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Fourier transform of Schwartz functions on the Heisenberg group

Francesca Astengo, Bianca Di Blasio, Fulvio Ricci (2013)

Studia Mathematica

Let H₁ be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space 𝓢ₘ(H₁) of Schwartz functions of type m and the space 𝓢(Σₘ) consisting of restrictions of Schwartz functions on ℝ² to a subset Σₘ of the Heisenberg fan with |m| of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H₁.

Fourier-Wigner transforms and Liouville's theorems for the sub-Laplacian on the Heisenberg group

Aparajita Dasgupta, M. W. Wong (2010)

Banach Center Publications

The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.

Fractional cartesian products of sets

Ron C. Blei (1979)

Annales de l'institut Fourier

Let E be a subset of a discrete abelian group whose compact dual is G . E is exactly p -Sidon (respectively, exactly non- p -Sidon) when ( * ) C E ( G ) ∧ ⊂ ℓ r holds if and only if r ∈ [ p , ∞ ] (respectively, r ∈ ( p , ∞ ) ). E is said to be exactly Λ β (respectively, exactly non- Λ β ) if E has the property ( * * ) every f ∈ L E 2 ( G ) satisfies ∫ G exp ( λ | f | 2 / α < ∞ , for all λ > 0 , if and only if α ∈ [ β , ∞ ) (respectively, α ∈ ( β , ∞ ) ).In this paper, for every p ∈ [ 1 , 2 ) and β ∈ [ 1 , ∞ ) , we display sets which are exactly p -Sidon, exactly non- p -Sidon, exactly Λ β and exactly non- Λ β .

Fredholm multipliers of semisimple commutative Banach algebras.

Pietro Aiena (1991)

Extracta Mathematicae

In some recent papers ([1],[2],[3],[4]) we have investigated some general spectral properties of a multiplier defined on a commutative semi-simple Banach algebra. In this paper we expose some aspects concerning the Fredholm theory of multipliers.

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