Multidimensional Integral Operators Research Articles

On the Bergmann Kernel Function in Hyperholomorphic Analysis

The hyperholomorphic Bergmann kernel function *B for a domain R is introduced as the special quaternionic "derivative" of the Green function for R. It is shown that *B is hyperholomorphic, Hermitian symmetric and reproduces hyperholornorphic functions. We obtain an integral representation of *B as a sum of two integrals. One of them gives a smooth function, and the other describes the behaviour of +B near a boundary. To investigate the hyperholomorphic Bergmann function for some fixed class of hyperholornorphic functions we have to use not only the properties of just this class but also those of some other classes. The second fact is completely unpredictable from the complex analysis point of view. The connection between the hyperholomorphic Bergmann projector (the integral operator with the kernel $B) and some classical multidimensional singular integral operators is established. I

Some Multidimensional Fractional Integral Operators Involving a General Class of Polynomials

In this paper the authors present a systematic investigation of two novel families of multidimensional fractional integral operators involving a general class of polynomials with essentially arbitrary coefficients. The various theorems established here involve compositions, inversion formulas, and multidimensional Mellin transforms, and convolutions of these general fractional integral operators. Each of the results obtained in this paper would unify and extend the corresponding (known or new) result for simpler families of fractional integral operators which were studied in several earlier works on the subject.

Fractional integral operators involving a general class of polynomials

In the present paper the authors derive a number of interesting expressions for the composition of certain multidimensional fractional integral operators involving a general class of polynomials with essentially arbitrary coefficients. It is shown how these fractional integral operators can be identified with elements of the algebra of functions having the multidimensional Mellin convolution as the product. Inversion formulas for the multidimensional fractional integrals are also established. The fractional integral operators studied here are fairly general in character, since (by suitably specializing the coefficients involved) the general class of polynomials can be reduced to each of the classical orthogonal polynomials, the Bessel polynomials, and numerous other classes of generalized hypergeometric polynomials studied in the literature.

Nonclassical multidimensional singular integral operators on R + n+1

Banach algebras of certain bounded operators acting on the half-spaceL p (R + +1 ,x 0 ) (1<p<∞, −1<α<p−1) are defined which contain for example Wiener-Hopf operators, defined by multidimensional singular convolution integral operators, as well as certain singular integral operators with fixed singularities. Moreover the symbol may be a positive homogeneous function only piecewise continuous on the unit sphere. Actually these multidimensional singular integral operators may be not Calderon-Zygmund operators but are built up by those in lower dimensions. This paper is a continuation of a joint paper of the author together with R.V. Duduchava [10]. The purpose is to investigate invertibility or Fredholm properties of these operators, while the continuity is given by definition. This is done in [10] forp=2 and −1<α<1, and in the present paper forL p (R + +1 ,x 0 α ) with 1<p<∞ and −1<α<p−1.

ON AN ALGEBRA CONNECTED WITH TOEPLITZ OPERATORS IN RADIAL TUBE DOMAINS

This article is a study of an algebra acting in and obtained by extending the classical algebra of multidimensional singular integral operators with the help of the orthogonal projection , where is the characteristic function of some cone in , and and are the direct and inverse Fourier transformations, respectively. Bibliography: 29 titles.

On the smoothness of the symbol of multidimensional singular integral operators

Пустьf — характерист ика, аΦ — соответству ющий символ n-мерного сингу лярного интегрального опера тора, которые являютс я однородными функция ми нулевой степени вRn,n≧2, с нулевыми средни ми значениями на един ичной сфереSn−1 пространств аRn. В статье установлива ются связи между хара ктеристикойf и символомϕ в термин ах наилучших равномерн ых приближений полин омами по сферическим гармони кам и сферических равномерных модулей непрерывности, котор ые позволяют установить связь так же в терминах пространст вWrHw(Sn−1). Доказывается, ч то установленные резул ьтаты в терминах пространствWrHw(Sn−1) неу лучшаемые.

Homotopic classification of noetherian systems of multidimensional bisingular integral operators

Homotopic classification of noetherian systems of multidimensional bisingular integral operators

ASYMPTOTIC BEHAVIOR OF THE SPECTRUM OF WEAKLY POLAR INTEGRAL OPERATORS

The principal term is calculated for an asymptotic expression for the spectrum of multidimensional symmetric integral operators of potential type.