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

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...

We deal with several classes of integral transformations of the form f(x)↦D∫R+2(e−vcosh⁡(x−u)±e−vcosh⁡(x+u))f(u)h(v)dudv, where D is a special differential operator. In case D is the identical operator, we prove a Young type theorem for a generalized convolution operator related to the Fourier and the Kontorovich–Lebedev integral transforms. In case D is a certain finite order differential operator, we study the unitary property of these transforms and apply them to solve a class of integro-differential equations. Several classes of system of integral equations are also considered.

We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.

We consider a second-order differential operator on an interval of the real line with integral boundary conditions. We show how to construct the adjoint operator. The differential operation of the adjoint operator can be loaded, and the domain of that operator can contain functions that, together with their derivatives, have jump discontinuities at countably many points. For the root functions of the adjoint operator, we obtain integral representations, in particular, a mean-value formula.

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 operators PaQb is 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.

In this article we present uniqueness, existence, and L p -estimates of the quasilinear stochastic partial differential equations driven by Levy processes of the type (0.1) d u = ( L u + F ( u ) ) d t + G k ( u ) d Z t k , where L is a pseudo-differential operator and Z k are independent Levy processes ( k = 1 , 2 , ⋯ ) . The operator L is random and may depend also on time and space variables. In particular, our results include an L p -theory of 2 m -order SPDEs with coefficients measurable in ( ω , t ) and continuous in x .

A couple of fractional powers of Hankel-type integral transformations are discussed on certain Zemainan type spaces. The operational formulae are developed. Two variants of fractional pseudo-differential operators (FrPDO) associated with the symbol a(x, y) are defined. Integral representation of FrPDO are obtained. The fractional powers of Hankel-type integral transformations are used in the solution of some partial differential equations.

In this paper, we study the boundedness results for a large class of pseudo-differential operators with smooth symbols on weighted Morrey and Weighted fractional Sobolev–Morrey spaces, respectively.

We give the determinants and upper bounds for the condition numbers of matrices of pseudo-differential operators $$T_\sigma $$ on $${\mathbb Z}_N$$ with separable symbols $$\sigma $$ on $${\mathbb Z}_N\times {\mathbb Z}_N.$$ For a time series z of size N, we give the computational complexity of $$T_\sigma z$$ . Numerical experiments are also given.

Characterizations of product of generalized pseudo-differential operators associated with symbol σ(x,ξ) ∈ Sm are discussed by exploiting the fractional Fourier transform.

International audience; We extend the results of our paper "Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0"written with Laure Saint-Raymond to the case of forced linear wave equations in any dimension. We prove that, in dimension 2,if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefrontsets of the limit Schwartz distribution using radial propagation estimates.

This article devoted to study periodic M-hypoelliptic pseudo-differential operators on torus $$\mathbb {T}^{n}= (\frac{\mathbb {R}}{2\pi \mathbb {Z}})^{n}.$$ More precisely, we consider toroidal M-hypoelliptic symbols and we construct the minimal and maximal extension on $$L^{p}(\mathbb {T}^{n}),$$ $$1 < p

This paper is the outcome of an attempt to understand the connection between singular traces and the Wodzicki residues of pseudo-differential operators on closed Riemannian manifolds as presented in the recent monograph of Lord, Sukochev, and Zanin. Employing my technique of dyadic representations of operators, I am able to replace the mountain tour performed by Sukochev and his coauthors through a walk in a park. The crucial point is that considerations about eigenvalues are no longer involved. To simplify understanding, the new approach is demonstrated by the example of pseudo-differential operators on the d-dimensional flat torus $${\mathbb{T}^d}$$ . In this special case it is possible to work with global symbols (which need not be smooth).

In this paper we find a formula for the adjoint of bounded pseudo-differential operators on $$L^2(\mathbb {S}^{1})$$ and also a formula for the adjoint of some bounded pseudo-differential operators on $$L^2(\mathbb {Z})$$ , which yealds a necessary and sufficient condition for self-adjointness of these operators.

The main purpose of this paper is to introduce new generalized differential operator Aμm, λ(α,β)f(z) defined in the open unit disc U = {z ∈ : |z| < 1}. We then, using this operator to introduce novel subclass Ωm*(δ,λ,α,β,b) by using the operator Aμm, λ(α,β)f(z). Then, we discuss coefficient estimates growth and distortion theorems, closure theorems and integral operator.

The aim of this work is to describe new interesting examples of non-smooth manifolds and elliptic pseudo-differential operators acting in functional spaces on such manifolds. Fredholm properties for these operators are studied by factorization methods, and these are based on several complex variables.

We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-adjoint differential operator which commute with the Fourier operator truncated on the considered finite symmetric interval.

The mapping properties of pseudo-differential operators associated with symbol \(\theta (z,\xi ), z= x+iy\) and \(\xi =u+it\) on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space are investigated. Motivated from Wong (An introduction to pseudo-differential operators. World Scientific, Singapore, 2014), \(L^{p}(\mathbb R^{n})\)-boundedness of pseudo-differential operators on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space is studied by using the fractional Fourier transform.

In the present paper we introduce sufficient conditions for subordination and superordination involving the operator DR λ m , n $DR_\lambda ^{m,n} $ and also we obtain sandwich-type results.

In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the pseudo-differential operator is obtained.

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$., Comment: 34

Substantive foundations are offered for solution of differential equations with fixed elements of coefficient system and continuous measurable functions of the right part.

In this paper, under the definition of fractional powers by Straub, we will give the representation of fractional powers of coercive differential operators by using pseudo differential operators.

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 \(\Delta _{2,\nu ,\mu ,\alpha }^{*}\).

Continuity of the first Hankel–Clifford transformation on the spaces of the type $$H_\mu $$ are investigated. Pseudo-differential operator $$h_{1,\mu ,a}$$ associated with Bessel type operator $$xD_{x}^{2}+(1-\mu )D_{x}$$ involving the symbol $$a(x,y)$$ whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain ultradifferentiable function spaces. It is shown that the operator $$h_{1,\mu ,a}$$ is a continuous linear mapping of one ultradifferentiable function spaces into another spaces of same type.

In this report we derive moments of several discrete distributions via a differential operator. The first part of the report is devoted to the moments of truncated classical distributions. In the second part we present the moments for inflated and deformed power series distributions. Finally we use a differential operator to derive moments of two modified negative binomial distributions.

We give large deviation estimates for a convolution semigroup, which is not Markovian and of Levy type, of big order.

Estimates in various Lebesgue spaces \(L_s(G)\), \(1\le s\le \infty \), are obtained for the root functions of an operator which relates to the differential operation \(-u''+p(x)u'+q(x)u\), \(x\in G=(a,b)\), with complex-valued singular coefficients. Among these estimates there are also the so-called anti-a priori estimates that link the root functions in the same chain. It is supposed that p(x) and q(x) belong locally to the spaces \(L_2\) and \(W_2^{-1}\), respectively, may have singularities at the end-points of G, and \(q(x)=q_1(x)+Q'(x)\) while \(Q(x), p(x), Q^2(x)w(x), p^2(x)w(x), q_1(x)w(x)\) are integrable on the whole interval G with \(w(x)=(x-a)(b-x)\).

We develop the local inequalities with new weights for the potential operator applied to differential forms. We also prove the global weighted norm inequalities for the potential operator in averaging domains and explore applications of our new results.

The Null Space Pursuit (NSP) algorithm based on the differential operator is an important method of signal denoising and signal separation. In this paper, we propose an arbitrary order differential operator and use it in a complex signal which is a sum of simple signal. By solving an optimization problem, we also estimate the parameters of the differential operator. Finally, we confirmed the practicability of the algorithm through experimental simulation.

In this work we establish the metric approximation property for Besov spaces defined on arbitrary compact Lie groups. As a consequence of this fact, we investigate trace formulae for nuclear Fourier multipliers on Besov spaces. Finally, we study the r-nuclearity, the Grothendieck-Lidskii formula and the (nuclear) trace of pseudo-differential operators in generalized H\"ormander classes acting on periodic Besov spaces. We will restrict our attention to pseudo-differential operators with symbols of limited regularity.

The pseudo-differential operator (p.d.o.) associated with the Jacobi differential operator in terms of a symbol is defined. It is shown that the product of two symbols is a symbol. Integral representations of p.d.o. are obtained. It is shown that the product of p.d.o.’s is bounded in a Sobolev type space associated with the Fourier Jacobi transform.

A brief introduction to the fractional Hankel transform and its basic properties is given.The fractional Hankel transform of tempered distributions is studied. A pseudo-differential operator involving fractional Hankel transform is introduced and some of its properties including boundedness are investigated in \(G_{\mu }^{s,2}(I)\).

We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients.

" PDF File" DOI : http://dx.doi.org/10.22342/jims.13.1.72.27-38

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator D in the numerator of the fraction has no effect on input functions (i.e., the derivative operation is removed) because we take the fraction as a whole part in the partial fraction expansion. The method in various variants is widely implemented in related fields in mechanics and engineering. We also point out that the same mistakes in the differential operator method are found in the related references [1-4].

A linear differential/integral operator is associated to a rational matrix in a natural way, and the behavior is defined to be the kernel of this operator. Conditions under which two rational matrices have the same behavior are derived.

We construct local ℒ-splines with uniform knots that preserve subsets from the kernel of a linear differential operator ℒ of order r with constant real coefficients and pairwise distinct roots of the characteristic polynomial. Pointwise estimates are found for the error of approximation by the constructed ℒ-splines on classes of functions defined by differential operators of orders smaller than r.