CHAPTER II - SIMPLEST PROPERTIES OF MULTIDIMENSIONAL SINGULAR INTEGRALS

Abstract

This chapter discusses the properties of multidimensional singular integrals. The singular integral, v (x) = ∫Emf(x, θ)/rmu(y) dy, characteristic f(x, θ) is a function of the point x and Q of which the first moves through the space Em or one of its regions and the other over a unit sphere. This is equivalent to the point, θ describing the whole of the space Em except the origin and infinitely distant points and the characteristic keeping a constant value when the point x is fixed and point θ moves along a ray passing through the origin. The chapter presents a theorem by which a singular integral satisfies a Lipschitz condition with exponent α in each of the parts Γ’j; and accordingly throughout the manifold Γ. Another theorem is presented that can be derived from this result; the theorem says if the kernel is a singular integral, taken over a closed Liapounov manifold, and is continuously differentiable for ξ ≠ η and satisfies inequality and the density of this integral satisfies in Γ a Lipschitz condition with exponent α < 1, then the singular integral satisfies a Lipschitz condition with the same exponent throughout the manifold Γ.