On multidimensional integral operators with homogeneous kernels and oscillatory radial coefficients

Abstract

in a domain G containing the point x = 0; the functions ak(x) and b(x) are assumed to be bounded in G (for details, see [1]). By now, criteria for invertibility and the Fredholm property have been obtained for integral operators whose kernels are homogeneous of degree (−n) and invariant under the rotation group SO(n); the Banach algebras generated by these operators have been described, and conditions for the projection method to apply have been found (e.g., see [2–6]). In addition, operators whose external coefficients “stabilize” at zero and infinity [5] have been studied. In the present paper, we consider operators with radial oscillating coefficients of the form |x| , where δ ∈ R. These coefficients play the same role in the theory of integral operators with homogeneous kernels as coefficients of the form e do in the theory of convolution operators. In what follows, we obtain a criterion for the Fredholm property and compute the index of integral operators with kernels homogeneous of degree (−n) and with coefficients of the form |x|. We use the following notation: R is the n-dimensional Euclidean space;