Distributions in spaces with thick points

We present a theory of distributions in a space with a thick point in dimensions n ≥ 2 , generalizing the theory of thick distributions in one variable given in Estrada and Fulling (2007) [8]. The higher dimensional situation is quite different from the one dimensional case. struct topological vector spaces of thick test functions and, by duality, spaces of thick distributions. We study several operations on these distributions, both algebraic and analytic, particularly partial differentiation. We introduce the notion of thick delta functions at the special point, not only of order 0 but of any integral order. We also consider the thick distributions constructed by the Hadamard finite part procedure. We give formulas for the derivatives of important thick distributions, including the finite part of power functions. We obtain the new formula ∂ ∗ 2 P f ( r − 1 ) ∂ x i ∂ x j = ( 3 x i x j − δ i j r 2 ) P f ( r − 5 ) + 4 π ( δ i j − 4 n i n j ) δ ∗ for the second order thick derivatives of the finite part of r − 1 in R 3 , where δ ∗ is a thick delta of order 0.