Schwartz Space Research Articles

Momentum ray transforms, II: range characterization in the Schwartz space

The momentum ray transform Ik integrates a rank m symmetric tensor field f over lines of with the weight tk: We give the range characterization for the operator on the Schwartz space of rank m smooth fast decaying tensor fields. In dimensions , the range is characterized by certain differential equations of order which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand–Helgason–Ludwig conditions.

On the anti-Wick symbol as a Gelfand-Shilov generalized function

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping S ( R n ) \mathcal {S}(\mathbb {R}^n) into S ′ ( R n ) \mathcal {S}’(\mathbb {R}^n) , which is generally not a tempered distribution, can still be defined as a Gel′fand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gel′fand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.

Spectrum of composition operators on [formula omitted] with polynomial symbols

We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to {0}, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origin. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

Tempered distributions and Schwartz functions on definable manifolds

We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands’ functoriality conjecture. As an evidence for their conjectures, Braverman and Kazhdan (2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in the work of Li (2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan’s later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan (2002).

The scattering problem for the three-dimensional cubic nonlinear Klein-Gordon equation with rapidly decreasing input data

The scattering problem for the three-dimensional cubic nonlinear Klein-Gordon equation is studied. It has been shown that the scattering operator S is well-defined on a neighborhood in the critical space H1/2(R3)⊕H−1/2(R3) of 0. We prove that if functions f− and g− are in the Schwartz space S(R3) and small in the sense of the critical space, then the corresponding output data (f+,g+):=S(f−,g−) also belongs to S(R3)⊕S(R3). Furthermore, we give sufficient conditions for (f−,g−) such that all order partial derivatives of f+ and g+ decay more rapidly than a same exponential function.

Real Paley-Wiener theorems in spaces of ultradifferentiable functions

We develop real Paley-Wiener theorems for classes Sω of ultradifferentiable functions and related Lp-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.

$L^p$ estimates for semi-degenerate simplex multipliers

Muscalu, Tao, and Thiele prove L^p estimates for the "Biest" operator defined on Schwartz functions by the map C^{1,1,1}: (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < \xi_3} \Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j } \Big] \,d \vec{\xi} via a time-frequency argument that produces bounds for all multipliers with non-degenerate trilinear simplex symbols. In this article we prove L^p estimates for a pair of simplex multipliers defined on Schwartz functions by the maps \begin{align*} C^{1,1,-2}:\ & (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < -{\xi_3}/{2}}\Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j } \Big] \,d \overrightarrow{\xi} \\ C^{1,1,1,-2}:\ & (f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 < \xi_2 < \xi_3 < -{\xi_4}/{2}} \Big[\prod_{j=1}^4 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j} \Big] d \overrightarrow{\xi} \end{align*} for which the non-degeneracy condition fails. Our argument combines the standard \ell^2 -based energy with an \ell^1 -based energy in order to enable summability over various size parameters. As a consequence, we obtain that C^{1,1,-2} maps into L^p for all 1/2 < p < \infty and C^{1,1,1,-2} maps into L^p for all 1/3 < p < \infty . Both target L^p ranges are shown to be sharp.

Principal Submodules in the Module of Entire Functions, Which is Dual to the Schwarz Space, and Weak Spectral Synthesis in the Schwartz Space

We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz space C∞ (a; b) are discussed.

Uniform in N estimates for a Bosonic system of Hartree–Fock–Bogoliubov type

We prove local in time, uniform in N, estimates for the solutions ϕ , Λ, and Γ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential v N ( x − y ) = N 3 β v ( N β ( x − y ) ) , with β < 1 and v a Schwartz function (satisfying additional technical requirements). The initial conditions are general functions in a Sobolev-type space, and the expected correlations in Λ develop dynamically in time. As shown in our previous work, as well as the work of Chong (both in the case β < 2 / 3 ), using the conserved quantities of the system of equations, this type of local in time estimates can be extended globally. Also, they can be used to derive Fock space estimates for the approximation of the exact evolution of a Bosonic system by quasi-free states of the form e N A ( ϕ ) e B ( k ) Ω . This will be addressed in detail in future work.

On the Inverse Ultrahyperbolic Klein-Gordon Kernel

In this work, we define the ultrahyperbolic Klein-Gordon operator of order α on the function f by T α ( f ) = W α ∗ f , where α ∈ C , W α is the ultrahyperbolic Klein-Gordon kernel, the symbol ∗ denotes the convolution, and f ∈ S , S is the Schwartz space of functions. Our purpose of this work is to study the convolution of W α and obtain the operator L α = T α − 1 such that if T α ( f ) = φ , then L α φ = f .

Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

Let L be a sub-Laplacian on a connected Lie group G of polynomial growth. It is well known that, if F:R→C is in the Schwartz class S(R), then the convolution kernel KF(L) of the operator F(L) is in the Schwartz class S(G). Here we prove a sort of converse implication for a class of groups G including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of KF(L) implies continuity of F.

Quasi-Nash varieties and Schwartz functions on them

We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that the classical properties of these spaces, that hold on Nash manifolds and real affine algebraic varieties, hold in this category as well.

Spectral Multipliers on 2-Step Stratified Groups, II

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

Solution of the Boussinesq equation using evolutionary vessels

In this work, we present a solution of the Boussinesq equation, which models shallow water waves. A powerful inverse scattering approach of Deift–Tomei–Trubowitz was published in 1982 for solving this equation in the Schwartz class. We rewrite the corresponding Lax pair in a matrix form and show that a recently developed theory of evolutionary nodes is applicable. As a result, we can locally solve this equation with analytic initial conditions on the line, providing a formula for the tau function, which defines where the solutions exist. More generally, arbitrary analytic solution is shown to arise from a vessel.

The massive Thirring system in the quarter plane

AbstractThe unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

Dynamical properties of weighted translation operators on the Schwartz space mathcal {S}(mathbb {R})

In this paper we investigate the dynamical properties of weighted translation operators acting on the Schwartz space mathcal {S}(mathbb {R}) of rapidly decreasing functions, i.e., operators of the form {T_w}:{mathcal {S}(mathbb {R})}rightarrow {mathcal {S}(mathbb {R})},~f(cdot )mapsto w(cdot )f(cdot +1). We characterize when those operators are hypercyclic, weakly mixing, mixing and chaotic. Several examples illustrate our results and show which of those classes are different.

Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$ is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable function to be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$ endowed with a suitable distance $\wh d$. This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.

Hörmander's multiplier theorem for the Dunkl transform

For a normalized root system R in RN and a multiplicity function k≥0 let N=N+∑α∈Rk(α). Denote by dw(x)=∏α∈R|〈x,α〉|k(α)dx the associated measure in RN. Let F stand for the Dunkl transform. Given a bounded function m on RN, we prove that if there is s>N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator Tmf=F−1(mFf) is of weak type (1,1), strong type (p,p) for 1<p<∞, and is bounded on a relevant Hardy space H1. To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if F is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function F is bounded on Lp(dw) for 1≤p≤∞. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.

Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

We consider the Cauchy problem of the nonlinear heat equation ut−Δu=ub,u(0,x)=u0, with b≥2 and b∈N. We prove that initial data u0∈S(Rn) (the Schwartz class) arbitrarily small in the scale invariant Besov-norm B˙n(b−1)b/2,q−2/b(Rn), can produce solutions that blow up in finite time. The case b=3 answers a question raised by Yves Meyer. Our result also proves that the smallness assumption put in an earlier work by C. Miao, B. Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.