Nonclassical multidimensional singular integral operators on R + n+1

Banach algebras of certain bounded operators acting on the half-spaceL_p(R₊ⁿ⁺¹,x₀) (1<p<, –1<<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 Calderón-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<<1, and in the present paper forL_p(R₊ⁿ⁺¹,x₀) with 1<p< and –1<<p–1.