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In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $ (\rho,\delta) $ (ρ , δ) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $ \rho =\delta. $ ρ = δ. Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated. [ABSTRACT FROM AUTHOR]

We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet elliptic equation driven by the (p , q) -Laplacian and a reaction which includes a singular term, a parametric convection and a Carathéodory perturbation. Under minimal conditions on the perturbation, using the theory of nonlinear operators of pseudomonotone type, we show that for all small values of the parameter, the problem has a positive solution and determines the properties of the solution set. [ABSTRACT FROM AUTHOR]

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate L 2 – L p boundedness of pseudo-differential operators on the Heisenberg motion group for the range 2 ≤ p ≤ ∞. We also provide a necessary and sufficient condition on the operator-valued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators. [ABSTRACT FROM AUTHOR]

We study properties of pseudodifferential operators which arise in their use in boundary value problems. Smooth domains as well as intersections of smooth domains are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol ς and two windows functions g 1 and g 2 . We first present some basic properties of the windowed Opdam–Cherednik transform. Then, we use modulation spaces associated with the Opdam–Cherednik transform as appropriate classes for symbols and windows and study the boundedness and compactness of the localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten–von Neumann class. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to enrich the theoretical system of the linear canonical Fourier transform (LCFT) by introducing the canonical potential J m , A and corresponding L p -Sobolev space. Moreover, the Schwartz-type space S A is introduced. Further, pseudo-differential operator (PDO) is defined and obtained its another integral representation. The L p -boundedness result for the pseudo-differential operator associated with the LCFT is discussed. Some applications of Sobolev-type spaces H A m , p and H A m , p are given. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in operators arising from the product of a weighted composition operator and a maximal differential operator on the Fock space. Let T max and C u , φ denote a maximal differential operator and a weighted composition operator, respectively. We characterize some C -selfadjoint operators C u , φ T max with respect to special conjugations C . Moreover, we obtain a partial characterization of hermitian operator C φ T max . [ABSTRACT FROM AUTHOR]

Let G be a compact Hausdorff group and H be a closed subgroup of G. In this paper, we show that every bounded linear operator T on L p (G / H) is a pseudo-differential operator with the symbol σ for 1 ≤ p < ∞. We present necessary and sufficient conditions on symbols to ensure that a bounded pseudo-differential operator on L 2 (G / H) is self-adjoint, normal and present explicit formula for their symbols. A necessary and sufficient condition is also given such that the bounded linear operators on L p (G / H) posses eigenvalues and eigenfunctions. [ABSTRACT FROM AUTHOR]

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the a-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, some elementary and useful terminologies associated to Lebedev–Skalskaya transform (LS-transform) are notified and then estimates for R κ 1 2 + i τ (x) , F κ 1 2 + i τ (x) , translation and convolution operators associated to LS-transform are obtained. The continuity of LS-transforms, translation and convolution operators on functions spaces S α , k , H α , k and G α , k are discussed. The continuity of pseudo-differential operators in terms of LS-transforms is discussed on functions spaces S α , k , H α , k , G α , k and Lebesgue space. At the end, an application of LS-transform is shown by solving a differential equation of convolution type. [ABSTRACT FROM AUTHOR]

Lutsky in 2008 and 2009, where they investigated the invertibility of parabolic pseudo-differential operators with rapidly growing symbols and obtained estimates of exponential behaviour of solutions of pseudo-differential operators with growing and discontinuous symbols. During several years which followed the defense of the PhD thesis there appeared Vladimir's publications presenting interesting results for quasi-elliptic pseudodifferential operators and on solvability of Cauchy and Goursat problems for parabolic pseudodifferential equations (1971-1979 and later on), on solvability difference-differential equations (1978-1985 and later on), on Fredholmness of elliptic pseudodifferential operators and associated boundary value problems on noncompact manifolds (papers of 1971-1979). In 1993 he defended his doctoral thesis "The method of limit operators in solvability of pseudo-differential equations and convolution equations". [Extracted from the article]

In this paper, we discuss a class of stochastic nonlinear partial differential equation of Burgers type driven by pseudo differential operator (Δ + Δ α) for α ∈ (0 , 2) , and fractional Brownian sheet. The existence and uniqueness of an Lp-valued (local) solution is established for the initial valued problem to the equation. [ABSTRACT FROM AUTHOR]

We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner. [ABSTRACT FROM AUTHOR]

In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator on some ultradifferentiable function spaces. [ABSTRACT FROM AUTHOR]

In this paper, some basic results and notations related to Lebedev–Skalskaya transforms (LS-transforms) are introduced and then estimates of translation and convolution operators associated to LS-transform are obtained. Continuity of LS-transforms on Lebesgue space as well as on function spaces S β , k and G α , k are discussed. Pseudo-differential operators in terms of LS-transforms are defined and their integral representations have been studied in various ways. Moreover, their estimates in Lebesgue space are obtained. At the end, some applications have been discussed. [ABSTRACT FROM AUTHOR]

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase. [ABSTRACT FROM AUTHOR]

This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations. [ABSTRACT FROM AUTHOR]

This paper surveys the recent theory of traces on ideals of Hilbert space operators. The main purpose is to summarize a new approach, which was developed by the author in the last decade. We sincerely hope that our simple and elegant methods will be helpful for many people working in non-commutative geometry, pseudo-differential operators, and mathematical physics, where traces have become useful tools. Historical comments and open problems should serve to complete the picture. [ABSTRACT FROM AUTHOR]

If M is a compact, contact manifold, with contact line bundle Σ, L a "sublaplacian" on M, characteristic on Σ ∖ 0 , and if { φ j } is an orthonormal basis of eigenfunctions of –L, with eigenvalues λ j ↗ + ∞ , then { φ j } concentrates microlocally on Σ ∖ 0. We study a microlocal Weyl formula, giving the limit as N → ∞ of N − 1 ∑ j = 1 N (A φ j , φ j) for a class of pseudodifferential operators A on M. Such a limit for A ∈ O P S 0 (M) , the algebra of classical pseudodifferential operators, was evaluated by Colin, Hillairet, and Trelat and used to study quantum ergodic theorems on compact 3 D contact manifolds. Here we take A to belong to a different pseudodifferential operator algebra, O P Ψ ˜ 0 (M) , tuned to the contact structure of M. This produces a Weyl formula exhibiting richer behavior than one sees for A ∈ O P S 0 (M). [ABSTRACT FROM AUTHOR]

We give the normality, self-adjointness and spectral invariance of pseudo-differential operators on finite abelian groups. We also give a formula for the determinant of every element in a group of pseudo-differential operators on a finite abelian group. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum of the operator (0.1) H : = (− Δ) M / 2 + V , M > 0 , on L 2 (R d / Γ) , with Γ a maximal dimension lattice in R d and V a pseudodifferential operator of order strictly smaller than M. We prove that most of its eigenvalues admit the asymptotic expansion (0.2) λ ξ = | ξ | M + Z (ξ) + O ( | ξ | − ∞) , where Z is a C ∞ (R d) function (symbol) and ξ ∈ Γ * (the dual lattice of Γ). [ABSTRACT FROM AUTHOR]

In this work we introduce and study a class of linear operators related to a finite measure space, for which its L 2 -Hilbert space is separable. These linear operators represent a generalization of pseudo-differential operators on S 1 , where S 1 is the unit circle with centre at the origin. We call these operators, S-operators or generalized pseudo-differential operators on S, where (S , B , m) is the corresponding finite measure space for which the L 2 -space is separable. We give some L 2 -boundedness and compactness results, and we also study the Hilbert–Schmidt property in connection with this class of linear operators. In the end we give some examples of finite measure spaces and corresponding S-operators for which we can apply our results. [ABSTRACT FROM AUTHOR]

In this work, we have introduced a pair of linear canonical Hankel transformations and investigated some of its properties on Zemanian-type spaces. Moreover two versions of pseudo-differential operator associated with canonical Hankel transformations are defined and discussed its integral representation. [ABSTRACT FROM AUTHOR]

In this paper we find a formula for the adjoint of bounded pseudo-differential operators on compact Lie groups, which yields a necessary and sufficient condition for self-adjointness of these operators. By a similar method we find necessary and sufficient conditions for bounded pseudo-differential operators on compact Lie groups to be normal or unitary and we give some examples of our results on . Then we show that similar results are true for some bounded pseudo-differential operators on . [ABSTRACT FROM AUTHOR]

The wavelet method based on the Meyer wavelet function and scaling function is a rather effective regularization method for solving some illposed problems. Recently, there are many works on this method limited to the a priori choice rule. The typical paper [H. Cheng and C.L. Fu, Wavelets and numerical pseudodifferential operator, Appl. Math. Model. 40 (2016), pp. 1776-1787] has systematically considered the a priori choice rule in the framework of the pseudodifferential operator (ψDO). In this paper, we will systematically consider the a posteriori choice rule for two kinds of ill-posed problems in the framework of the ψDO, and construct the convergence error estimates between the exact solution and its regularized approximation. [ABSTRACT FROM AUTHOR]

This paper focus on a simplified viscoelastic problem subjected to unilateral boundary conditions. The problem is reduced to a pseudodifferential linear complementarity problem using Fourier techniques. Under appropriate regularity assumptions on the data, the diagonal process is employed to prove the existence result to this pseudodifferential linear complementarity problem. [ABSTRACT FROM AUTHOR]

The paper treats boundary value problems for the fractional Laplacian (-∆)a,a > 0, and its generalizations to pseudodifferential operators (ψdo's) Pof order 2awith evensymbol, applied to functions on a smooth subset Ω of Rn. There are several meaningful localboundary conditions, such as the Dirichlet and Neumann conditions γa-1ku=ϕ,k= 0,1,where γa-1ku=ck∂kn(u/da-1)|∂Ω,d(x) = dist(x, ∂Ω). We show a new Green's formula(P u, v)Ω-(u, P ∗v)Ω= (s0γa-11u+Bγa-10u, γa-10v)∂Ω-(s0γa-10u, γa-11v)∂Ω,where Bis a first-order ψdo on ∂Ω.Moreover, we show in the elliptic case how the Poisson-like solution operator KDfor thenonhomogeneous Dirichlet problem is constructed from P+in the factorization P∼P-P+obtained in earlier work. The Dirichlet-to-Neumann operator SD N =γa-11KDis derivedfrom this as a first-order ψdo on ∂Ω, with an explicit formula for the principal symbol. Thisleads to a characterization of those operators Pfor which the Neumann problem is Fredholmsolvable. [ABSTRACT FROM AUTHOR]

We prove a criterion for a "magnetic" Weyl operator to be trace-class by extending a method developed by Cordes, Kato and Arsu. Using the Calderon-Vaillancourt type Theorem for magnetic Weyl operators and an interpolation argument we also give a criterion for a "magnetic" Weyl operator to be in a Schatten-von Neumann class with 1 < p < ∞. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to find out the estimates of convolution for zero-order Mehler–Fock transform with various approaches. Pseudo-differential operator in terms of zero-order Mehler–Fock transform is defined and obtained its another integral representation. Further, its estimate in Lebesgue space has been studied. At the end some applications of zero-order Mehler–Fock transform and of convolution are discussed. [ABSTRACT FROM PUBLISHER]

In this paper, we study necessary optimality conditions for local Pareto and weak Pareto solutions of multiobjective problems involving inequality and equality constraints in terms of convexificators. We develop the enhanced Karush–Kuhn–Tucker conditions and introduce the associated pseudonormality and quasinormality conditions. We also introduce several other new constraint qualifications which entirely depend on the feasible set. Then a connecting link between these constraint qualifications is presented. Moreover, we provide several examples that clarify the interrelations between the different results that we have established. [ABSTRACT FROM PUBLISHER]

We seek to extend work by Christianson–Hassell–Toth [3] on restrictions of Neumann data of Laplacian eigenfunctions to interior hypersurfaces to a general semiclassical setting. In the semiclassical regime the appropriate generalization is to study the restrictions of the functionv = ν(x,hD)uwhereν(x,hD) is the operator defined by quantizing the normal velocity observable. For the Laplacianwhereνis the normal to the hypersurface. We find thatprovideduis anquasimode of the semiclassical pseudodifferential operatorp(x,hD). This statement should be interpreted as a statement of non-concentration for the quantization of normal velocity. [ABSTRACT FROM AUTHOR]

We outline some simplified and more general method for constructing parametrices on higher singular spaces. We also outline basic ideas on operators on manifolds with conical or edge singularities. [ABSTRACT FROM AUTHOR]

In this article, an intelligent method to detect, locate and quantify structural damages is presented via an optimization model. To predict the damage location and severity, the theory of pseudo-residual force vector (RFV) is applied. The proposed method can identify damages based on only a few mode shapes of the structure that can be easily obtained by a dynamic test. The objective function is defined as a minimum difference between the numerical and experimental variables in the RFV and the gravitational search algorithm is employed as a meta-heuristic technique for solving this optimization problem. The efficiency of the proposed method is investigated through the numerical examples with different damage scenarios. In the examples, the experimental data were simulated numerically using a finite element model of the structure and as demonstrated, it is possible to identify the damages with a reasonable level of accuracy while considering noise effects. [ABSTRACT FROM PUBLISHER]

We prove that Hörmander’s global wave front set and Nakamura’s homogeneous wave front set of a tempered distribution coincide. In addition, we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura’s homogeneous wave front set. [ABSTRACT FROM PUBLISHER]

A pair of pseudo-differential operators (p.d.o.) involving the Kontorovich–Lebedev transform (KL-transform) as well as its adjoint is defined. Estimates for translation and convolution function associated with the Kontorovich–Lebedev transform are obtained. Continuity of KL-transform and p.d.o. onandare discussed. [ABSTRACT FROM AUTHOR]

We obtain a global version in theN-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator involving the Kontorovich–Lebedev transform is defined. The boundedness of pseudo differential operator in a certain Sobolev-type space is shown. Composition of pseudo-differential operatorsis defined in terms of the two symbols, and also its boundedness in certain Sobolev-type space is given. Some special cases are also discussed. Finally, the Kontorovich–Lebedev transform is used in the solution of some partial differential equations. [ABSTRACT FROM PUBLISHER]

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures. [ABSTRACT FROM AUTHOR]

LetMbe a compact manifold with or without boundary andH⊂Mbe a smooth, interior hypersurface. We study the restriction of Laplace eigenfunctions solvingtoH. In particular, we study the degeneration ofu|Has one microlocally approaches the glancing set by finding the optimal powers0, so thatremains uniformly bounded inL2(H) ash→0. Moreover, we show that this bound is saturated at everyh-dependent scale near glancing using examples on the disk and sphere. We give an application of our estimates to quantum ergodic restriction theorems. [ABSTRACT FROM AUTHOR]

In this article, a three-level resolution Vernier delay line time-to-digital converter (TDC) was proposed. The proposed TDC core was based on the pseudo-differential digital architecture that made it insensitive to nMOS and pMOS transistor mismatches. It also employed a Vernier delay line (VDL) in conjunction with an asynchronous read-out circuitry. The time interval resolution was equal to the difference of delay between buffers of upper and lower chains. Then, via the extra chain included in the lower delay line, resolution was controlled and power consumption was reduced. This method led to high resolution and low power consumption. The measurement results of TDC showed a resolution of 4.5 ps, 12-bit output dynamic range, and integral nonlinearity of 1.5 least significant bits. This TDC achieved the consumption of 68.43 µW from 1.1-V supply. [ABSTRACT FROM AUTHOR]

Two versions of pseudodifferential operators (pdo) involving fractional powers Hankel–Clifford integral transformations are defined. The composition of first and second fractional pdo is defined. We show that the pdo and composition of pdo are bounded in a certain Sobolev-type space associated with the fractional powers of Hankel–Clifford integral transformations. Some special cases are also discussed. [ABSTRACT FROM PUBLISHER]

We consider the scattering theory for a pair of operatorsH0andH = H0 + VonL2(M,m), whereMis a Riemannian manifold,H0is a multiplication operator onM, andVis a pseudodifferential operator of order − μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces. [ABSTRACT FROM AUTHOR]

A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators. [ABSTRACT FROM AUTHOR]

The article discusses the ability of out-of-sample forecasting test in safeguarding agaist data mining. Topics mentioned include the important role of Diebold-Mariano (DM) test in the annals of forecast evaluation, the difficulty to trick pseudo-out-of-sample procedures than popular full-sample procedures, and the results are more reliable than in-sample forecasting.