Nonclassical multidimensional singular integral operators on R + n+1

Abstract

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.