Multiplication operator

For a measurable scalar coefficient $a$ , the multiplication operator $M_a$ acts by $M_af=af$ on functions for which the product belongs to the chosen function space. On $L^p$ , a bounded coefficient defines a bounded operator with

\|M_af\|_p\le\|a\|_\infty\|f\|_p.

These are Lebesgue norms.

Domains

For an unbounded coefficient, one must specify the domain, such as $\{f\in L^p:af\in L^p\}$ . Multiplication of a distribution instead requires a smooth coefficient, or additional hypotheses that make the product meaningful.