Schatten–von Neumann properties for Hörmander classes on compact Lie groups

Symmetrized trace and symmetrized determinant of odd class pseudo-differential operators

Let G be an arbitrary compact Lie group. In this work we apply the method of the analytic continuation of traces in order to compute the Wodzicki residue for a classical pseudo-differential operator on G in terms of its matrix-valued symbol (which is globally defined on the non-commutative phase space $$G\times \widehat {G},$$ with $$\widehat {G}$$ being the unitary dual of G). Our main theorem is complementary to the results in Cardona et al. (Dixmier traces, Wodzicki residues, and determinants on compact Lie groups: the paradigm of the global quantisation. arXiv:2105.14949), where we remove the ellipticity hypothesis when the operators belong to the Hörmander classes on G defined by local coordinate systems.

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We describe all multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold, that are continuous along continuous paths and the restriction to zero order operators of which is of class C1. This boils down to a description of all traces on zero order classical pseudodifferential operators, which turn out to be linear combinations of the Wodzicki residue [W] and leading symbol traces introduced in [PR1], both of which are continuous. Consequently, multiplicative determinants are parametrized by the residue determinant [W87, Sc] and a new “leading symbol determinant”, both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator.

On the reduction of certain pseudodifferential operators with noninvolutive characteristics

This chapter concerns the relation between the boundary integral operators and classical pseudodifferential operators. A large class of boundary integral operators including those presented in the previous chapters belong to the special class of classical pseudodifferential operators on compact manifolds. We are particularly interested in strongly elliptic systems of pseudodifferential operators providing Gårding's inequality, see Theorem 9.1.4.

We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig–Ponce–Rolvung–Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola–Metcalfe–Tataru in the case of elliptic principal term.A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig–Ponce–Rolvung–Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola–Metcalfe–Tataru to the present case of nonelliptic principal term.

It is shown that two classes of generalized elliptic pseudo-differential operators, GEL(X) and REL(X), selected by the author from the class of classical linear pseudo-differential operators coincide. It is also shown that for any operators A, B ∈GEL(X) their composition AB and their global parametrices PA, PB belong to GEL(X). An operator A belongs to GEL(X) independently of the choice of a basis in E and of the weighted order of A. Some properties of the classes EFL(U) and REL(U) arising in microalocal analysis of generalized elliptic operators are studied. Bibliography: 12 titles.

We introduce a new class of selfadjoint compact pseudodifferential operators, which is analogous to a class of elliptic unbounded pseudodifferential operators and is, therefore, suitable for obtaining upper and lower estimates on the eigenvalues of operators in this class. We prove such estimates and, as an application, we show that any operator from this class belongs to the Schatten-von Neuman class if and only if its symbol belongs to the Lorentz space.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

It is well known that the space of classical pseudo-differential operators is invariant under conjugation by classical Fourier integral operators. However, the Weyl-Hormander calculus [Ho1] [Ho2] provides a much larger framework for the theory of pseudo-differential operators. Any riemannian metric g on the phase space R n x × R n ξ, satisfying the conditions of definition 1.1, defines a graded algebra of pseudo-differential operators. The classical theory corresponds to a particular metric, namely g(dx,dξ) = dx 2 + dξ2/〈ξ〉2.

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Let B B be a compact Riemannian manifold, let Ω \Omega denote the cylinder R × B \mathbb {R}\times B , Δ Ω \Delta _\Omega its Laplace operator and Λ = ( 1 − Δ Ω ) − 1 / 2 \Lambda =(1-\Delta _\Omega )^{-1/2} . Let A \mathfrak {A} denote the C*-algebra of bounded operators on L 2 ( R × B ) L^2(\mathbb {R}\times B) generated by all the classical pseudodifferential operators on R × B \mathbb {R}\times B of the form L Λ N L\Lambda ^N , N N a nonnegative integer and L L an N N -th order differential operator whose (local) coefficients approach 2 π 2\pi -periodic functions at + ∞ +\infty and − ∞ -\infty . Let E \mathfrak {E} denote the kernel of the continuous extension of the principal symbol to A \mathfrak {A} . The problem of computing the K-theory index map δ 1 ( K 1 ( A / E ) ) → K 0 ( E ) ≃ Z 2 \delta _1(K_1(\mathfrak {A}/\mathfrak {E}))\to K_0(\mathfrak {E})\simeq \mathbb {Z}^2 on an element of K 1 ( A / E ) K_1(\mathfrak {A}/\mathfrak {E}) is reduced to the problem of computing the Fredholm indices of two elliptic operators on the compact manifold S 1 × B S^1\times B . For B = S 1 B=S^1 , Hess went further and proved in her thesis that K 0 ( A ) ≅ Z 5 K_0(\mathfrak {A})\cong \mathbb {Z}^5 and K 1 ( A ) ≅ Z 4 K_1(\mathfrak {A})\cong \mathbb {Z}^4 .

In recent years, the use of Peter-Weyl theory (the theory of Fourier analysis on compact Lie groups) to define so-called “global symbols” of operators on compact Lie groups has emerged as a fruitful technique to study pseudo-differential operators. The aim of this thesis is to discuss similar techniques in the setting of compact homogeneous spaces. The approach is to relate operators on homogeneous spaces to those on compact Lie groups, and then to utilize the recently developed techniques on such groups. Two methods of associating operators on homogeneous spaces with those on compact Lie groups, called projective and horizontal lifting, along with their properties, merits and problems are considered. A key tool used in this analysis is the notion of a difference operator. This thesis includes a detailed study of such operators and their properties, combined with comprehensive calculations involving such operators on the homogeneous spaces Sn−1 = SO(n)/ SO(n− 1). This thesis concludes with a generalization of the symbolic calculus on compact Lie groups developed by M. Ruzhansky and V. Turunen together with a collection of conjectures, which if proven would relate the generalization to pseudo-differential theory on homogeneous spaces.

Global functional calculus for operators on compact Lie groups