1,721,012 research outputs found
Algebras of differential operators on Lie groups and spectral multipliers
Full text linkLet (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint
operators on L2(X, μ), which commute strongly pairwise, i.e., which
admit a joint spectral resolution E on Rn. A joint functional calculus is then
defined via spectral integration: for every Borel function m : Rn → C,
m(L) = m(L1, . . . ,Ln) =
∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called
the joint spectral multiplier associated to m(L) - is (E-essentially) bounded.
However, the abstract theory of spectral integrals does not tackle the following
problem: to find conditions on the multiplier m ensuring the boundedness of
m(L) on Lp(X, μ) for some p ≠ 2.
We are interested in this problem when the measure space is a connected Lie
group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential
operators on G. In fact, the question has been studied quite extensively in the
case of a single operator, namely, a sublaplacian or a higher-order analogue.
On the other hand, for multiple operators, only specific classes of groups and
specific choices of operators have been considered in the literature.
Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential
operators on a connected Lie group G, which commute pairwise (as operators
on smooth functions). Under the assumption that the algebra generated by
L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98],
including positive elliptic operators, sublaplacians and Rockland operators---we
prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on
L2(G). Moreover, we perform an abstract study of such a system of operators,
in connection with the algebraic structure and the representation theory of G,
similarly as what is done in the literature for the algebras of differential operators
associated with Gelfand pairs.
Under the additional assumption that G has polynomial volume growth,
weighted L1 estimates are obtained for the convolution kernel of the operator
m(L) corresponding to a compactly supported multiplier m satisfying some
smoothness condition. The order of smoothness which we require on m is related
to the degree of polynomial growth of G. Some techniques are presented,
which allow, for some specific groups and operators, to lower the smoothness
requirement on the multiplier.
In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous
operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending
the result for a single operator of [Chr91] and [MM90]. Further, a product theory
is developed, by considering several homogeneous groups Gj , each of which with
its own system of operators; a non-conventional use of transference techniques
then yields a multiplier theorem of Marcinkiewicz type, not only on the direct
product of the Gj , but also on other (possibly non-homogeneous) groups, containing
homomorphic images of the Gj . Consequently, for certain non-nilpotent
groups of polynomial growth and for some distinguished sublaplacians, we are
able to improve the general result of [Ale94]
A sharp multiplier theorem for solvable extensions of Heisenberg and related groups
Full text linkLet G be the semidirect product N R, where N is a stratified Lie group and R acts on N via
automorphic dilations. Homogeneous left-invariant sub-Laplacians on N and R can be lifted
to G, and their sum is a left-invariant sub-Laplacian on G. In previous joint work of Ottazzi,
Vallarino and the first-named author, a spectral multiplier theorem of Mihlin–Hörmander type
was proved for , showing that an operator of the form F() is of weak type (1, 1) and
bounded on L p(G) for all p ∈ (1,∞) provided F satisfies a scale-invariant smoothness
condition of order s > (Q + 1)/2, where Q is the homogeneous dimension of N. Here we
show that, if N is a group of Heisenberg type, or more generally a direct product of Métivier
and abelian groups, then the smoothness condition can be pushed down to the sharp threshold
s > (d + 1)/2, where d is the topological dimension of N. The proof is based on lifting to
G weighted Plancherel estimates on N and exploits a relation between the functional calculi
for and analogous operators on semidirect extensions of Bessel–Kingman hypergroups
Riesz transforms on groups
Full text linkWe prove the -boundedness for all of the first-order
Riesz transforms associated with the Laplacian
on the -group ; here and are left-invariant vector
fields on in the directions of the factors and
respectively. This settles a question left open in previous work of Hebisch and
Steger (who proved the result for ) and of Gaudry and Sj\"ogren (who
only considered ). The main novelty here is that we can treat the case
and include the Riesz transform in the direction of
; an operator-valued Fourier multiplier theorem on
turns out to be key to this purpose. We also establish a weak type
endpoint for the adjoint Riesz transforms in the direction of .
By transference, our results imply the -boundedness for
of the first-order Riesz transforms associated with the Schr\"odinger operator
on the real line.Comment: 40 page
Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups
Full text linkLet G = N ⋊ A, where N is a stratified group and A = ℝ acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian Δ on G. We prove a theorem of Mihlin–Hörmander type for spectral multipliers of Δ. The proof of the theorem hinges on a Calderón–Zygmund theory adapted to a sub-Riemannian structure of G and on L1-estimates of the gradient of the heat kernel associated to the sub-Laplacian Δ
L'eco-innovazione per la transizione ecologica
Full text linkL’attuale configurazione delle politiche pubbliche sovranazionali e nazionali è contraddistinta dalla fissazione di obiettivi che guardano alla transizione ecologica e alla neutralità climatica. Ciò presuppone una corposa attività di trasformazione del modello economico in chiave ecologica, sicché le attività antropiche riducano progressivamente il proprio impatto sull’ambiente fino a neutralizzarne gli effetti. A tal fine, la domanda di innovazione si fa sempre più evidente, giacché le soluzioni innovative, frutto dell’attività creativa dell’innovatore, apportano efficienza. Questa specifica caratteristica è solitamente ricondotta a un incremento di utilità di ordine economico e quantitativo, ma può essere declinata anche in senso qualitativo, per esempio a tutela dell’ambiente. Così, la cosiddetta eco-innovazione diviene uno strumento utile e flessibile con il quale riorientare il paradigma economico verso parametri quali la sostenibilità, la circolarità o la rigenerazione. Vista la strategicità del tema e le diverse implicazioni a esso sottese, l’indagine in questione si propone di verificare la sua rilevanza giuridica, a livello nazionale ed europeo, e di individuare le politiche e gli strumenti giuridici attraverso cui gli ordinamenti incentivano, sviluppano o adottano eco-innovazioni. Difatti, sebbene l’innovazione, anche nella sua accezione ambientale, sia centrale nelle riflessioni scientifiche delle scienze, economica e applicate, la scienza giuridica non ha ancora dedicato un’attenzione significativa a tale argomento. Pertanto, l’analisi prende le mosse dalla ricostruzione del concetto di innovazione ed eco-innovazione in chiave sistematica all’interno delle fonti giuridiche euro-unitarie e nazionali. Il riconoscimento della sua rilevanza giuridica multilivello è infatti funzionale alla disamina degli strumenti giuridici, su tutti quelli tradizionalmente preposti alla tutela dell’ambiente, che più o meno direttamente sono in grado di intercettare le soluzioni eco-innovative al fine di impiegarle per il raggiungimento degli obiettivi citati. Dal momento che l’emersione di profili giuridici dell’eco-innovazione si coglie sia da un punto di vista ermeneutico che pratico, l’analisi trova la sua conclusione nell’esemplificazione di un’eco-innovazione. A tal proposito, si guarda all’economia circolare quale forma completa e compiuta di eco-innovazione, vista la condivisione con quest’ultima delle caratteristiche essenziali e degli strumenti giuridici serventi
Singular integrals on ax+b hypergroups and an operator-valued spectral multiplier theorem
No full textLet Lν = −∂
2
x − (ν − 1)x−1∂x be the Bessel operator on the halfline Xν = [0, ∞) with measure x
ν−1 dx. In this work we study singular integral
operators associated with the Laplacian ∆ν = −∂
2
u +e
2uLν on the product Gν
of Xν and the real line with measure du. For any ν ≥ 1, the Laplacian ∆ν is
left-invariant with respect to a noncommutative hypergroup structure on Gν,
which can be thought of as a fractional-dimension counterpart to ax+b groups.
In particular, equipped with the Riemannian distance associated with ∆ν, the
metric-measure space Gν has exponential volume growth. We prove a sharp Lp
spectral multiplier theorem of Mihlin–H ̈ormander type for ∆ν, as well as the
Lp-boundedness for p ∈ (1, ∞) of the associated first-order Riesz transforms.
To this purpose, we develop a Calder ́on–Zygmund theory `a la Hebisch–Steger
adapted to the nondoubling structure of Gν, and establish large-time gradient
heat kernel estimates for ∆ν. In addition, the Riesz transform bounds for
p > 2 hinge on an operator-valued spectral multiplier theorem, which we prove
in greater generality and may be of independent interes
Uniform pointwise estimates for ultraspherical polynomials
Full text linkWe prove pointwise bounds for two-parameter families of Jacobi polynomials. Our bounds imply estimates for a class of functions arising from the spectral analysis of distinguished Laplacians and sub-Laplacians on the unit sphere in arbitrary dimension, and are instrumental in the proof of sharp multiplier theorems for those operators
From refined estimates for the spherical harmonics to a sharp multiplier theorem on the Grushin sphere
Full text linkWe 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
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