The Schwartz-Sobolev Theory of Distributions

Let R n be a real n-dimensional space in which we have a Cartesian system of coordinates such that a point P is denoted by x = (x 1, x 2,…, x n ) and the distance r, of P from the origin, is r = |x | = (x 1 2 + x 2 2 + … + x n 2 )1/2. Let k be an n-tuple of nonnegative integers, k = (k 1, k 2,…, k n ), the so-called multiindex of order n; then we define 1 $$ \begin{gathered} \left| k \right| = k_1 + k_2 + \cdot \cdot \cdot k_n , x^k = x_1 ^{k_1 } x_2^{k_2 } \cdot \cdot \cdot x_n^{k_n } , \hfill \\ k! = k_1 !k_2 \cdot \cdot \cdot k_n !, \left( {\begin{array}{*{20}c} k \\ p \\ \end{array} } \right) = \frac{{k!}} {{k!\left( {k - p} \right)}} \hfill \\ and \hfill \\ D_k = \frac{{\partial ^{\left| k \right|} }} {{\partial x_1^{k_1 } \partial x_2^{k_2 } \cdot \cdot \cdot \partial x_n^{k_n } }} = \frac{{\partial ^{k_1 + k_2 + \cdot \cdot \cdot + k_n } }} {{\partial x_1^{k_1 } \partial x_2^{k_2 } \cdot \cdot \cdot \partial x_n^{k_n } }} = D_2^{k_2 } \cdot \cdot \cdot D_n^{k_n } , \hfill \\ \end{gathered} $$ where D j = ∂/∂x j , j = 1,2,…, n. For the one-dimensional case D k reduces to d/DK. Furthermore, if any component of k is zero, the differentiation with respect to the corresponding variable is omitted. For instance, in R 3, with k = (3, 0, 4), we have 2 $$ D^k = \partial ^7 /\partial x_1^3 \partial x_3^4 = D_1^3 D_3^4 . $$