Modern Trends in Pseudo-Differential Operators

1. Integral transforms on a sphere.- 1. The generalized kernels (xy)?+-, (+-xy + iO)?.- 2. The operator E(?), its relation with the Fourier and Mellin transform.- 3. The action of E(?) on spherical functions.- 4. Operators related with the transform E(?).- 5. The spaces Hs(?, Sn-1). The operator E(?) on the spaces Hs(?, Sn-1).- 6. An analog of the Paley-Wiener theorem for the operator E(?).- 2. The Fourier transform and convolution operators on spaces with weighted norms.- 1. The spaces H?s(?n).- 2. The Fourier transform on the spaces Hs?(?n).- 3. Convolution operator on the spaces Hs?(?n).- 4. The spaces Hs?(?m,?m-n).- 5. Transversal operators and special representations.- 6. Estimates for the convolution operator on the spaces H?s(?m, ?m-n).- 3. Meromorphic pseudodifferential operators.- 1. Canonical meromorphic pseudodifferential operators.- 2. Operations on canonical meromorphic pseudodifferential operators.- 3. General meromorphic pseudodifferential operators.- 4. Traces of meromorphic pseudodifferential operators.- 5. Meromorphic pseudodifferential operators on strongly oscillating functions.- 6. Estimates for meromorphic pseudodifferential operators.- 7. Periodic meromorphic pseudodifferential operators.- 8. Change of variables in meromorphic pseudodifferential operators.- 4. Pseudodifferential operators with discontinuous symbols on manifolds with conical singularities.- 1. Pseudodifferential operators on ?n.- 2. Pseudodifferential operators on a conic manifold.- 3. Pseudodifferential operators on manifolds with conical points.- 4. Algebras generated by pseudodifferential operators of order zero.- 5. The spectrum of a C* -algebra of pseudodifferential operators with discontinuous symbols on a closed manifold.- 1. Results from the theory* of C* -algebras.- 2. The spectrum of a C* -algebra of pseudodifferential operators with discontinuities of the first kind in the symbols on a smooth closed manifold (statement of the main theorem).- 3. Representations of the algebra $$ \mathfrak{G} $$(?) generated by the operators E(?)-1F(o, ?)E(?).- 4. Representations of an algebra $$ \mathfrak{G} $$(lx).- 5. Proof of theorem 2.1.- 6. Ideals in the algebra of pseudodifferential operators with discontinuous symbols.- 7. Spectra of C* -algebras of pseudodifferential operators on a manifold with conical points.- 8. The spectrum of a C* -algebra of pseudodifferential operators with oscillating symbols.- 6. The spectrum of a C* -algebra of pseudodifferential operators on a manifold with boundary.- 1. The algebras $$ \mathfrak{G} $$c(?).- 2. The algebras $$ \mathfrak{G} $$(?).- 3. The algebras $$ \mathfrak{G} $$c(l?).- 4. The algebras $$ \mathfrak{G} $$(l?).- 5. The spectrum of an algebra of pseudodifferential operators on a manifold with boundary.- Bibliographical sketch.- References.

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces

Pseudo differential operators and integrable systems in AXIOM

We discuss a sub-algebra ¸ of Pseudo Differential Operators of which is Fredholm-closed and commutative modulo the ideal of compact operators. The algebra ¸ contains in particular all LP - Fourier-multipliers of Mihlin type. is shown to be an algebra with symbol, and its symbol space is determined. As an application we get simple criteria for an operator in ¸to be Fredholm. Furthermore, we regard algebras ¸ on Sobolev spaces , which are isometrically isomorphic to ¸ and have equivalent properties. Finally we conclude Fredholm criteria for mroe general Differential and Pseudo Differential Operators.

We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces M p, q , acting on a given Lebesgue space L r . Namely, we find the full range of triples (p, q, r), for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space W(L r , L s ) and even on modulation spaces M r, s . Finally, the action of pseudodifferential operators with symbols in is also investigated.

Point symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tN. The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known W∞ algebra of KP symmetries and into the free fermion algebra ĝl(∞). The corresponding action on the tau-function is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a P∞ algebra, based on the algebra of pseudodifferential operators.

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S∈Sp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

We consider the problem of the asymptotics of tr f(Tα) where f is a “general” function, Tα, the pseudodifferential operator with symbol σ(x,ξ,α). Here α is a large parameter, and σ is a rapidly decreasing function. If σ smooth, techniques which are by now standard yield complete asymptotic expansions, with computable coefficients, as α→∞. Here we consider also cases where σ is not smooth; it may jump across a hypersurface in x-space or a hyperplane in x, ξ-space. We prove the existence of complete asymptotic expansions and show how to compute the coefficients for polynomial f. Of central importance is the analytic continuations of certain distributions acting on spaces of non-smooth test functions.

Using his formulation of the potential theoretic notion of balayage and his deep results about this idea, Beurling gave sufficient conditions for Fourier frames in terms of balayage. The analysis makes use of spectral synthesis, due to Wiener and Beurling, as well as properties of strict multiplicity, whose origins go back to Riemann. In this setting and with this technology, with the goal of formulating non-uniform sampling formulas, we show how to construct frames using pseudo-differential operators. This work fits within the context of the short time Fourier transform (STFT) and time-frequency analysis.

In this work, the quadratic-phase Fourier transform (QPFT) on Schwartz space is defined and its necessary results are derived, including adjoint formula, Parseval identity, and continuity property. Furthermore, the continuity property on tempered distribution space is discussed, and also an example of the QPFT is provided. The present analysis is applied to develop a new class of pseudo-differential operators and prove an important theorem on Schwartz space. The boundary value problems of generalized partial differential equations (i.e. generalized telegraph, wave, and heat equations) are discussed using QPFT, and graphical plots are provided.

Sufficient conditions for stability on a finite time interval are derived for a class of evolution variational inequalities with the use of Lyapunov functions and frequency-domain conditions. These inequalities are considered for Hilbert spaces and Sobolev spaces of infinite order in the sense of Yu.A. Dubinskii. We show how to use the stability result on a finite interval to characterize bifurcations. An algorithm for finding observation functionals is presented that uses an isomorphism between the algebra of pseudodifferential operators with constant coefficients whose symbols are real-analytic functions in some domain and an algebra of analytic matrix-valued functions.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.