Distributions. Definition and Basic Properties

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Let X \subset \mathbb{R}^d be an open set. A linear form \nu \colon C_c^\infty(X) \to \mathbb{C} is called a distribution if, for every compact set K \subset X, there is a real number C \geq 0 and a non-negative integer N such that for all \phi \in C_c^\infty(X) with \text{supp } \phi \subset K,

\lvert \langle \nu, \phi \rangle \rvert \leq C \displaystyle{\sum_{\lvert \alpha \rvert \leq N} \sup \lvert \partial^\alpha \phi \rvert}

The vector space of distributions on X is denoted by \mathcal{D}'(X).

The support of a distribution \nu \in \mathcal{D}'(X), written \text{supp }\nu, is by definition the complement of the set \big\{ x \in X : \nu = 0 \text{ in a neighborhood of } x \big\}.

Sequencial continuity.

A linear form \nu on C_c^\infty(X) is a distribution if and only if \lim_n \langle \nu, \phi_n \rangle = 0 for every sequence \big\{ \phi_n \big\}_{n \in \mathbb{N}} which converges to zero in C_c^\infty(X).

Localization

Let X \subset \mathbb{R}^d be an open set, and let \big\{ X_\lambda \big\}_{\lambda \in \Lambda} be an open cover of X (where \Lambda is an index set).  Suppose that, for each \lambda \in \Lambda there is a distribution \nu_\lambda \in \mathcal{D}'(X_\lambda) and that

\nu_\lambda = \nu_\mu on X_\lambda \cap X_\mu if X_\lambda \cap X_\mu \neq \emptyset.

Then there is a unique distribution \nu \in \mathcal{D}'(X) such that \nu = \nu_\lambda in X_\lambda for each \lambda \in \Lambda.

Influence of supports

Let \nu \in \mathcal{D}'(X) and let \phi \in C_c^\infty(X). If the supports of \nu and \phi are disjoint, then \langle \nu, \phi \rangle = 0.

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