Publications (38)33.07 Total impact
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ABSTRACT: The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calder\'onZygmund operators with different homogeneities, such as operators that occur in subelliptic problems and those arising in elliptic problems. For example, one would like to describe the algebras containing the operators related to the KohnLaplacian for appropriate domains, or those related to inverses of H\"ormander subLaplacians, when these are composed with the more standard class of pseudodifferential operators. The algebras we study can be characterized in a number of different but equivalent ways, and consist of operators that are pseudolocal and bounded on $L^{p}$ for $1<p<\infty$. While the usual class of Calder\'onZygmund operators is invariant under a oneparameter family of dilations, the operators we study fall outside this class, and reflect a multiparameter structure.  [Show abstract] [Hide abstract]
ABSTRACT: We prove various Hardytype and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto .^{\alpha} L^{\alpha/2} f$, where $.$ is a homogeneous norm, $0 < \alpha < Q/p$, and $L$ is the subLaplacian, are bounded on the Lebesgue space $L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the HeisenbergPauliWeyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $.^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$.  [Show abstract] [Hide abstract]
ABSTRACT: We prove several PaleyWienertype theorems related to the spherical transform on the Gelfand pair $\big(H_n\rtimes U(n),U(n)\big)$, where $H_n$ is the $2n+1$dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in ${\mathbb R}^2$, we prove that spherical transforms of $ U(n)$invariant functions and distributions with compact support in $H_n$ admit unique entire extensions to ${\mathbb C}^2$, and we find realvariable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.  [Show abstract] [Hide abstract]
ABSTRACT: Let H1 be the 3dimensional 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 jmj of the halflines removed. This result is then applied to study the case of general Schwartz functions on H1.  [Show abstract] [Hide abstract]
ABSTRACT: Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of leftinvariant and Kinvariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N). Fixing a system D of d formally selfadjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpotent Gelfand pair satisfying Vinberg's condition, the spherical transform establishes an isomorphism from the space of $K$invariant Schwartz functions on N and the space of restrictions to S_D of Schwartz functions in R^d.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a leftinvariant and U(n)invariant Riemannian metric. For $0\le k\le 2n+1$, let $\Delta_k$ denote the Hodge Laplacian restricted to $k$forms. Our first main result shows that $L^2\Lambda^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\V_\nu$ with the properties: {itemize} $\dom \Delta_k$ splits along the $\V_\nu$'s as $\sum_\nu(\dom\Delta_k\cap \V_\nu)$; $\Delta_k:(\dom\Delta_k\cap \V_\nu)\longrightarrow \V_\nu$ for every $\nu$; for each $\nu$, there is a Hilbert space $\cH_\nu$ of $L^2$sections of a U(n)homogeneous vector bundle over $H_n$ such that the restriction of $\Delta_k$ to $\V_\nu$ is unitarily equivalent to an explicit scalar operator. {itemize} Next, we consider $L^p\Lambda^k$, $1<p<\infty$, and prove that the same kind of decomposition holds true. More precisely we show that: {itemize} the Riesz transforms $d\Delta_k^{\half}$ are $L^p$bounded; the orthogonal projection onto $\cV_\nu$ extends from $(L^2\cap L^p)\Lambda^k$ to a bounded operator from $L^p\Lambda^k$ to the the $L^p$closure $\cV_  [Show abstract] [Hide abstract]
ABSTRACT: Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1<p<\infty$.  [Show abstract] [Hide abstract]
ABSTRACT: This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of Kinvariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for Kinvariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as a direct sum of [n,n] and a Kinvariant irreducible subspace.  [Show abstract] [Hide abstract]
ABSTRACT: The dispersive properties of the wave equation u tt +Au=0 are considered, where A is either the Hermite operator −Δ+x2 or the twisted Laplacian −(∇ x −iy)2/2−(∇ y +ix)2/2. In both cases we prove optimal L 1−L ∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ν ) associated to fractional powers of the twisted Laplacian for 0<ν<1.  [Show abstract] [Hide abstract]
ABSTRACT: The spectrum of a Gelfand pair of the form \({(K\ltimes N,K)}\), where N is a nilpotent group, can be embedded in a Euclidean space \({{\mathbb R}^d}\). The identification of the spherical transforms of Kinvariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on \({{\mathbb R}^d}\) has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free twostep nilpotent Lie group with three generators, with K = SO3 (Astengo et al. in J Funct Anal 251:772–791, 2007; Astengo et al. in J Funct Anal 256:1565–1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143–2168, 2009). We prove that the same identification holds for all pairs in which the Korbits in the centre of N are spheres. In the appendix, we produce bases of Kinvariant polynomials on the Lie algebra \({{\mathfrak n}}\) of N for all Gelfand pairs \({(K\ltimes N,K)}\) in Vinberg’s list (Vinberg in Trans Moscow Math Soc 64:47–80, 2003; Yakimova in Transform Groups 11:305–335, 2006).  [Show abstract] [Hide abstract]
ABSTRACT: We give various equivalent formulations to the (partially) open problem about $L^p$boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.  [Show abstract] [Hide abstract]
ABSTRACT: Let Hn be the (2n+1)dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K⋉Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of Kinvariant Schwartz functions on Hn and the space of Schwartz function on a closed subset of Rs homeomorphic to the Gelfand spectrum of the Banach algebra of Kinvariant integrable functions on Hn.  [Show abstract] [Hide abstract]
ABSTRACT: We review recent results proved jointly with B. Di Blasio and F. Astengo. On the Heisenberg group Hn, consider the two commuting self adjoint operators L and i 1T , where L is the sublaplacian and T is the central derivative. Their joint L2spectrum is the socalled Heisenberg fan, contained in R2. To any bounded Borel function m on the fan, we associate the operator m(L;i 1T ). The main result that we describe says that the convolution kernel of m(L;i 1T ) is a Schwartz function if and only if m is the restriction of a Schwartz function on R2. We point out that this result can be interpreted in terms of the spherical transform for the convolution algebra of U(n)invariant functions on Hn. We also describe extensions to more general situations.  [Show abstract] [Hide abstract]
ABSTRACT: The spectrum of a Gelfand pair $(K\ltimes N, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the free twostep nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.  [Show abstract] [Hide abstract]
ABSTRACT: In this note we describe the dual and the completion of the space of finite linear combinations of $(p,\infty)$atoms, $0<p\leq 1$ on ${\mathbb R}^n$. As an application, we show an extension result for operators uniformly bounded on $(p,\infty)$atoms, $0<p < 1$, whose analogue for $p=1$ is known to be false. Let $0 < p <1$ and let $T$ be a linear operator defined on the space of finite linear combinations of $(p,\infty)$atoms, $0<p < 1 $, which takes values in a Banach space $B$. If $T$ is uniformly bounded on $(p,\infty)$atoms, then $T$ extends to a bounded operator from $H^p({\mathbb R}^n)$ into $B$.  [Show abstract] [Hide abstract]
ABSTRACT: 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.  [Show abstract] [Hide abstract]
ABSTRACT: In its simpler form, the Heisenberg–Pauli–Weyl inequality says that In this paper, we extend this inequality to positive selfadjoint operators L on measure spaces with a “gauge function” such that (a) measures of balls are controlled by powers of the radius (possibly different powers for large and small balls); (b) the semigroup generated by L satisfies ultracontractive estimates with polynomial bounds of the same type. We give examples of applications of this result to subLaplacians on groups of polynomial volume growth and to certain higherorder leftinvariant hypoelliptic operators on nilpotent groups. We finally show that these estimates also imply generalized forms of local uncertainty inequalities.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.  [Show abstract] [Hide abstract]
ABSTRACT: We study the spectrum of the Hodge Laplacian Δ 1 acting on 1forms on the (2n+1)dimensional Heisenberg group ℍ n , by finding the eigenvalues of the image of Δ 1 in the Bargmann representations. As a consequence, we determine explicitly the eigenvalues for Δ 1 on some compact quotients of ℍ n . This note is part of a larger project [the authors, Geom. Funct. Anal. 17, No. 3, 852–886 (2007; )Zbl 1148.43008], in which we study the question of the boundedness of spectral multipliers of Δ 1 on ℍ n .  [Show abstract] [Hide abstract]
ABSTRACT: We prove that, if \(\Delta_{1}\) is the Hodge Laplacian acting on differential 1forms on the (2n + 1)dimensional Heisenberg group, and if m is a Mihlin–Hörmander multiplier on the positive halfline, with L 2order of smoothness greater than \(n + \frac{1}{2}\) , then m(Δ1) is L p bounded for 1 < p < ∞. Our approach leads to an explicit description of the spectral decomposition of \(\Delta_{1}\) on the space of L 2forms in terms of the spectral analysis of the subLaplacian L and the central derivative T, acting on scalarvalued functions.
Publication Stats
454  Citations  
33.07  Total Impact Points  
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Institutions

20022013

Scuola Normale Superiore di Pisa
Pisa, Tuscany, Italy


2006

ChristianAlbrechtsUniversität zu Kiel
Kiel, SchleswigHolstein, Germany


2005

University of Bonn
 Mathematical Institute
Bonn, North RhineWestphalia, Germany


19952003

Politecnico di Torino
Torino, Piedmont, Italy
