H be a hyperbolic space over R; C or H; and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K∩H-invariant functions.

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