On the convolution product of the distributional kernel K α , β , γ , ν

We introduce a distributional kernel K α , β , γ , ν which is related to the operator ⊕ k iterated k times and defined by ⊕ k = [ ( ∑ r = 1 p ∂ 2 / ∂ x r 2 ) 4 − ( ∑ j = p + 1 p + q ∂ 2 / ∂ x j 2 ) 4 ] k , where p + q = n is the dimension of the space ℝ n of the n -dimensional Euclidean space, x = ( x 1 , x 2 , … , x n ) ∈ ℝ n , k is a nonnegative integer, and α , β , γ , and ν are complex parameters. It is found that the existence of the convolution K α , β , γ , ν ∗ K α ′ , β ′ , γ ′ , ν ′ is depending on the conditions of p and q .