The Weyl transform and bounded operators on Lp(Rn)

A multiplier theorem for the Weyl transform is proved. This theorem is used to derive sufficient conditions for the boundedness of a general operator on L^p($\text{R}$ⁿ). An application to multipliers of the Hermite expansion is given.     

The multiplier for the ball and radial functions

It is shown that the multiplier for the ball is restricted weak type on radial functions in L^p($\text{R}$ⁿ) when $\text{p =}\text{2n}\text{(n + 1)}$. Interpolation then yields a theorem of Herz.     

Contractive Korovkin approximations

Our results are related to $\text{L}$¹-shadows in L_p-spaces. For p = 1 we will complete the characterization of $\text{L}$¹-shadows and $\text{L}$^1,1-shadows. For 1 < p < ∞ S. J. Bernau has shown that the $\text{L}$¹-shadow of a set in L_p is the range of a contractive projection. We will show that the corresponding theorem is not true for all reflexive spaces, but is true for locally uniformly convex reflexive spaces.     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$     

Multipliers of Sobolev spaces

Let W^m,p denote the Sobolev space of functions on $\text{R}$ⁿ whose distributional derivatives of order up to m lie in L^p($\text{R}$ⁿ) for 1 ? p ? ∞. When 1 < p < ∞, it is known that the multipliers on W^m,p are the same as those on L^p. This result is true for p = 1 only if n = 1. For, we prove that the integrable distributions of order ?1 whose first order derivatives are also integrable of order ?1, belong to the class of multipliers on W^m,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on L^p, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on W^m,l are necessarily integrable distributions of order ?1 or ?2 accordingly as m is odd or even. We have obtained the multipliers from L¹($\text{R}$ⁿ) into W^m,p, 1 ? p ? ∞, and the multiplier space of W^m,1 is realised as a dual space of certain continuous functions on $\text{R}$ⁿ which vanish at infinity.     

Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group $\text{K =}\text{G}\text{R}$). We show how to explicitly construct sequences {U_n} of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L^p(X, μ) ↓ L^p(X, μ) on an L^p space, 1 <p < ∞, and any f ∈ L^p(X, μ), the averages

$\text{A}{}_{\mathrm{n}}\text{f=}\text{1}\text{|U}{}_{\mathrm{n}}\text{|}\text{∫}\text{U}{}_{\mathrm{n}}\text{T}{}_{\mathrm{g}}{}^{\mathrm{?1f}}\text{dg (|E|=}\text{left Haar measure in}\text{G)}$

converge in L^p norm, and pointwise μ-a.e. on X, to G-invariant functions $\text{f1}$ in L^p(X, μ). A single sequence {U_n} in G works for all L^p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.     

Regularity properties of Schrödinger and Dirichlet semigroups

First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ on the various Lebesgue's spaces L^q=L^q$\text{R}$ⁿ, dx, whenever the negative part of V is in L^∞ + L^p for some $\text{p >}\text{max}\text{{1,}\text{n}\text{2}\text{}}$. Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups.     

A note on k-plane integral transforms

Let Π be a k-dimensional subspace of Rⁿ, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π^⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by $\text{L?(Π, x″) = ∝Π?(x′, x″) dx′}$. In this article certain a priori inequalities are established which show in particular that if $\text{? ? L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{)}$, $\text{1 ? p}\text{\$}\text{?}\text{n}\text{k}$, then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces L^p(Rⁿ), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rⁿ are also obtained.     

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by $\text{C}$^p(G)(0<p?2), (ii) the K-biinvariant elements in $\text{C}$^p(G) by $\text{I}$^p(G), (iii) the positive definite (zonal) spherical functions by $\text{P}$, and (iv) the spherical transform on $\text{C}$^p(G) by ? → $\text{}\text{?}$gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto $\text{C}$^l(G), (ii) there exists a unique measure μ of polynomial growth on $\text{P}$ such that T[ψ]=∫pψdμ for all ψ?I¹(G) (iii) all measures μ of polynomial growth on $\text{P}$ are obtained in this way, and (iv) T may be extended to a particular $\text{I}$^p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of $\text{P}$. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.     

Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.     

Thin presentation of knots in lens spaces and RP3-conjecture

This Note concerns knots in a lens space L that produce S³ by Dehn surgery. We introduce the thin presentation of knots in L, with respect to a standard spine. We prove that among such knots, those having a thin presentation with only maxima, are 0-bridge or 1-bridge braids in L. In the case $\text{L=}\text{R}\text{P}{}^3$, we deduce that minimally braided knots in $\text{R}\text{P}{}^3$ cannot yield S³ by Dehn surgery. To cite this article: A. Deruelle, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Radial bounds for perturbations of elliptic operators

Elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$, α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}$ or a quotient space $\text{R}{}^{\mathrm{n}}\text{R}{}^{\mathrm{n}}\text{U}{}_{\mathrm{\alpha }}\text{, U}{}_{\mathrm{\alpha }}\text{? R}{}^{\mathrm{n}}$ are considered. It is shown that the L^p-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L¹ radial convolution kernel. Some consequences are: A can be closed in all L^p (1 ? p ? ∞), and is essentially self-adjoint in L² if it is symmetric; A generates an analytic semigroup e^?tA in the right half plane, strongly L^p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability $\text{φ(εA)?→}{}_{\mathrm{\epsilon \; \to \; 0}}\text{φ(0) ?}$, with φ analytic, is proved for $\text{? ? L}{}^{\mathrm{p}}$, with convergence in L^p and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients.     

The functions operating on multiplier algebras

Let 1 < p < ∞ with p ≠ 2. Let G denote one of the groups $\text{T}$ⁿ, $\text{R}$ⁿ, or $\text{Z}$ⁿ. We show that only entire functions operate in certain algebras of multipliers on L_p(G).     

On the functional dimension of the space of solutions of partial differential equations

Given a differential polynomial P(D) in Rⁿ with constant coefficients, consider the functional dimension df$\text{N}$ of the space $\text{N}$ = {u∈C(Rⁿ):P(D)u = 0} endowed with the topology of uniform convergence on compact subsets of Rⁿ. If P(D) is elliptic then df$\text{N}$ = n, by a theorem of Y. Kōmura. We prove the converse: If df$\text{N}$ = n then the differential polynomial P(D) must be elliptic.     

Oscillatory properties of strongly superlinear differential equations with deviating arguments

We regard a graph G as a set {1,…, v} together with a nonempty set E of two-element subsets of {1,…, v}. Let p = (p₁,…, p_v) be an element of $\text{R}$^nv representing v points in $\text{R}$ⁿ and consider the realization G(p) of G in $\text{R}$ⁿ consisting of the line segments [p_i, p_j] in $\text{R}$ⁿ for {i, j} ?E. The figure G(p) is said to be rigid in $\text{R}$ⁿ if every continuous path in $\text{R}$^nv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ? $\text{R}$^nv which is the image (Tp₁,…, Tp_v) of p under an isometry T of $\text{R}$ⁿ. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in $\text{R}$² is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in $\text{R}$³.     

On compactness of commutators of multiplications and convolutions,and boundedness of pseudodifferential operators

Commutators [a(M), b(D)] of a multiplication (a(M)u)(x) = a(x) u(x) and a convolution b(D) = F^?1b(M)F (F = Fourier transform) are L²-compact if only the continuous functions a and b are bounded and for c = a and c = b we have $\text{lim}{}_{\mathrm{¦x¦\to \infty }}\text{sup}\text{{¦ c(x + h) ? c(x)¦ : ¦ h ¦ ? 1} = 0}$. An improvement of a result by Calderon and Vaillancourt of boundedness of pseudodifferential operators is discussed (including an independent proof). Similar results on L^p-compactness and L^p-boundedness, 1 < p < ∞, using the Hoermander-Mihlin boundedness theorem on $\text{R}$ⁿ-Fourier-multipliers, and with conditions and proofs different from the case of L².