This article is cited in 13 papers

Multidimensional Integral Operators with Homogeneous Kernels of Compact Type and Multiplicatively Weakly Oscillating Coefficients

V. M. Deundyak

Southern Federal University

Abstract: In the space $L_p(\mathbb R^n)$, $1<p<+\infty$, we consider a new class of integral operators with kernels homogeneous of degree $-n$, which includes the class of operators with homogeneous $SO(n)$-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.

Keywords: multidimensional integral operator, operators with multiplicatively weakly oscillating coefficients, homogeneous kernel, convolution operator, the space $L_p(\mathbb R^n)$.

UDC: 517.9

Received: 15.05.2009

DOI: 10.4213/mzm8717

 English version: Mathematical Notes, 2010, 87:5, 672–686

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