$(M,\omega)$ is a compact Kaehler manifold and $f\_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V\_s$. My question is whether the function $\frac{f\_{t,s}^\* \omega^n}{\omega^n}$ is a bounded independent of $t$ and $s$. If not, can we control the $L^p$($p>1$) norm of $\frac{f\_{t,s}^\* \omega^n}{\omega^n}$ uniformly with respect to $t$ and $s$?



Volume form under holomorphic automorphisms
There is no such bound. As a counterexample take $(M,\omega)=(\mathbb{CP}^1, \omega\_{FS})$. Identify $\mathbb{CP}^1$ with $\mathbb{C}\cup \infty$; in such co-ordinates
$$\omega\_{FS}=\frac{i dz\wedge d\overline z}{(1+|z|^2)^2}.$$
The radial vector field $V=z\partial/\partial z$ on $\mathbb{C}$ extends smoothly to a holo vector field on $\mathbb{CP}^1$, and the pullbacks of $\omega\_{FS}$ by the family of diffeomorphisms generated by $V$ are
$$\omega\_{\lambda}=\frac{i dz\wedge d\overline z}{(\lambda^{-1}+\lambda|z|^2)^2},$$
where $\lambda$ varies in $\mathbb{R}^+$. The ratio
$$
\frac{\omega\_\lambda}{\omega\_1}=\frac{(1+|z|^2)^2}{(\lambda^{-1}+\lambda|z|^2)^2}
$$
is clearly unbounded. The $L^p$ norm of this quantity is also unbounded ($p>1$), since for $\lambda$ large,
\begin{align\*}
\int\_\mathbb{C}\left[\frac{\omega\_\lambda}{\omega\_1}\right]^{2p}\omega\_1
&=\int\_\mathbb{C}\frac{(1+|z|^2)^{2p-2}}{(\lambda^{-1}+\lambda|z|^2)^{2p}}i dz\wedge d\overline z\\
&\geq\int\_{|z|<\lambda^{-1}}\frac{(1+|z|^2)^{2p-2}}{(\lambda^{-1}+\lambda|z|^2)^{2p}}i dz\wedge d\overline z\\
&\geq\int\_{|z|<\lambda^{-1}}\frac{i dz\wedge d\overline z}{(2\lambda^{-1})^{2p}}\\
&=C\lambda^{2p-2}.
\end{align\*}
One remark: I don't understand why there are two parameters $t$ and $s$ in your notation -- usually the notation for a 1-parameter subgroup features only one parameter! Did I misunderstand your question?


