product of distributions with a smooth function




Let nn \in \mathbb{N} and consider n\mathbb{R}^n the Cartesian space of dimension nn.


(product of a distribution with a smooth function)

For u𝒟( n)u \in \mathcal{D}'(\mathbb{R}^n) a distribution, and fC ( n)f \in C^\infty(\mathbb{R}^n) a smooth function, their product

fuC ( n) f \cdot u \;\in\; C^\infty(\mathbb{R}^n)

is the distribution given on a compactly supported smooth function gC cp ( n)=𝒟( n)g \in C^\infty_{cp}(\mathbb{R}^n) = \mathcal{D}(\mathbb{R}^n) by

(fu)(g)u(fg), (f\cdot u)(g) \;\coloneqq\; u\left( f \cdot g\right) \,,

where on the right we have the application of uu regarded as a continuous linear functional u:𝒟( n)u \colon \mathcal{D}(\mathbb{R}^n) \to \mathbb{C} to the ordinary pointwise product of smooth functions fgf \cdot g.



(product of a distribution with a non-singular distributions is product of distribution with a smooth function)

The wave front set of a non-singular distribution u fu_f corresponding to a smooth function fC ( n)f \in C^\infty(\mathbb{R}^n), is empty (this prop.). Therefore the product of distributions (def. ) of a non-singular distribution with any distribution uu is defined, and given by the product of distributions with smooth functions according to def. :

u fu=fu=u(f()). u_f \cdot u = f \cdot u = u(f\cdot (-)) \,.


See also

Created on November 7, 2017 at 11:26:25. See the history of this page for a list of all contributions to it.