Pseudo-differential operator in the framework of linear canonical transform domain

The primary aim is to develop a new class of pseudo-differential operators involving Linear Canonical Transform (LCT) and some of its essential results in Schwartz space and tempered distribution space. The secondary aim is to find applications of the LCT while constructing new general partial differential equations (Heat and Wave) and derive their solutions. Further, we have discussed particular cases of these partial differential equations. Furthermore, we have also shown the resolution of the solution of these equations graphically.

In this thesis, a pseudodifferential calculus for a degenerate hyperbolic Cauchy problem is developed. The model for this problem originates from a certain observation in fluid mechanics, and is then extended to a more general class of hyperbolic Cauchy problems where the coefficients degenerate like a power of $t + |x|^2$ as $(t,x) \to (0,0)$. Symbol classes and pseudodifferential operators are introduced. In this process, it becomes apparent that exactly in the origin, these operators are of type (1,1). Although these operators are not $L^2$-continuous in general, a proof of continuity in $\mathscr C([0,T],L^2(\mathbb R^d))$ is given for a suitable subclass. An adapted scale of function spaces is defined, where at $t = 0$ these spaces coincide with 2-microlocal Sobolev spaces with respect to the Lagrangian $\dot T^*_0\mathbb R^d$. In these spaces, energy estimates are derived, so that a symbolic approach can be applied to prove wellposedness of the Cauchy problem.

Linear canonical transformations (LCTs) were introduced almost simultaneously during the early 1970s by Stuart A. Collins Jr. in paraxial optics, and independently by Marcos Moshinsky and Christiane Quesne in quantum mechanics, to understand the conservation of information and of uncertainty under linear maps of phase space. Only in the 1990s did both sources begin to be referred jointly in the growing literature, which has expanded into a field common to applied optics, mathematical physics, and analogic and digital signal analysis. In this introductory chapter we recapitulate the construction of the LCT integral transforms, detailing their Lie-algebraic relation with second-order differential operators, which is the origin of the metaplectic phase. Radial and hyperbolic LCTs are reviewed as unitary integral representations of the two-dimensional symplectic group, with complex extension to a semigroup for systems with loss or gain. Some of the more recent developments on discrete and finite analogues of LCTs are commented with their concomitant problems, whose solutions and alternatives are contained the body of this book.

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.

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1<p<\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0<p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1<p<\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0<p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Two variants of fractional powers of Hankel–Clifford transformations of order \(\nu \ge 0\) with parameter \(\alpha \) are introduced and studied their properties. The operational formulas are developed. We study pseudo-differential operators associated with the symbol \(a(x,y)\) and their integral representations. Some properties of Sobolev type spaces are studied. Fractional powers of Hankel–Clifford transformations are used in the solution of partial differential equation involving Kepinski type operators \(\Delta _{1,\nu ,\mu ,\alpha }^{*}\) and \(\Delta _{2,\nu ,\mu ,\alpha }^{*}\).

Pseudodifferential operators on ultra-modulation spaces

This book develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. The first chapter studies classes of pseudodifferential operators whose symbols have a limited degree of regularity; the second chapter shows how paradifferential operators yield sharp estimates on the action of various nonlinear operators on function spaces. The third chapter applies this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients. Michael Taylor is the author of several well-known books on topics in PDEs and pseudodifferential operators. His Noncommutative Harmonic Analysis, Volume 22 in the Mathematical Surveys and Monographs series published by the AMS, is a good introduction to the use of Lie groups in linear analysis and PDEs. The present book, Tools for PDE, is suitable as a text for advanced graduate students preparing to concentrate in PDE and/or harmonic analysis.

This article is a survey of Louis Nirenberg’s contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.

Linear Partial Differential Equations with Multiple Involutive Characteristics O. Liess, L. Rodino. Gevrey and Analytic Hypoellipticity D.S. Tartakoff. Higher Microlocalization and Propagation of Singularities O. Liess. Conormality and Lagrangian Properties in Diffractive Boundary Value Problems P. Laubin. Parametrized Pseudodifferential Operators and Geometric Invariants G. Grubb. Boundary Value Problems and Edge Pseudodifferential Operators B.-W. Schulze. Wodzicki's Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conical Singularities E. Schrohe. Lower Bounds for Pseudodifferential Operators C. Parenti, A. Parmeggiani. Weyl Formula for Globally Hypoelliptic Operator in Rn E. Buzano. Splitting in Large Dimension and Infrared Estimates B. Helffer. Microlocal Exponential Estimates and Applications to Tunneling A. Martinez. A Trace Formula and Review of Some Estimates for Resonances J. Sjostrand.

We discuss spaces of Sobolev type which are defined by the operator with singularity: D = d / d x − ( c / x ) R \mathcal {D} = d/dx - (c/x)R , where R u ( x ) = u ( − x ) Ru(x) = u(-x) and c > 1 c > 1 . This operator appears in a one-dimensional harmonic oscillator governed by Wigner’s commutation relations. We study smoothness of u u and continuity of u / x β u / x^{\beta } ( β > 0 \beta > 0 ) where u u is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.

In the present work we consider the class of symbols and associated pseudo-differential operators defined in ([3], [4], [6]). We get results related to Theorem A.3 and Lemma 9·1 in [2]; we use similar methods.

Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two functions in the Schwartz class $\mathcal{S}(\mathfrak{g}^{*})$, where $\vee$ and $\wedge$ are the Abelian Fourier transforms on the Lie algebra $\mathfrak{g}$ and on the dual $\mathfrak{g}^{*}$. In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order generalization of the Weyl calculus for pseudodifferential operators of Hormander. The idea of such a calculus consists in describing the product $f \# g$ for some classes of symbols. We find a formula for $D^{\alpha}(f \# g)$ for Schwartz functions $f,g$ in the case of two-step nilpotent Lie groups, that includes the Heisenberg group. We extend this formula to the class of functions $f,g$ such that $f^{\vee}, g^{\vee}$ are certain distributions acting by convolution on the Lie group, that includes usual classes of symbols. In the case of the Abelian group $R^{d}$ we have $f \# g = fg$, so $D^{\alpha}(f \# g)$ is given by the Leibniz rule.

For a long time, numerous problems in various applied fields [1-3] have been investigated with the use of singular integral equations. For example, some problems of aerodynamics [1], diffraction of electromagnetic and acoustic waves [2], elasticity [3], etc. can be reduced to the solution of such equations. The theory of singular integral equations was developed long ago [3], but related applied problems required the solution of these equations in the class of absolutely integrable functions. It was mentioned in [4] that, under specific conditions imposed on the coefficients of a singular integral equation, one can find solutions in the class of fimctions with nonintegrable singularity of the form 1/x as x -* 0. But the absence of applied problems leading to such solutions, numerical methods for their construction, and a theory of distributions developed for singular integral equations hindered the investigation of such singular solutions. However, in the framework of aerodynamics [5, 6], the mechanics of wings has recently been analyzed with the use of ejection of an external flow. It was shown in [6] that, in the case of a thin profile, this problem can be reduced to solving a singular integral equation of the first kind on a closed interval in the class of fimctions that have a nonintegrable singularity of the above-mentioned type at the ejection point. By [7], the investigation of the diffraction of an electromagnetic wave on an ideal conductor with a current thread on the surface leads to a similar mathematical problem. Such problems were numerically solved in [1, 7, 8] in sorne special cases with the use of the discrete vortex and discrete current methods. However, the integral operators constructed in these papers are continuously invertible in related pairs of spaces but have the property that their squares map the entire subspace of singular solutions to zero. In the present paper, in view of [9, 10], we give another mathematical treatment of the same applied problems, which permits one to consider them in the framework of the theory of pseudodifferential operators on a closed curve. Then we describe the construction of Sobolev type weighted spaces and develop the theory of linear operators (similar to pseudodifferential ones) in these spaces for open curves. We present examples illustrating the difference between the notions of integral in the sense of the Cauchy principal value and pseudodifferential operators on distributions that are singular solutions. Similar linear operators are constructed for singular integral equations of the second kind with Hilbert kernel and constant coefficients on a closed interval. We indicate the possibility of the generalization of the suggested construction of linear operators to some spatial problems.

This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to 'algebraize' the index theory by means of pseudo-differential operators and methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the student to work efficiently with the principal symbols of the elliptic and boundary operators on the boundary. This book is ideal for use in graduate-level courses on partial differential equations, elliptic systems, pseudo-differential operators and matrix analysis. Since many new methods and results are introduced and used throughout the book, all the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises.