Pseudodifferential Calculus Research Articles

Boundedness of Metaplectic Operators Within $$L^p$$ Spaces, Applications to Pseudodifferential Calculus, and Time–Frequency Representations

AbstractHausdorff–Young’s inequality establishes the boundedness of the Fourier transform from $$L^p$$ L p to $$L^q$$ L q spaces for $$1\le p\le 2$$ 1 ≤ p ≤ 2 and $$q=p'$$ q = p ′ , where $$p'$$ p ′ denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the $$L^p-L^q$$ L p - L q boundedness of metaplectic operators, which play a significant role in harmonic analysis. We demonstrate that metaplectic operators are bounded on Lebesgue spaces if and only if their symplectic projection is either free or lower block triangular. As a byproduct, we identify metaplectic operators that serve as homeomorphisms of $$L^p$$ L p spaces. To achieve this, we leverage a parametrization of the symplectic group by Dopico and Johnson. We use our findings to provide boundedness results within $$L^p$$ L p spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplectic Wigner distributions, which are essential to measure local phase-space concentration of signals. Using the factorization by Dopico and Johnson, we infer a decomposition law for metaplectic operators on $$L^2({\mathbb {R}^{2d}})$$ L 2 ( R 2 d ) in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in $$\mathop {\mathrm {{Sp}}}\limits (2d,\mathbb {R})$$ Sp ( 2 d , R ) , and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces $$M^p(\mathbb {R}^d)$$ M p ( R d ) .

Factorizations for quasi-Banach time–frequency spaces and Schatten classes

We deduce factorization properties for Wiener amalgam spaces WLp,q, an extended family of modulation spaces M(ω,ℬ), and for Schatten symbols spw in pseudo-differential calculus under e.g. convolutions, twisted convolutions and symbolic products. Here M(ω,ℬ) can be any quasi-Banach Orlicz modulation space. For example we show that WL1,r∗WLp,q=WLp,q and WL1,r#spw=spw when r∈(0,1], r≤p,q<∞. In particular we improve Rudin’s identity L1∗L1=L1.

Well-posedness for a class of pseudo-differential hyperbolic equations on the torus

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus.

Multiresolution kernel matrix algebra

We propose a sparse algebra for samplet compressed kernel matrices to enable efficient scattered data analysis. We show that the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. The compression can be performed in cost and memory that scale essentially linearly with the number of data points for kernels of finite differentiability. The same holds true for the addition and multiplication of S-formatted matrices. We prove that the inverse of a kernel matrix, given that it exists, is compressible in the S-format as well. The use of selected inversion allows to directly compute the entries in the corresponding sparsity pattern. Moreover, S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as Aα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{A}}^\\alpha $$\\end{document} or exp(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp ({\\varvec{A}})$$\\end{document} of a matrix A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{A}}$$\\end{document}. The matrix algebra is justified mathematically by pseudo differential calculus. As an application, we consider Gaussian process learning algorithms for implicit surfaces. Numerical results are presented to illustrate and quantify our findings.

Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.

$C^{\ast}$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a \ast -homomorphism of C^{\ast} -algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes trace theorem for compact Riemannian manifolds.

Hypoelliptic Operators in Geometry

The workshop titled Hypoelliptic Operators in Geometry was coorganized by Davide Barilari (Padova), Xiaonan Ma (Paris), Nikhil Savale (K¨oln) and Yi Wang (Baltimore). It was well attended by 55 participants, with 45 of them being present in person and 10 being online. The participants came from several continents, age groups and included male as well as female researchers. Several interesting themes were discussed including: analysis around Kohn’s Laplacian in CR geometry, analogous covariant operators arising in conformal geometry, the spectral theory of the sub-Riemannian Laplacian, pseudodifferential calculi in non-commutative geometry and the geometric applications of Bismut’s hypoelliptic Laplacians.

Pseudo-differential calculi and entropy estimates with Orlicz modulation spaces

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. For example we generalize the classical result by Cordero and Nicola that if1p′+1q′⩽1p1+1p2,1p′+1q′⩽1q1+1q2,pj,qj⩽q′,q⩽p and a∈Mp,q, then the pseudo-differential operator Op(a) is continuous from Mp1,q1 to Mp2′,q2′.We also show that the entropy functional Eϕ possess suitable continuity properties on a suitable Orlicz modulation space MΦ satisfying Mp⊆MΦ⊆M2, though Eϕ is discontinuous on M2=L2.

Full Double Hölder Regularity of the Pressure in Bounded Domains

Abstract We consider Hölder continuous weak solutions $u\in C^{\gamma }(\Omega )$, $u\cdot n|_{\partial \Omega }=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega \subset{\mathbb{R}}^{d}$. If $\Omega $ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2\gamma }_{*}(\Omega )$ in the case $\gamma \in (0,\frac{1}{2}]$, where $C^{2\gamma }_{*}$ is the Hölder–Zygmund space, which coincides with the usual Hölder space for $\gamma &amp;lt;\frac 12$. This result, together with our previous one in [ 11] covering the case $\gamma \in (\frac 12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma }_{*}$ estimate for the pressure on the whole space ${\mathbb{R}}^{d}$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma =\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood–Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step that plays a major role in our approach.

Positivity results for Weyl’s pseudo-differential calculus on the Wiener space

This paper deals with positivity properties for a pseudo-differential calculus, generalizing Weyl’s classical quantization, and set on an infinite dimensional configuration space, the Wiener space. In this frame, we show that a positive symbol does not, in general, give a positive operator. In order to measure the nonpositivity, we establish a Gårding’s inequality, which holds for the symbol classes at hand. Nevertheless, for symbols with radial aspects, additional assumptions ensure the positivity of the associated operator.

Hypoelliptic and spectral estimates for some linear operators of Landau type

We are interested in the inhomogeneous Landau equation which describes the evolution of a particle density f = f (t, x, v) representing at time t $\ge$ 0, the density of particles at position x $\in$ R 3 and velocity v $\in$ R 3. The study is motivated by the linearization of the Landau equation near Maxwellian distribution. In this article, we establish hypoelliptic estimates, a localization of the spectrum and estimates of the resolvent of the linear Landau operator with hard potentials and Maxwellian molecules. The proof is based on a multiplier method and requires fine pseudo-differential calculus tools.

Continuity of pseudodifferential operators with nonsmooth symbols on mixed-norm Lebesgue spaces

Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as can be seen from recent interest in the continuity of certain classes of pseudodifferential operators on these spaces. In this paper, we use some recent advances in the pseudodifferential calculus for nonsmooth symbols to prove the boundedness of pseudodifferential operators with such symbols on mixed-norm Lebesgue spaces.

On Kato's smoothing effect for a fractional version of the Zakharov-Kuznetsov equation

In this work, we study some regularity properties associated with the initial value problem (IVP)$\begin{align} \left\{ \begin{array}{ll} \partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u = 0, \quad 0&lt; \alpha\leq 2, &amp; \\ u(x, 0) = u_{0}(x), \quad x = (x_{1}, x_{2}, \dots, x_{n})\in \mathbb{R}^{n}, \, n\geq 2, \quad t\in\mathbb{R}, &amp; \ \end{array} \right. \end{align}\ \ \ \ \ (1)$where $ (-\Delta)^{\alpha/2} $ denotes the $ n- $dimensional fractional Laplacian.We show that solutions to the IVP (1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of $ \frac{\alpha}{2} $ derivatives, almost everywhere in time. One of the main difficulties that emerge when trying to obtain this regularizing effect underlies that the operator in consideration is non-local, and the property we are trying to describe is local, so new ideas are required. Nevertheless, to avoid these problems, we use a perturbation argument replacing $ (-\Delta)^{\frac{\alpha}{2}} $ by $ (I-\Delta)^{\frac{\alpha}{2}}, $ that through the use of pseudo-differential calculus allows us to show that solutions become locally smoother by $ \frac{\alpha}{2} $ of a derivative in all spatial directions.As a by-product, we use this particular smoothing effect to show that the extra regularity of the initial data on some distinguished subsets of the Euclidean space is propagated by the flow solution with infinity speed.

On a class of pseudodifferential operators on the product of compact Lie groups

AbstractIn this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved.

A calculus for magnetic pseudodifferential super operators

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators L̂=−iopA(h),⋅ defined in terms of a magnetic pseudodifferential operator opA(h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.

Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems

Abstract We introduce and analyse various regularized combined field integral equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyse OS methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators, which is also the basis of high-order Nyström quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.

Graded hypoellipticity of BGG sequences

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

Global Existence of Non-Cutoff Boltzmann Equation in Weighted Sobolev Space

This article presents a new approach of semigroup analysis and pseudo-differential calculus for deriving the regularizing estimate on non-cutoff linearized Boltzmann equation. We are able to obtain regularizing estimate of semigroup \(e^{tB}\) that is continuous from weighted Sobolev space \(H(a^{-1/2})H^m_x\) to \(H(a^{1/2})H^m_x\) with a sharp large time decay. With these properties, we prove the existence of global-in-time unique solution to the non-cutoff Boltzmann equation for hard potential on the whole space with weak regularity assumption on initial data. We consider the hard potential case since \(H(a^{1/2})\) can be embedded in \(L^2\). This work develops the application of pseudo-differential calculus, spectrum analysis and semigroup theory to non-cutoff Boltzmann equation.

Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibered boundary (aka ϕ-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibers are points. The new feature in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate ‘split’ pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on ϕ-manifolds.

The fractional Schrödinger equation on compact manifolds: global controllability results

The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g). To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the relation between the geometric control condition and the controllability for the fractional Schrödinger equation.