Multidimensional Integral Operators Research Articles

C*-ALGEBRA GENERATED BY INTEGRAL OPERATORS WITH HOMOGENEOUS KERNELS AND OSCILLATING COEFFICIENTS OF VARIOUS TYPES

We consider the C*-algebra, generated by multidimensional integral operators with homogeneous degree \((-n)\) kernels and by the operators of multiplication by radial weak oscillating functions and by oscillating functions of the form \(|x|^{i\alpha }\). For this algebra, an operator symbolic calculus is constructed. In terms of this calculus, necessary and sufficient conditions for the Fredholm property of an operator are obtained.

Frames of solutions and discrete analysis of pseudo‐differential equations

We construct discrete analogs of multidimensional singular integral operators and study their invertibility. Moreover, we give a comparison between continual and discrete case. We give the theory of periodic Riemann problem also, because it is needed for studying invertibility of so‐called paired equations. For more general case of pseudo‐differential operators, we construct the solvability theory for discrete pseudo‐differential equations in discrete analogs of Sobolev–Slobodetskii spaces. Some comparison results for discrete and continuous solutions are given also in appropriate discrete normed spaces.

Об условиях нётеровости и индексе некоторых двумерных сингулярных интегральных операторов

The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space E_n was constructed by S.G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces L_p (D) (considered over the field of real numbers), 1<p<∞, and formulas for calculating indices are given. The method developed by R.V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.

ON POINTWISE APPROXIMATION OF FUNCTIONS OF SEVERAL VARIABLES BY SINGULAR INTEGRALS

We prove a theorem on weighted pointwise convergence of \ multidimensional integral operators with radial kernels to generating function of several variables, which are in general non-integrable in $n$-dimensional Euclidean space $E_{n}$ in the sense of Lebesgue$.$ Main result holds at almost every point of $E_{n}.$

ON THE NOETHER THEORY OF TWO-DIMENSIONAL SINGULAR OPERATORS AND APPLICATIONS TO BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF FOURTH-ORDER ELLIPTIC EQUATIONS

It is known that the general theory of multidimensional singular integral operators over the entire space Em was constructed by S. G. Mikhlin. It is shown that in the two-dimensional case, if the operator symbol does not turn into zero, then the Fredholm theory holds. As for operators over a bounded domain, in this case the boundary of the domain significantly affects the solvability of such operator equations. In this paper we consider two-dimensional singular operators with continuous coefficients over a bounded domain. Such operators are widely used in many problems of the theory of partial differential equations. In this regard, it would be interesting to find criteria of Noetherity of such operators as explicit conditions for its coefficients. Depending on the 2m+ 1 connected components, necessary and sufficient conditions of Noetherity for such operators are obtained and a formula for the evaluation of the index is given. The results are applied to the Dirichlet problem for general fourth-order elliptic systems.

Multidimensional singular integrals and integral equations in fractional spaces, II

ABSTRACT In this paper, boundedness, noetherity and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations in Besov spaces are studied.

Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces

The case of one-dimensional and multidimensional non-convolutional integral operators in Lebesgue spaces is considered in this paper. The convergence in the norm and almost everywhere of non-convolution integral operators in Lebesgue spaces was insufficiently studied. The kernels <img src=image/13420650_01.gif> of non-convolutional integral operators do not need to have a monotone majorant, so the well-known results on the convergence almost everywhere of convolutional averages are not applicable here. The kernels <img src=image/13420650_01.gif> of nonconvolutional integral operators take into account different behaviors at <img src=image/13420650_02.gif> and <img src=image/13420650_03.gif> depending on <img src=image/13420650_04.gif> (which is important in applications) and cover the situation in the particular case of convolutional and non-convolutional integral operators. We are interested in the behavior of function <img src=image/13420650_05.gif> as <img src=image/13420650_06.gif>. Theorems on convergence almost everywhere in the case of one-dimensional and multidimensional nonconvolution integral operators in Lebesgue spaces are proved. The theorems proved are more general ones (including for convolutional integral operators) and cover a wide class of kernels.

Multidimensional singular integrals and integral equations in fractional spaces I

The boundedness, invertibility and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations are investigated in Besov spaces.

Matrix representations of multidimensional integral and ergodic operators

We provide a representation of the C*-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. The method can be useful in various applications, since many discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: (1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, (2) we find a wave function and spectral estimates for 3D discrete Schrödinger equation with planar, guided, local potential defects, and point sources. The accuracy of approximation of continuous multi-kernel integral operators by the operators with piecewise constant kernels is also discussed.

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators belong to this algebra. For the piecewise constant kernels, we provide an explicit representation of the algebra as a direct product of simple matrix algebras. This representation allows us to compute the inverse operators and to find the spectrum explicitly. Moreover, explicit traces and determinants of such operators are also constructed. Generally speaking, the analysis of integral operators is reduced to the analysis of matrices.

Multidimensional butterfly factorization

This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size $N\times N$ with a product of $\O(\log N)$ sparse matrices, each of which contains $\O(N)$ nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in $\O(1)$ operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.

Convergence of multi-dimensional integral operators and applications

In this paper we investigate a general multi-dimensional integral operator \(V_{T}\). Under the condition that the kernel function of \(V_{T}\) is in a suitable Herz space, we get several convergence theorems about norm and almost everywhere convergence and convergence at Lebesgue points. The multi-dimensional convergence is investigated over cones and cone-like sets. As special cases we consider three multi-dimensional integral operators, the \(\theta \)-summation of Fourier transforms and Fourier series and the discrete wavelet transforms. The convergence results are formulated for functions from the Wiener amalgam spaces and variable Lebesgue spaces, too.

Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation

In this paper we present a novel numerical algorithm for the space-homogeneous multicomponent (multidimensional) Smoluchowski coagulation equation, the number of components is considered as dimensionality. The new methodology is based on the classical finite-difference predictor-corrector scheme. In a straightforward implementation of this scheme, however, one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid. We propose to use special low-parametric representations for the grid functions and as well for the coagulation kernel. The corresponding multidimensional arrays are approximated by low-rank tensor-train decompositions reducing them to combinations of small low-dimensional arrays, eventually to matrices for which we can use fast algorithms of linear algebra. Instead of O ( N 2 d ) operations in the classical scheme, we propose a new method that requires only O ( d 2 N log ? N ) operations, where N is the number of nodes per axis in the space grid and d is the number of components. In this work we accelerate the predictor-corrector time-scheme and use the trapezoidal rule for the computation of multidimensional integral operators. Thus, the accuracy of the new method is O ( h 2 + ? 2 ) , where h is the space grid step and ? is the time step.

C*-algebra of integral operators with homogeneous kernels and oscillating coefficients

We consider the C*-algebra generated by multidimensional integral operators with (−n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.

Some sharp inequalities for multidimensional integral operators with homogeneous kernel: an overview and new results

One goal of this paper is to point out the fact that a big number of inequalities provedfrom time to time in journal publications, both one-dimensional and multi-dimensional, are particularcases of ...

Multidimensional integral operators with homogeneous kernels and with coefficients oscillating at infinity

We consider multidimensional integral operators with homogeneous kernels and with oscillating coefficients of the form e iδ|x|. For such operators, we obtain a criterion for the Fredholm property and an index formula.

Calderón–Zygmund Type Singular Operators in Weighted Generalized Morrey Spaces

We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces \(\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),\) defined by a function \(\varphi (x,r)\) and radial type weight \(w(|x-x_0|), x_0\in {\mathbb {R}}^{n}.\) These conditions are given in terms of inclusion into \(\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),\) of a certain integral constructions defined by \(\varphi \) and w. In the case of \(\varphi =\varphi (r)\) we also provide easy to check sufficient conditions for that in terms of indices of \(\varphi \) and w.

On some Multidimensional Fractional Integral Operators Involving Multivariable I-Function

In the present paper, we study certain multidimensional fractional integral operators involving a general I-function in their kernel. We give five basic properties of these operators, and then establish two theorems and two corollaries, which are believed to be new. These basic theorems exhibit structural relationships between the multidimensional integral transforms. The one- and two-dimensional analogues of these results, which are new and of interest in themselves, can easily be deduced. Special cases of these latter theorems will give rise to certain known results obtained from time to time by several earlier authors.

Projection method for integral operators with homogeneous kernels perturbed by one-sided multiplicative shifts

We obtain the applicability criteria of the projection method to multidimensional integral operators with homogeneous kernels perturbed by operators of one-sided multiplicative shifts.

On an algebra of multidimensional integral operators with homogeneous-difference kernels

A Banach algebra generated by multidimensional integral operators with homogeneous-difference kernels is considered. A symbolic calculus is constructed for this algebra and, in terms of this calculus, necessary and sufficient conditions for the invertibility of an operator are obtained.