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Fulvio Ricci

Publications and source records attributed to Fulvio Ricci.

At least 37 records · Page 2Linked to original sources
arXiv2017

Nilpotent gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L^1(N)^K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)^K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L^1(N)^K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N. We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

math.FA↗
arXiv2016

A maximal restriction theorem and Lebesgue points of functions in F(L^p)

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q-norm of the restriction of the Fourier transform of a function f in L^p (say, on Euclidean space) to a given subvariety S, endowed with a suitabel measure. Such estimates allow to define the restriction Rf of the Fourier transform of an L^p-function to S in an operator theoretic sense. In this article, we begin to investigate the question what is the "intrinsic" pointwise relation between Rf and the Fourier transform of f, by looking at curves in the plane, for instance with non-vanishing curvature. To this end, we bound suitable maximal operators, including the Hardy-Littlewood maximal function of the Fourier transform of f restricted to S.

math.CA↗
arXiv2016

Spherical analysis on homogeneous vector bundles

Given a Lie group $G$, a compact subgroup $K$ and a representation $τ\in\hat K$, we assume that the algebra of $\text{End}(V_τ)$-valued, bi-$τ$-equivariant, integrable functions on $G$ is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of $G$-invariant differential operators on the homogeneous bundle $E_τ$ over $G/K$. In particular, we observe that, under the above assumptions, $(G,K)$ is a Gelfand pair and show that the Gelfand spectrum for the triple $(G,K,τ)$ admits homeomorphic embeddings in $\mathbb C^n$. In the second part, we develop in greater detail the spherical analysis for $G=K\ltimes H$ with $H$ nilpotent. In particular, for $H=\mathbb R^n$ and $K\subset SO(n)$ and for the Heisenberg group $H_n$ and $K\subset U(n)$, we characterize the representations $τ\in \hat K$ giving a commutative algebra. \end{abstract}

math.RT↗
arXiv2015

Algebras of singular integral operators with kernels controlled by multiple norms

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 Calderón-Zygmund operators with different homogeneities, such as operators that occur in sub-elliptic problems and those arising in elliptic problems. For example, one would like to describe the algebras containing the operators related to the Kohn-Laplacian for appropriate domains, or those related to inverses of Hörmander sub-Laplacians, when these are composed with the more standard class of pseudo-differential operators. The algebras we study can be characterized in a number of different but equivalent ways, and consist of operators that are pseudo-local and bounded on $L^{p}$ for $1<p<\infty$. While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators we study fall outside this class, and reflect a multi-parameter structure.

math.FA↗
arXiv2015

Archimedes: a feasibility study of an experiment to weigh the electromagnetic vacuum

Archimedes is a feasibility study of a future experiment to ascertain the interaction of vacuum fluctuations with gravity. The experiment should measure the force that the earth's gravitational field exerts on a Casimir cavity by using a small force detector. Here we analyse the main parameters of the experiment and we present its conceptual scheme, which overcomes in principle the most critical problems.

gr-qc↗
arXiv2013

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_α: f \mapsto |.|^{-α} L^{-α/2} f$, where $|.|$ is a homogeneous norm, $0 < α< Q/p$, 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 $|.|^βf$ and the $L^r$ norm of $L^{δ/2} f$.

math.FA↗
arXiv2013

Paley--Wiener Theorems for the U(n)--spherical transform on the Heisenberg group

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.

math.FA↗
arXiv2012

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition

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.

math.FA↗
arXiv2012

Duality in spaces of finite linear combinations of atoms

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 $B$.

math.FA↗
arXiv2012

Analysis of the Hodge Laplacian on the Heisenberg group

We consider the Hodge Laplacian $Δ$ 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 $Δ_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first main result shows that $L^2Λ^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\V_ν$ with the properties: {itemize} $\dom Δ_k$ splits along the $\V_ν$'s as $\sum_ν(\domΔ_k\cap \V_ν)$; $Δ_k:(\domΔ_k\cap \V_ν)\longrightarrow \V_ν$ for every $ν$; for each $ν$, there is a Hilbert space $\cH_ν$ of $L^2$-sections of a U(n)-homogeneous vector bundle over $H_n$ such that the restriction of $Δ_k$ to $\V_ν$ is unitarily equivalent to an explicit scalar operator. {itemize} Next, we consider $L^pΛ^k$, $1<p<\infty$, and prove that the same kind of decomposition holds true. More precisely we show that: {itemize} the Riesz transforms $dΔ_k^{-\half}$ are $L^p$-bounded; the orthogonal projection onto $\cV_ν$ extends from $(L^2\cap L^p)Λ^k$ to a bounded operator from $L^pΛ^k$ to the the $L^p$-closure $\cV_

math.FA↗
arXiv2011

Opportunity to Test non-Newtonian Gravity Using Interferometric Sensors with Dynamic Gravity Field Generators

We present an experimental opportunity for the future to measure possible violations to Newton's 1/r^2 law in the 0.1-10 meter range using Dynamic gravity Field Generators (DFG) and taking advantage of the exceptional sensitivity of modern interferometric techniques. The placement of a DFG in proximity to one of the interferometer's suspended test masses generates a change in the local gravitational field that can be measured at a high signal to noise ratio. The use of multiple DFGs in a null experiment configuration allows to test composition independent non-Newtonian gravity significantly beyond the present limits. Advanced and third-generation gravitational-wave detectors are representing the state-of-the-art in interferometric distance measurement today, therefore we illustrate the method through their sensitivity to emphasize the possible scientific reach. Nevertheless, it is expected that due to the technical details of gravitational-wave detectors, DFGs shall likely require dedicated custom configured interferometry. However, the sensitivity measure we derive is a solid baseline indicating that it is feasible to consider probing orders of magnitude into the pristine parameter well beyond the present experimental limits significantly cutting into the theoretical parameter space.

gr-qc↗
arXiv2011

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets

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.

math.AC↗
arXiv2010

Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions I. Rank-One Actions on the Centre

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)

math.FA↗
arXiv2008

Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_{3,2}

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.

math.FA↗
arXiv2008

Gelfand pairs on the Heisenberg group and Schwartz functions

Let $\Hn$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a compact group of automorphisms of $\Hn$ such that $(K\ltimes \Hn,K)$ 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 $\R^s$ homeomorphic to the Gelfand spectrum of the Banach algebra of $K$-invariant integrable functions on $\Hn$.

math.FA↗
arXiv2005

L^p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group

We prove that, if Δ_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-Hörmander multiplier on the positive half-line, with L^2-order of smoothness greater than n+1/2, then m(Δ_1) is L^p-bounded for 1<p<\infty. Our approach leads to an explicit description of the spectral decomposition of Δ_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.

math.CA↗