Paired Integral Operators with Homogeneous Kernels Perturbated by Operators of Multiplicative Shift

Avsyankin O. G. , Koval'chuk A. M.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 1.

Abstract:
In the space \(L_p(\mathbb{R}^n)\), where \(1\leqslant p\leqslant \infty\), we consider an operator \(B\), which is the sum of two terms. The first term is a paired multidimensional integral operator, whose kernels are homogeneous of degree \((-n)\) and invariant with respect to the rotation group of \(\mathbb{R}^n\)-space, and the second term is a series, convergent in the operator norm, composed of multidimensional multiplicative shift operators with complex coefficients. We impose some additional conditions on the kernels and coefficients of the operator \(B\), and these conditions ensure the boundedness of this operator in the space of summable functions. The main aim of the paper is to study the invertibility of the operator \(B\). To solve this problem we use a special method that allows the reduction of the multidimensional paired operator to an infinite sequence of one-dimensional paired operators \(B_m\), where \(m \in\mathbb{Z}_+\). It is shown that the operator \(B\) is invertible if and only if all the operators \(B_m\) are invertible, where \(m\) runs through all values from zero to some finite number \(m_0\). In turn, the operators \(B_m\) reduce to integral-difference convolution operators whose theory is well known. All this allowed us to determine the symbol of the operator \(B\). This symbol represents the pair of functions \((\beta_1(m,\xi), \beta_2(m,\xi))\), defined on the set \(\mathbb{Z}_+ \times \mathbb{R}\). If the symbol is non-degenerate, then we define in a natural way a real number \(\nu\) and integers \(\varkappa_m\), where \(m \in\mathbb{Z}_+\). Numbers \(\nu\) and \(\varkappa_m\) are called indices. The main result of the work is the invertibility criterion of the multidimensional paired operator \(B\) in the space \(L_p(\mathbb{R}^n)\). According to this criterion, the operator \(B\) is invertible if and only if its symbol is non-degenerate, and all its indices are zero.

Keywords: paired operator, integral operator, homogeneous kernel, multiplicative shift, invertibility, spherical harmonics

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