Fulvio Ricci

Scuola Normale Superiore di Pisa, Pisa, Tuscany, Italy

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Publications (41)41.71 Total impact

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    ABSTRACT: We prove various Hardy-type 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 sub-Laplacian, 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 Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $|.|^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$.
    Advances in Mathematics 08/2013; 277. DOI:10.1016/j.aim.2014.12.040 · 1.35 Impact Factor
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    ABSTRACT: We prove several Paley--Wiener-type 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 real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.
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    ABSTRACT: The authors consider the Schwartz space S(H 1 ) on the (three-dimensional) Heisenberg group H 1 . They characterise S(H 1 ) via sequences of Schwartz functions defined on ℝ 2 . The definition of these sequences relies on the decomposition given by the representation theory of the group H 1 (Plancherel formula) as well as certain special features of the action of the torus 𝕋 1 on functions of H 1 . This generalises the results in [J. Funct. Anal. 251, No. 2, 772–791 (2007; Zbl 1128.43009); ibid. 256, No. 5, 1565–1587 (2009; Zbl 1167.43008)] by the same authors in the case of 𝕋 1 -invariant functions in S(H 1 ). The proofs are based on the results in [loc. cit.] as well as Whitney’s extension properties and a deep understanding of the Gelfand spectrum in this context.
    Studia Mathematica 01/2013; 214(3). DOI:10.4064/sm214-3-1 · 0.63 Impact Factor
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    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 left-invariant and K-invariant 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 self-adjoint 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.
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    ABSTRACT: We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant 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_
    Memoirs of the American Mathematical Society 06/2012; 233(1095). DOI:10.1090/memo/1095 · 1.78 Impact Factor
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    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 $1<p<\infty$.
    Revista Matematica Iberoamericana 07/2011; 28(3). DOI:10.4171/RMI/688 · 0.54 Impact Factor
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    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 K-invariant 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 K-invariant 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 K-invariant irreducible subspace.
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    ABSTRACT: The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K lx N, K) in Vinberg's list [27, 30]. (The references numbers refers to the bibliography at the end of the article) Comment: 29 pages
    Mathematische Zeitschrift 02/2010; 271(1-2). DOI:10.1007/s00209-011-0861-3 · 0.68 Impact Factor
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    ABSTRACT: The dispersive properties of the wave equation u tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|2 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. KeywordsWave equation-Strichartz estimates-Decay estimates-Dispersive equations-Schrödinger equation-Harmonic analysis-Almost periodicity Mathematics Subject Classification (2000)35L05-35Q40-58J45-11K70-11L03
    Journal of Fourier Analysis and Applications 01/2010; 16(2):294-310. DOI:10.1007/s00041-009-9104-y · 1.08 Impact Factor
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    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.
    Journal für die reine und angewandte Mathematik (Crelles Journal) 03/2009; DOI:10.1515/CRELLE.2010.072 · 1.30 Impact Factor
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    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 K-invariant 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 K-invariant integrable functions on Hn.
    Journal of Functional Analysis 03/2009; 256(5-256):1565-1587. DOI:10.1016/j.jfa.2008.10.008 · 1.15 Impact Factor
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    Fulvio Ricci
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    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 L2-spectrum is the so-called 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.
    Revista de la Unión Matemática Argentina 01/2009; · 0.27 Impact Factor
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    Veronique Fischer, Fulvio Ricci
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    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 two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.
    Annales- Institut Fourier 10/2008; DOI:10.5802/aif.2486 · 0.64 Impact Factor
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    Fulvio Ricci, Joan Verdera
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    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 $B$.
    Transactions of the American Mathematical Society 10/2008; 363(3). DOI:10.1090/S0002-9947-2010-05036-6 · 1.10 Impact Factor
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    Adam Korányi, Fulvio Ricci
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    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.
    Colloquium Mathematicum 05/2008; 118(1). DOI:10.4064/cm118-1-3 · 0.42 Impact Factor
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    Paolo Ciatti, Fulvio Ricci, Maddala Sundari
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    ABSTRACT: In its simpler form, the Heisenberg–Pauli–Weyl inequality says that In this paper, we extend this inequality to positive self-adjoint 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 sub-Laplacians on groups of polynomial volume growth and to certain higher-order left-invariant hypoelliptic operators on nilpotent groups. We finally show that these estimates also imply generalized forms of local uncertainty inequalities.
    Advances in Mathematics 11/2007; 215(2-215):616-625. DOI:10.1016/j.aim.2007.03.014 · 1.35 Impact Factor
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    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.
    Journal of Functional Analysis 10/2007; 251(2-251):772-791. DOI:10.1016/j.jfa.2007.06.010 · 1.15 Impact Factor
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    ABSTRACT: We study the spectrum of the Hodge Laplacian 1 acting on 1-forms on the (2n+1)-dimensional Heisenberg group Hn, by finding the eigenvalues of the image of 1 in the Bargmann representations. As a consequence, we determine explicitely the eigenvalues for 1 on some compact quotients of Hn. This note is part of a larger project (MPR), in which we study the question of the boundedness of spectral multipliers of 1 on Hn.
    Collectanea Mathematica 01/2006; 2006. · 0.61 Impact Factor
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    ABSTRACT: We prove that, if \Delta_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-H\"ormander multiplier on the positive half-line, with L^2-order of smoothness greater than n+1/2, then m(\Delta_1) is L^p-bounded for 1<p<\infty. Our approach leads to an explicit description of the spectral decomposition of \Delta_1 on the space of L^2-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.
    Geometric and Functional Analysis 09/2005; 17(3). DOI:10.1007/s00039-007-0612-0 · 1.32 Impact Factor
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    Herbert Koch, Fulvio Ricci
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    ABSTRACT: Let n >= 1, d = 2n, and let (x, y) is an element of R-n x R-n be a generic point in R-2n, The twisted Laplacian\ [GRAPHICS] has the spectrum {n + 2k = lambda(2) : k a nonnegative integer}. Let P-lambda be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent rho(p) in the estimate [GRAPHICS] for all p is an element of [2, infinity], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for rho(p) is [GRAPHICS]
    Studia Mathematica 01/2005; 180(2). DOI:10.4064/sm180-2-1 · 0.63 Impact Factor

Publication Stats

592 Citations
41.71 Total Impact Points

Institutions

  • 2003–2013
    • Scuola Normale Superiore di Pisa
      Pisa, Tuscany, Italy
  • 2005
    • University of Bonn
      • Mathematical Institute
      Bonn, North Rhine-Westphalia, Germany
  • 2004
    • Université d'Orléans
      Orléans, Centre, France
  • 1986–2003
    • Politecnico di Torino
      Torino, Piedmont, Italy