72 research outputs found
Some remarks on harmonic projection operators on spheres
We give a survey of recent works concerning the mapping properties of joint harmonic projection operators, mapping the space of square integrable functions on complex and quaternionic spheres onto the eigenspaces of the Laplace-Beltrami operator and of a suitably defined subLaplacian. In particular, we discuss similarities and differences between the real, the complex and the quaternionic framework
A Restriction Theorem for M\'etivier Groups
In the spirit of an earlier result of M\"uller on the Heisenberg group we
prove a restriction theorem on a certain class of two step nilpotent Lie
groups. Our result extends that of M\"uller also in the framework of the
Heisenberg group.Comment: Corrected typos, introduction revised. Final version, to appear in
Advances in Mathematic
L^p-summability of Riesz means for the sublaplacian on complex spheres
In this paper we study the L^p-convergence of the Riesz means for the
sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We
show that the Riesz means of order delta of a function f converge to f in
L^p(S^{2n-1}) when delta>delta(p):=(2n-1)|1\2-1\p|. The index delta(p) improves
the one found by Alexopoulos and Lohoue', 2n|1\2-1\p|, and it coincides with
the one found by Mauceri and, with different methods, by Mueller in the case of
sublaplacian on the Heisenberg group.Comment: Rapporto interno Politecnico di Torino, Novembre 200
On the norms of quaternionic harmonic projection operators
As a consequence of integral bounds for three classes of quaternionic spherical harmon-ics, we prove some bounds from below for the (Lp,L2) norm of quaternionic harmonic projectors, for p between 1 and 2
From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere
We prove a sharp multiplier theorem of Mihlin-H\"ormander type for the
Grushin operator on the unit sphere in , and a corresponding
boundedness result for the associated Bochner-Riesz means. The proof hinges on
precise pointwise bounds for spherical harmonics.Comment: 32 page
The maximal operator of a normal Ornstein--Uhlenbeck semigroup is of weak type (1,1)
Consider a normal Ornstein\u2013Uhlenbeck semigroup in R^n, whose co- variance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where the matrix defining the covariance is I and the drift matrix is diagonal
On the maximal operator of a general Ornstein-Uhlenbeck semigroup
If is a real, symmetric and positive definite matrix, and
a real matrix whose eigenvalues have negative real parts, we
consider the Ornstein--Uhlenbeck semigroup on with covariance
and drift matrix . Our main result says that the associated maximal
operator is of weak type with respect to the invariant measure. The
proof has a geometric gist and hinges on the "forbidden zones method"
previously introduced by the third author.Comment: 21 pages. Introduction revised. Some changes in Sections 3 and
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