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In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the n-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space Hq(Rn). [ABSTRACT FROM AUTHOR]

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev–Slobodecky spaces. [ABSTRACT FROM AUTHOR]

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As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse problem. The correct (vector) order can be found utilizing the available data. In this paper we offer an new method of solution of this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution U ^ (t , ξ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{U}(t, \xi)$$\end{document} evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of a system of governing pseudo-differential equations. [ABSTRACT FROM AUTHOR]

Abstract Let T a , φ $T_{a,\varphi }$ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions a ( x , ξ ) ∈ L ∞ S ρ n ( ρ − 1 ) / 2 ( ω ) $a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$ and φ ∈ L ∞ Φ 2 $\varphi \in L^{\infty }\varPhi ^{2}$ , the authors show that T a , φ $T_{a,\varphi }$ is bounded from L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ to L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ .

For a function ϕ satisfying some suitable growth conditions, consider the following general dispersive equation defined by { i ∂ t u + ϕ (− Δ) u = 0 , (x , t) ∈ ℝ n × ℝ , u (x , 0) = f (x) , f ∈ S (ℝ n) , where ϕ (− Δ) is a pseudo-differential operator with symbol ϕ(∣ξ∣). In the present paper, when the initial data f belongs to Sobolev space, we give the local and global weighted Lq estimate for the global maximal operator S ϕ ∗ ∗ defined by S ϕ ∗ ∗ f (x) = sup t ∈ ℝ | S t , ϕ f (x) | , where S t , ϕ f (x) = (2 π) − n ∫ ℝ n e i x ⋅ ξ + i t ϕ (| ξ |) f ^ (ξ) d ξ is a formal solution of the equation (*). [ABSTRACT FROM AUTHOR]

In this work we obtain sharp L p -estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on R n . The main result extends the classical Fefferman's sharp theorem on the L p -boundedness of pseudo-differential operators for Hörmander classes on R n to general graded Lie groups, also adding the borderline ρ = δ case. [ABSTRACT FROM AUTHOR]

In this paper we study large-time behavior evolution problems on the n-dimensional torus T n , n ≥ 1 . Here we analyze the solutions to these problems, studying their regularity and obtaining estimates of them. The main tools we use is the toroidal Fourier transform, together with Fourier series and a version of the Hardy-Littlewood inequality, applied to our case of the n-dimensional torus T n . We use this inequality to find an estimate of solutions to evolution problems. [ABSTRACT FROM AUTHOR]

This paper deals with the study of some mapping properties of fractional Hankel transform on some suitably constructed Gelfand‐Shilov spaces of type W. Further pseudo‐differential operator associated with fractional Hankel transform is defined and studied its bounded ness properties on some suitably designed Gelfand‐Shilov spaces of type W. [ABSTRACT FROM AUTHOR]

Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study properties of the linear canonical transform (LCT) on Schwartz-type space A (ℝ). The symbol class ρ , σ m , A is introduced. The pseudo-differential operator (p.d.o.) involving LCT is defined and also some of its properties including boundedness are investigated in Sobolev-type space. Kernel and integral representation of p.d.o. are obtained. Some applications of LCT to generalized partial differential equations and canonical convolution integral equation have been solved. [ABSTRACT FROM AUTHOR]

In this paper, we prove a reducibility result for a relativistic Schrödinger equation on torus with time quasi-periodic unbounded perturbations of order 1/2. As far as we known, this is the first reducibility result for the relativistic Schrödinger equation. [ABSTRACT FROM AUTHOR]

Let − (n + 1) < m ⩽ − (n + 1) (1 − ϱ) and let T a ∈ ℒ ϱ m , δ be pseudo-differential operators with symbols a(x, ξ) ∈ ℝn × ℝn, where 0 < ϱ ⩽ 1,0 ⩽ δ < 1 and δ ⩽ ϱ. Let μ, λ be weights in Muckenhoupt classes Ap, ν = (μλ−1)1/p for some 1 < p < ∞. We establish a two-weight inequality for commutators generated by pseudo-differential operators Ta with weighted BMO functions b ∈ BMOν, namely, the commutator [b, Ta] is bounded from Lp(μ) into Lp(λ). Furthermore, the range of m can be extended to the whole m ⩽ − (n + 1)(1 − ϱ). [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the onedimensional perturbation L = H+V, where H is the harmonic oscillator and V is a decreasing scalar potential. It is well known that the eigenvalues of L can be expressed as 2k + 1 + μk. The main result of the paper is to describe an asymptotic formula for the fluctuation {μk} when k tends to infinity. [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 article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process. [ABSTRACT FROM AUTHOR]

Characterizations of product of generalized pseudo-differential operators associated with symbol σ(x, ξ) ∊ Sm are discussed by exploiting the fractional Fourier transform. [ABSTRACT FROM AUTHOR]

In the context of manifolds of bounded geometry, we show that the properties of proper uniform pseudo-differential operators (PUPDOs) constructed by Kordyukov, Meladze, and Shubin carry over to PUPDOs whose local representations have symbols belonging to the (weighted) class M ρ , Λ m introduced by Garello and Morando. Under the M-ellipticity assumption, we show that the minimal and maximal extensions of such PUPDOs in the L p -spaces, where 1 < p < ∞ , coincide. [ABSTRACT FROM AUTHOR]

We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller–Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo-differential operators having scalar symbols. [ABSTRACT FROM AUTHOR]

We give a rigorous formulation of phase space path integrals on the torus T d = (R / 2 π Z) d for fundamental solutions of higher-order parabolic equations. Especially, by using the pseudo-differential operators on the torus, we prove that the time slicing approximation of the phase space path integral converges on compact subsets of T d × R d . [ABSTRACT FROM AUTHOR]

Let T a , φ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions a (x , ξ) ∈ L ∞ S ρ n (ρ − 1) / 2 (ω) and φ ∈ L ∞ Φ 2 , the authors show that T a , φ is bounded from L 2 (R n) to L 2 (R n) . [ABSTRACT FROM AUTHOR]

In this paper we study the perturbation L = H + V , where H = - d 2 m d x 2 m + x 2 m on R , m ∈ N ∗ and V is a decreasing scalar potential. Let λ k be the k th eigenvalue of H. We suppose that the eigenvalues of L around λ k can be written in the form λ k + μ k . The main result of the paper is an asymptotic formula for fluctuation { μ k } which is given by a transformation of V. In the case m = 1 we recover a result on the harmonic oscillator. [ABSTRACT FROM AUTHOR]

Let X be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of H X (R n) , the Hardy space associated with X, via the Littlewood–Paley g-functions and g λ ∗ -functions. Moreover, the authors obtain the boundedness of Calderón–Zygmund operators on H X (R n) . For the local Hardy-type space h X (R n) associated with X, the authors also obtain the boundedness of S 1 , 0 0 (R n) pseudo-differential operators on h X (R n) via first establishing the atomic characterization of h X (R n) . Furthermore, the characterizations of h X (R n) by means of local molecules and local Littlewood–Paley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz–Hardy space, the Lorentz–Hardy space, the Morrey–Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the g λ ∗ -function characterization obtained in this article improves the known results via widening the range of λ . [ABSTRACT FROM AUTHOR]

This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of R d . On this approximate eigenbasis, the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as O (n m 2) (initial) and O (m 3) (hyperparameter learning) with m basis functions and n data points. Furthermore, the basis functions are independent of the parameters of the covariance function, which allows for very fast hyperparameter learning. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. We also show that the convergence rate of the truncation error is independent of the input dimensionality provided that the differentiability order of the covariance function increases appropriately, and for the squared exponential covariance function it is always bounded by ∼ 1 / m regardless of the input dimensionality. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data. [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 article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calderón–Vaillancourt-type theorems for these operators. [ABSTRACT FROM AUTHOR]

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. For example we generalize the classical result by Cordero and Nicola that if 1 p ′ + 1 q ′ ⩽ 1 p 1 + 1 p 2 , 1 p ′ + 1 q ′ ⩽ 1 q 1 + 1 q 2 , p j , q j ⩽ q ′ , q ⩽ p and a ∈ M p , q , then the pseudo-differential operator Op (a) is continuous from M p 1 , q 1 to M p 2 ′ , q 2 ′ . We also show that the entropy functional E ϕ possess suitable continuity properties on a suitable Orlicz modulation space M Φ satisfying M p ⊆ M Φ ⊆ M 2 , though E ϕ is discontinuous on M 2 = L 2. [ABSTRACT FROM AUTHOR]

Abstract We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral.

In recent papers and books, a global quantization has been developed for unimodular groups of type I. It involves operator-valued symbols defined on the product between the group G and its unitary dual G ^ , composed of equivalence classes of irreducible representations. For compact or for graded Lie groups, this has already been developed into a powerful pseudo-differential calculus. In the present article we extend the formalism to arbitrary locally compact groups of type I, making use of the Fourier theory of non-unimodular second countable groups. The unitary dual and its Plancherel measure being quite abstract in general, we put into evidence situations in which concrete forms are available. Kirillov theory and parametrizations of large parts of G ^ allow rewriting the basic formulae in a manageable form. Some examples of completely solvable groups are worked out. [ABSTRACT FROM AUTHOR]

We define and study coherent states, a Berezin–Toeplitz quantization and covariant symbols on the product Ξ : = G × g ♯ between a connected simply connected nilpotent Lie group and the dual of its Lie algebra. The starting point is a Weyl system codifying the natural canonical commutation relations of the system. The formalism is meant to complement the quantization of the cotangent bundle T ♯ G ≅ G × g ♯ by pseudo-differential operators, to which it is connected in an explicit way. Some extensions are indicated, concerning τ -quantizations and variable magnetic fields. [ABSTRACT FROM AUTHOR]

For a connected simply connected nilpotent Lie group G with Lie algebra g and unitary dual G ^ one has (a) a global quantization of operator-valued symbols defined on G × G ^ , involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on G × g ∗ , taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits { Ω ξ ∣ ξ ∈ G ^ } . We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families of Weyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When G is also graded, we make a short presentation of the symbol classes S ρ , δ m , transferred from G × G ^ to G × g ∗ by means of the connection mentioned above. [ABSTRACT FROM AUTHOR]

We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent s ≤ 0 for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space ℝ 3 . [ABSTRACT FROM AUTHOR]

We define pseudo-differential operators on a locally compact, Hausdorff and abelian group G as natural extensions of pseudo-differential operators on R n . In particular, for pseudo-differential operators with symbols in L 2 (G × G ^) , where G ^ is the dual group of G, we give explicit formulas for the products and adjoints, characterize them as Hilbert–Schmidt operators on L 2 (G) and prove that they form a C ∗ -algebra, which is also a H ∗ -algebra. We give a characterization of trace class pseudo-differential operators in terms of symbols lying in a subspace of L 1 (G × G ^) ∩ L 2 (G × G ^) . [ABSTRACT FROM AUTHOR]

Let X = (Xt)t≥0 be a stochastic process which has a (not necessarily stationary) independent increment on a probability space (Ω, ℙ). In this paper, we study the following Cauchy problem related to the stochastic process X: (*) ∂u/∂t (t, x) = A(t)u(t, x) + f(t, x), u(0,·) = 0, (t, x) ∈ (0, T) × Rd, where f ∈ Lp((0, T); Lp(Rd)) = Lp((0, T);Lp) and A(t)u(t, x) = lim h↓0 E[u(t, x + Xt+h − Xt) − u(t, x)]/h. We provide a sufficient condition on X (see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in Lp([0, T];Hpᛰ), where Hpᛰ is a ᛰ-potential space on Rd (see Definition 2.9). Furthermore we show that for this solution, ||u||Lp([0,T ];Hpᛰ) ≤ N ||f||Lp([0,T ];Lp), where N is independent of u and f. [ABSTRACT FROM AUTHOR]

In this note we announce our investigation on the Lp properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators by showing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.