Given a function $b(t,x), x\in \mathbb{R}^d, t \in[0,T]$, we can define $$ b_n(t,x) = \frac{1}{1 + n^{-a} \left | b(t,x) \right | }b(t,x) $$ and we can see that $ \left | b_n(t,x) \right | \leq \min(n^a, \left | b(t,x) \right |) $ , meaning that $$ \left | b_n(t,x) \right | \leq n^a , \; \forall x\in \mathbb{R}^d, t \in[0,T] . $$
If we now define $$g(t,x) = b(t,x) + f(x)\; $$ and $$\;g_n(t,x) = b_n(t,x) + f(x), $$ where $f(x)$ is Lipschitz continuous $\forall x\in \mathbb{R}^d \;$, is there any way we can find a simillar bound for $g_n(t,x)$? I can't seem to find any way to bound it since $f(x)$ has at most linear growth ( from Lipschitz continuity ). The only thing I managed to come up with is $$ \left | g_n(t,x) \right | \leq n^a \left | b(t,x) \right | + n^a\left | f(x) \right | +\left | b(t,x) \right |\left| f(x) \right | $$ or $$ \left | g_n(t,x) \right | \leq n^a \left | b(t,x) \right |\left| f(x) \right| \left(\ \frac{1}{n^a}+\frac{1}{\left | b(t,x) \right |}+\frac{1}{\left| f(x) \right|} \right). $$ Note:
$b(t,x)$ is one-sided Lipschitz and has superlinear growth, $a \in (0,1/2]$.