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1
Content available remote

A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)

100%
Godoy T., Saal L.
Colloquium Mathematicum
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2006
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tom 106
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nr 2
231-255
EN
Let 𝓢(Hₙ) be the space of Schwartz functions on the Heisenberg group Hₙ. We define a spherical transform on 𝓢(Hₙ) associated to the action (by automorphisms) of U(p,q) on Hₙ, p + q = n. We determine its kernel and image and obtain an inversion formula analogous to the Godement-Plancherel formula.
2
Content available remote

The type set for homogeneous singular measures on ℝ ³ of polynomial type

100%
Ferreyra E., Godoy T.
Colloquium Mathematicum
|
2006
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tom 106
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nr 2
161-175
EN
Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $μ(E) = ∫_{D} χ_{E}(x,φ(x))dx$ with D = {x ∈ ℝ ²:|x| ≤ 1} and let $T_{μ}$ be the convolution operator with the measure μ. Let $φ = φ₁^{e₁} ⋯ φₙ^{eₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_{i} ≠ m/2$ for each $φ_{i}$ of degree 1, then the type set $E_{μ}: = {(1/p,1/q) ∈ [0,1] × [0,1]: ||T_{μ}||_{p,q} < ∞}$ can be explicitly described as a closed polygonal region.
3
Content available remote

On the relative fundamental solutions for a second order differential operator on the Heisenberg group

100%
Godoy T., Saal L.
Studia Mathematica
|
2001
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tom 145
|
nr 2
143-164
EN
Let Hₙ be the (2n+1)-dimensional Heisenberg group, let p,q ≥ 1 be integers satisfying p+q=n, and let $L = ∑_{j=1}^{p} (X²_{j} + Y²_{j}) -∑_{j=p+1}^{n} (X²_{j} + Y²_{j})$, where {X₁,Y₁,...,Xₙ,Yₙ,T} denotes the standard basis of the Lie algebra of Hₙ. We compute explicitly a relative fundamental solution for L.
4
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$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group

100%
Godoy T., Rocha P.
Colloquium Mathematicum
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2013
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tom 132
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nr 1
101-111
EN
We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $ν(E) = ∫_{ℂⁿ} χ_{E}(w,φ(w)) η(w)dw$, where $φ(w) = ∑_{j=1}^{n} a_{j}|w_{j}|²$, w = (w₁,...,wₙ) ∈ ℂⁿ, $a_{j} ∈ ℝ$, and η(w) = η₀(|w|²) with $η₀ ∈ C_{c}^{∞}(ℝ)$. We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^{p}(ℍⁿ) - L^{q}(ℍⁿ)$ bounded. We also obtain $L^{p}$-improving properties of measures supported on the graph of the function $φ(w) = |w|^{2m}$.
5
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Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)

100%
Godoy T., Saal L.
Colloquium Mathematicum
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2000
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tom 86
|
nr 2
177-187
EN
Let $H_{n}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_{n}$. So $L^{2}(H_{n})$ has a natural structure of G-module. We obtain a decomposition of $L^{2}(H_{n})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^{2}(H_{n})$ where $L=\sum_{j=1}^{p} (X_{j}^{2}+Y_{j}^{2}) - \sum_{j=p+1}^{n} (X_{j}^{2}+Y_{j}^{2})$, and where ${X_{1},Y_{1},...,X_{n},Y_{n},T}$ denotes the standard basis of the Lie algebra of $H_{n}$. Finally, we obtain a spectral characterization of the bounded operators on $L^{2}(H_{n})$ that commute with the action of G.
6
Content available remote

Endpoint bounds for convolution operators with singular measures

81%
Ferreyra E., Godoy T., Urciuolo M.
Colloquium Mathematicum
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1998
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tom 76
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nr 1
35-47
EN
Let $S\subset \R^{n+1}$ be the graph of the function $\varphi:[ -1,1]^n\rightarrow \R $ defined by $\varphi ( x_1,\dots,x_n) =\sum_{j=1}^n| x_j|^{\beta_j},$ with 1<$\beta_1\leq \dots \leq \beta_n,$ and let $\mu $ the measure on $\R^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu $ is $L^p$-$L^q$ bounded.
7
Content available remote

On some singular integral operatorsclose to the Hilbert transform

81%
Godoy T., Saal L., Urciuolo M.
Colloquium Mathematicum
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1997
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tom 72
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nr 1
9-17
EN
Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p(ℝ)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_m f(x) = p.v. \int k(x-y) m(x+y) f(y)dy$ where $k(x) = \sum_{j ∈ ℤ} 2^j φ _j (2^j x)$ for a family of functions ${φ_j}_{j∈ℤ}$ satisfying conditions (1.1)-(1.3) given below.
8
Content available remote

Convolution operators with anisotropically homogeneous measures on $ℝ^{2n}$ with n-dimensional support

81%
Ferreyra E., Godoy T., Urciuolo M.
Colloquium Mathematicum
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2002
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tom 93
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nr 2
285-293
EN
Let $α_i,β_i > 0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t • x = (t^{α₁}x₁,..., t^{αₙ}xₙ)$, $t ∘ x = (t^{β₁}x₁,..., t^{βₙ}xₙ)$ and $||x|| = ∑_{i = 1}^{n} |x_i|^{1/α_i}$. Let φ₁,...,φₙ be real functions in $C^∞(ℝⁿ-{0}) $ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on $ℝ^{2n}$ given by $μ(E) = ∫_{ℝⁿ} χ_E(x,φ(x)) ||x||^{γ-α} dx$, where $α = ∑_{i=1}^{n} α_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_μf = μ ∗ f$ and let $||T_μ||_{p,q}$ be the operator norm of $T_μ$ from $L^{p}(ℝ^{2n})$ into $L^q(ℝ^{2n})$, where the $L^{p}$ spaces are taken with respect to the Lebesgue measure. The type set $E_μ$ is defined by $E_μ = {(1/p,1/q): ||T_μ||_{p,q} < ∞, 1 ≤ p,q ≤ ∞}$. In the case $α_i ≠ β_k$ for 1 ≤ i,k ≤ n we characterize the type set under certain additional hypotheses on φ.
9
Content available remote

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

81%
Ferreyra E., Godoy T., Urciuolo M.
Studia Mathematica
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2004
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tom 160
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nr 3
249-265
EN
Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = {(x,φ(x)): |x| ≤ 1} and let σ be the Borel measure on Σ defined by $σ(A) = ∫_{B} χ_{A}(x,φ(x))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p}(ℝ³)$ to $L^{q}(Σ,dσ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.
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