  It is known that local zeta functions associated with real analytic functions
can be analytically continued as meromorphic functions to the whole complex
plane. But, in the case of general ($C^{\infty}$) smooth functions, the
meromorphic extension problem is not obvious. Indeed, it has been recently
shown that there exist specific smooth functions whose local zeta functions
have singularities different from poles. In order to understand the situation
of the meromorphic extension in the smooth case, we investigate a simple but
essentially important case, in which the respective function is expressed as
$u(x,y)x^a y^b +$ flat function, where $u(0,0)\neq 0$ and $a,b$ are nonnegative
integers. After classifying flat functions into four types, we precisely
investigate the meromorphic extension of local zeta functions in each cases.
Our results show new interesting phenomena in one of these cases. Actually,
when $a<b$, local zeta functions can be meromorphically extended to the
half-plane ${\rm Re}(s)>-1/a$ and their poles on the half-plane are contained
in the set $\{-k/b:k\in\mathbb{N}$ with $k<b/a\}$.
