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A study on the vanishing moments of wavelets on p-adic fields is carried out in this paper. The p-vanishing moments and discrete p-vanishing moments are defined on a p-adic field and the relation between them is investigated. The p-vanishing moments of Haar-type and non-Haar type wavelet functions are computed. Also, the connection between p-vanishing moment of non-Haar type wavelet functions and the approximation order of the indicator function of the compact open subgroup B 0 (0) of Q p are established. A characterization of the p-vanishing moments of nonorthogonal wavelets is done. [ABSTRACT FROM AUTHOR]

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups. [ABSTRACT FROM AUTHOR]

This paper is devoted in investigations concerning the study of the coupled potential operator J s α , β and corresponding L p -Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space S α , β is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The L p norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator J s α , β is defined as a pseudo-differential operator related with a precise symbol. The operator J s α , β is extended to a space of distributions. An L p -Sobolev boundedness result for the operator J s α , β is shown. The spaces H p m , α , β and H p m , α , β introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces. [ABSTRACT FROM AUTHOR]

A study on the vanishing moments of wavelets on p-adic fields is carried out in this paper. The p-vanishing moments and discrete p-vanishing moments are defined on a p-adic field and the relation between them is investigated. The p-vanishing moments of Haar-type and non-Haar type wavelet functions are computed. Also, the connection between p-vanishing moment of non-Haar type wavelet functions and the approximation order of the indicator function of the compact open subgroup B 0 (0) of Q p are established. A characterization of the p-vanishing moments of nonorthogonal wavelets is done. [ABSTRACT FROM AUTHOR]

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo‐differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo‐differential operators are derived. It is shown that pseudo‐differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

The novel fractional Stockwell transform is extended to the suitably defined Schwartz space and the generalized fractional Stockwell transform is introduced. The pseudo-differential operator associated with the fractional Stockwell transform is defined and it is proved to be continuous on the Schwartz space. Examples are illustrated for the generalized fractional Stockwell transform. The generalized convection-diffusion equation is solved using the fractional Stockwell transform. [ABSTRACT FROM AUTHOR]

In this paper, a pair of quadratic-phase Hankel transformations are discussed on certain Zemanian-type spaces. Some estimates of quadratic-phase Hankel (QPH) translation and convolution are obtained. Moreover, two versions of pseudo-differential operators (p.d.o.) associated with quadratic-phase Hankel transformations are defined on some spaces. Furthermore, we have discussed the integral representation of the p.d.o. Using the quadratic-phase Hankel convolution it is shown that p.d.o. satisfy some norm inequalities. Finally, these Hankel-type transformations are utilized to solve some partial differential equations. [ABSTRACT FROM AUTHOR]

Regularization is a method for providing a stable approximate solution to ill-posed operator equations, and it involves the regularization parameter which plays an important role in the convergence of the method. In this article, we propose a class of a posteriori parameter choice rules for filter-based regularization methods and establish the optimal rate of convergence O (δ ν ν + 1 ) from the proposed rules. We study these methods along with the proposed parameter choice rules in the context of pseudo-differential operator equations as well as the analytic continuation problem. The numerical implementation of the pseudo-differential operator equation and analytic continuation problem is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain Fourier multiplier theorem, the approximation characterization and embedding for B ω u type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability. Then, we characterize these spaces via Peetre’s maximal functions, the Lusin area function, and the Littlewood–Paley g λ ∗ -function. Finally, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel–Lizorkin-type spaces B ˙ p , q s , τ (W) and F ˙ p , q s , τ (W) . In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón–Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel–Lizorkin spaces B ˙ p , q s (W) and F ˙ p , q s (W) . In particular, even when reverting to the boundedness of Calderón–Zygmund operators on unweighted Triebel–Lizorkin spaces F ˙ p , q s , these results are still better. [ABSTRACT FROM AUTHOR]

In this note, we deal with the fractional logarithmic Schrödinger operator (I + (- Δ) s) log and the corresponding energy spaces for variational study. The fractional (relativistic) logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, log (1 + | ξ | 2 s) , s > 0 . We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of R N. We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid SBP-SAT/pseudo-spectral method is proposed for solving the time-dependent Wigner equation. High-order summation-by-parts (SBP) operators are utilized to discretize the Wigner equation spatially, where the inflow boundary conditions are weakly imposed by adding simultaneous approximation terms (SATs) to the semi-discretized Wigner equation. The pseudo-differential term, governing the quantum effect, is discretized in a pseudo-spectral manner with spectral accuracy. L 2 -stabilities of both the semi-discretized (excluding time) and fully discretized systems are thoroughly discussed, with the inclusion of an arbitrary-stage explicit Runge–Kutta scheme for time integration. Numerical experiments are conducted, including simulations of a harmonic oscillator, a Gaussian wave packet, and a typical RTD with its I–V characteristic curves. The numerical results demonstrate: (1) the accuracy order of the numerical scheme in discretizing the Wigner equation in phase space matches the theoretical value; (2) observation of typical quantum effects, including tunneling and negative resistance; and (3) rapid convergence of numerical solutions relative to the accuracy order of SBP operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

Pseudo-differential operators (PDO) Q (x , ℒ a , x) and Q (x , ℒ a , x) involving the index Whittaker transform are defined. Estimates for these operators in Hilbert space L2a(ℝ+;ma(x)dx) are obtained. A symbol class Ω is introduced. Later product and commutators for the PDO are investigated and their boundedness results are discussed. [ABSTRACT FROM AUTHOR]

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation. [ABSTRACT FROM AUTHOR]

Systems of fractional order differential and pseudo-differential equations are used in modeling of various dynamical processes. In the analysis of such models, including stability analysis, asymptotic behaviors, etc., it is useful to have a representation formulas for the solution. In this paper we prove the existence and uniqueness theorems and derive representation formulas for the solution of general systems of fractional multi-order linear pseudo-differential equations through the matrix-valued Mittag-Leffler function. Examples illustrating the obtained results are also provided. [ABSTRACT FROM AUTHOR]

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method. [ABSTRACT FROM AUTHOR]

Consider a pseudo-differential operator where the symbol a is in the rough Hörmander class with m ∈ ℝ and ρ ∈ [0, 1]. In this note, when 1 ≤ p ≤ 2, if m < n (ρ − 1) p and , then for any f ∈ S(ℝn) and x ∈ ℝn, we prove that where M is the Hardy-Littlewood maximal operator. Our theorem improves the known results and the bound on m is sharp, in the sense that n (ρ − 1) p can not be replaced by a larger constant. [ABSTRACT FROM AUTHOR]

The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic α -stable stochastic process with the power α ∈ (1 , 2 ] . This generator is the fractional Laplacian of the order α . Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order β is a positive number less than α . The jump theorem is known in the case of β = α - 1 . We present here the corresponding results in the cases of β < α - 1 and β > α - 1 . In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier. [ABSTRACT FROM AUTHOR]

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem 0.1 ∂ t u (t , x) = ψ (t , - i ∇) u (t , x) + f (t , x) , (t , x) ∈ (0 , T) × R d , u (0 , x) = 0 , x ∈ R d , in (Muckenhoupt) weighted L p -spaces with time-measurable pseudo-differential operators 0.2 ψ (t , - i ∇) u (t , x) : = F - 1 ψ (t , ·) F [ u ] (t , ·) (x). More precisely, we find sufficient conditions of the symbol ψ (t , ξ) (especially depending on the smoothness of the symbol with respect to ξ ) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted L p -spaces. Here the symbol ψ (t , ξ) is merely measurable with respect to t, and the sufficient smoothness of ψ (t , ξ) with respect to ξ is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to equation (0.1), 0.3 ∫ 0 T ∫ R d | (- Δ) γ / 2 u (t , x) | p (t 2 + | x | 2 ) α / 2 d x d t ≤ N ∫ 0 T ∫ R d | f (t , x) | p (t 2 + | x | 2 ) α / 2 d x d t and 0.4 ∫ 0 T ∫ R d | (- Δ) γ / 2 u (t , x) | p | x | α 2 d x q / p t α 1 d t ≤ N ∫ 0 T ∫ R d | f (t , x) | p | x | α 2 d x q / p t α 1 d t , where p , q ∈ (1 , ∞) , - d - 1 < α < (d + 1) (p - 1) , - 1 < α 1 < q - 1 , - d < α 2 < d (p - 1) , and γ is the order of the operator ψ (t , - i ∇) . [ABSTRACT FROM AUTHOR]

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

In this note, we consider a pseudo-differential operator T a defined as T a f (x) = ∫ R n ∫ R n e 2 π i (x - y) · ξ a (x , y , ξ) f (y) d ξ d y. If 0 ≤ ρ , δ < 1 and a (x , y , ξ) satisfies that sup y , ξ ∈ R n (1 + | ξ |) n (1 - ρ) + ρ N - δ M ‖ ∇ ξ N ∇ y M a (· , y , ξ) ‖ L ∞ (R n) < + ∞ for any N ≤ n + 1 and M ≤ 1 , then we show that the pseudo-differential operator T a is bounded from H 1 to L 1 . However, this result is not true if ρ = 1 or δ = 1 . [ABSTRACT FROM AUTHOR]

We establish the Fredholmness of a pseudo-differential operator whose symbol is of class C0,σ, 0 < σ < 1, in the spatial variable. Our work here refines the work of H. Abels, C. Pfeuffer (2020). [ABSTRACT FROM AUTHOR]

The Laplacian on the Heisenberg group as a sum of squares of the standard vector fields on the Heisenberg group is first made explicit to be a Laplace–Beltrami operator on the Heisenberg group, which is well known to be a Riemannian manifold. The dissipation of the heat generated by the Laplacian on the Heisenberg group is then given in terms of estimates reminiscent of the Hausdorff–Young inequalities for the Euclidean Fourier transform. The evolution of the quantum wave driven by the Laplacian on the Heisenberg group is given. Weyl transforms, which are pseudo-differential operators in disguise, are instrumental in deriving the explicit formula for the heat kernel of the Laplacian on the Heisenberg group. [ABSTRACT FROM AUTHOR]

We investigate the Gabor spherical mean transform and demonstrate the quantitative Shapiro dispersion uncertainty principle and the umbrella theorem specifically tailored for this transform. Next, we define the Toeplitz operators L S g 1 , g 2 in connection with two window functions g 1 and g 2 , as well as the symbol S . We establish the boundedness and compactness of these operators. Ultimately, we introduce the Schatten–von Neumann class S p , where p ∈ [ 1 , + ∞ ] , and show that the Toeplitz operators are members of the class S p . Additionally, we provide a proof for a trace formula. [ABSTRACT FROM AUTHOR]

In this note we study the pseudo-differential operator T a f (x) = ∫ R n e i x · ξ a (x , ξ) f ^ (ξ) d ξ. For 0 ≤ δ < 1 , it is well-known that T a is bounded on L 2 when the symbol a is in the Hörmander class S ρ , δ n (ρ - 1) 2 . But it is not bounded on L 2 or L ∞ in general when a ∈ S ρ , 1 n (ρ - 1) 2 . Here we give a special critical Hörmander class S ρ m (ln 6 L) such that S ρ , δ m ⫋ S ρ m (ln 6 L) ⫋ S ρ , 1 m for any 0 ≤ δ < 1 . We show that the pseudo-differential operator T a is bounded from L ∞ to BMO if 0 ≤ ρ < 1 and a ∈ S ρ n (ρ - 1) 2 (ln 6 L) . This result is an improvement of all known related theorems. [ABSTRACT FROM AUTHOR]

We study the boundedness properties of pseudo-differential operators associated with a symbol on rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies Hörmander-type conditions near infinity. We use a generalized transference principle by Coifmann-Weiss to establish a connection between the L p operator norm of the local part of the pseudo-differential operators on symmetric space with the Euclidean pseudo-differential operators. We also define a class of pseudo-differential operators with symbols having no regularity conditions in the space variable and explore their L p -boundedness properties. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Direct Discontinuous Galerkin (DDG) method for solving a time dependent partial differential equation with convection-diffusion terms and a nonlocal term which is a pseudo-differential operator of order α ∈ (1 , 2). This kind of equation was first introduced to describe morphodynamics of dunes and was then used for signal processing methods. We consider the DDG method which is based on the direct weak formulation of the PDE into the DG function space for both numerical solution and test functions. Suitable numerical fluxes for all operators are then introduced. We prove nonlinear stability estimates along with convergence results. Finally numerical experiments are given to illustrate qualitative behaviors of solutions and to confirm convergence results. [ABSTRACT FROM AUTHOR]

This paper introduces the quaternion Schwarz type space, and quaternion linear canonical transform (QLCT) mapping properties are also discussed. Further, the quaternion pseudo‐differential operator (QPDO) associated with QLCT is described. Some of its characteristics, including estimates, boundedness, and integral representation in quaternion Sobolev type space, are derived. Some applications of QLCT, quaternion differential equations, are also discussed. [ABSTRACT FROM AUTHOR]

Refinable functions play an important role in the construction and properties of wavelets. Basically, most of the wavelets are generated from refinable functions. In this paper, a study on the approximation properties of refinable functions on p -adic fields is carried out with necessary theoretical background. Various equivalent forms of approximation order and the connection between the approximation order and the Strang–Fix condition are derived. Finally a characterization for the approximation order of a refinable function is given in terms of order of the sum rules associated with the refinement mask. [ABSTRACT FROM AUTHOR]

In this article, we consider a harmonic analysis of locally compact groups and introduce a generalization of the classical cross-Wigner distribution defined on G x Ĝ by ... We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show that assume Λ and Γ are closed cocompact subgroups of G and Ĝ, respectively, then, for a given window φ ∈ L² (G), either both systems ..., and ... are Gabor systems in L2 (G), simultaneously, with the same upper bound, or neither ... nor ... comprises a Gabor system. Finally, pseudo-differential operators on locally compact groups are studied, we establish that assuming a pseudo-differential operator Aa corresponds to the symbol a ... then Aa is bounded operator Wτp,q (G) → Wτp,q (G). [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of linear operators defined on a separable Hilbert space H . More precisely, we define the generalized Fourier multipliers T σ A : H → H , where σ are symbols which belong to L p (Z) , 1 ≤ p ≤ ∞ and A : H → H is a bounded linear operator which satisfies some suitable conditions. Moreover, we also define function of generalized Fourier multipliers F T σ A : H → H where F : C → C is a continuous function. We give some boundedness and compactness results and we also study Schatten-von Neumann properties for these classes of linear operators mentioned above. Further, we study the existence and the regularity of the solutions of the linear equations F T σ A f + λ f = g , f , g ∈ H , where F : R → R + is a real-valued non-negative continuous function and λ is a real positive number. At last, an example of function of generalized Fourier multipliers in connection to Weyl transform is given. [ABSTRACT FROM AUTHOR]

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to generalize the Barut-Girardello transform (BG-transform) in two different ways as a pair of Barut-Girardello type transforms (BG-type transforms). Certain differential operators related to it are introduced and then continuity of the differential operators as well as both BG-type transforms on Zemanian type function spaces are discussed. Translation and convolution operators associated with BG-type transforms are introduced and obtained their estimates in Lebesgue space. Moreover, pseudo-differential operators involving BG-type transforms are defined and discussed their continuity on Zemanian type function spaces. At the end, some applications have been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [16] , which provides more elaborate and natural weighted estimates in the multilinear setting. [ABSTRACT FROM AUTHOR]

In this article, we explore the further study of filter based schemes along with parameter choice rules and then apply to pseudo-differential operator equations and the analytic continuation problem. As we know that the stability of the regularization schemes depends on the regularization parameter and here we derive an a posteriori parameter choice rule which is best among all other parameter choice rules; and this parameter choice rule minimizes an upper bound of the approximation error ‖ x α , δ − x † ‖. Numerical experiments are also provided to validate the proposed theory. • The parameter choice rule considered here is the best among all existing parameter choice rules. • We studied filter-based regularization schemes along with the parameter choice rule. • The parameter choice rule provides optimal rate of convergence for filter-based schemes. • We studied the parameter choice rule in the context of two practical examples. [ABSTRACT FROM AUTHOR]

Let T be a linear operator that, for some p1∈(1,∞), is bounded on Lp1(w~) for all w~∈Ap1(Rd) and in addition compact on Lp1(w1) for some w1∈Ap1(Rd). Then T is bounded and compact on Lp(w) for all p∈(1,∞) and all w∈Ap(Rd). This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the Lp scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers. [ABSTRACT FROM AUTHOR]

We provide general sets of functionals for which parabolic phase space Feynman path integrals on the torus T d ≡ (R / 2 π Z) d have a mathematically rigorous meaning. More exactly, for each functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of T d × Z d to some function of the ending point of position paths and the starting point of momentum paths. Each set of functionals is closed under addition, multiplication, translation, invertible integer linear transformation, and functional differentiation. As a result, we can create a large number of path integrable functionals. Though we must exercise caution when using phase space path integrals, several properties comparable to those of conventional integrals are applicable. [ABSTRACT FROM AUTHOR]

We prove that certain classes of pseudo-differential operators with symbols that are nonsmooth in the space variable are subscript 퐿 푝 -bounded on the 푚 -dimensional torus for 1 푝 . [ABSTRACT FROM AUTHOR]

This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

A 1D hyperbolic equation is derived for the Rayleigh wave induced by prescribed surface loading. The wave operator turns out to be independent of the vertical coordinate, which appears only in the right hand side of the equation as a parameter within the pseudo-differential operator acting on the given load. It is shown that in case of the classical 2D Lamb problem this operator causes smoothening of the surface concentrated impulse as the depth increases. The suggested formulation enables revealing of the peculiarities of surface elastic wave. [ABSTRACT FROM AUTHOR]

In this paper we introduce and investigate new 2-microlocal spaces associated with Besov type and Triebel–Lizorkin type spaces. We establish characterizations of these function spaces via the φ –transform, the atomic and molecular decomposition and the wavelet decomposition. As applications we consider boundedness of the Calder o ´ n–Zygmund operator and the pseudo–differential operator on the function spaces. [ABSTRACT FROM AUTHOR]

In this paper, the weighted boundedness for some multilinear operators generated by the pseudo-differential operators and the weighted Lipschitz functions are obtained.

This work presents a new methodology to model elastic lattice systems through a two-step approach that permits to reliably capture its dynamic behavior with a lower computational cost than modeling the lattice explicitly. The first step consists of a non-standard continualization accounting for scale effects. Several methods are explored to derive new continuum models, whose dispersive and vibrational behaviors are compared to that of the lattice, considered as a reference. Non-classical models with micro-inertia reveal high accuracy, not presenting physical inconsistencies. The second step follows a FEM spatial discretization of the developed continua, accounting for micro inertia terms in the mass matrix. Finally, the FEM formulation allows the use of element sizes larger (up to four times) than the physical length scale of the lattice system, thus significantly reducing the computational cost while maintaining accuracy and enabling a versatile application to materials, geometries and boundary conditions. The methodology is tested here for a 2D system with displacements in the plane, but can be extended to other lattice typologies as well. [Display omitted] • A novel approach for dynamic FEM in rectangular plate lattice systems is developed. • Different continualization method are applied persuading new micro-inertia models. • The continuum models are assessed by dynamic comparisons with the discrete one. • The occurrence of physical inconsistences in some new continuum models is assessed. • The FEM significantly reduces the computational cost and provides reliable results. [ABSTRACT FROM AUTHOR]