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1
Content available remote

Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$

100%
Marletta G., Ricci F.
Studia Mathematica
|
1998
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tom 130
|
nr 1
53-65
EN
Given a hypersurface $x_n = Ꮁ(x_1...,x_{n-1})$ in $ℝ^n$, where Ꮁ is homogeneous of degree d>0, we define the two-parameter maximal operator $ Mf(x) = sup_{a,b>0} ∫_{s∈ℝ^{n-1},|s| < 1} $ |f(x - (as, bᎱ(s)))|ds$. We prove that if d ≠ 1 and the hypersurface has non-vanishing Gaussian curvature away from the origin, then M is bounded on $L^p$ if and only if p>n/(n-1). If d = 1, i.e. if the surface is a cone, the same conclusion holds in dimension n ≥ 3 if the surface has n-1 non-vanishing principal curvatures away from the origin and it intersects the hyperplane $x_n = 0$ only at the origin.
2
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A unified approach to compact symmetric spaces of rank one

100%
Korányi A., Ricci F.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 1
43-87
EN
A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.
3
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Spectral projections for the twisted Laplacian

100%
Koch H., Ricci F.
Studia Mathematica
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2007
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tom 180
|
nr 2
103-110
EN
Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = -1/2 ∑_{j=1}^{n} [(∂_{x_{j}} + iy_{j})² + (∂_{y_{j}} - ix_{j})²]$ has the spectrum {n + 2k = λ²: k a nonnegative integer}. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $||P_{λ}u||_{L^{p}(ℝ^{d})} ≲ λ^{ϱ(p)} ||u||_{L²(ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1) ≤ p ≤ ∞.
4
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Fourier transform of Schwartz functions on the Heisenberg group

81%
Astengo F., Di Blasio B., Ricci F.
Studia Mathematica
|
2013
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tom 214
|
nr 3
201-222
EN
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.
5
Content available remote

Two-parameter maximal functions associated with degenerate homogeneous surfaces in ℝ³

81%
Marletta G., Ricci F., Zienkiewicz J.
Studia Mathematica
|
1998
|
tom 130
|
nr 1
67-75
EN
We consider the two-parameter maximal operator $Mf(x)= sup_{a,b>0}$ ʃ_{|s| < 1} |f(x-(as,bΓ(s)))|ds$ on a homogeneous surface $x_3 = Γ(x_1,x_2)$ in $ℝ^3$. We assume that the curvature of the level set $Γ(x_1,x_2) = 1$ has a degeneracy of finite order k at a given point. We prove that the operator M is bounded on $L^p$ if and only if $p > max{3/2, 2k/(k+1)}$.
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