Criteria to prove that a map is a tempered distribution

Question

There is any simple sufficient condition to determine if a function is a tempered distribution? For example, given the map : $$ F \phi = \int_\epsilon^\infty \! \frac {\phi(x)}{\sqrt{x}} \, \mathrm{d}x. $$ I can prove that this is a tempered distribution with consideration about topology (for every epsilon this distribution is associated with an $L^2$ function, ecc..), but how can I do it using the definition?