I want to bound the moments of stochastic integrals as
$$E\left|\int_0^1 f(s)d L_s\right|^\alpha,\alpha\in[0,1],$$
where $(L_s)_{s\ge0}$ is a Lévy process with Gaussian part $\sigma^2$ and Lévy measure $\nu$ supported on $[-1,1]$, and $f:[0,1]\to\mathbb{R}$ is bounded and measurable. I want estimates as $$E|\int_0^1 f(s)d L_s|^\alpha\le C\int_0^1 |f(s)|^\delta ds,\delta>0.$$ The only thing I can derive is something like \begin{eqnarray*} E\left|\int_0^1 f(s)d L_s\right|^\alpha&\le& \left(E\left|\int_0^1 f(s)d L_s\right|^2\right)^{\alpha/2}\\ &\le& \left(E\left|\int_0^1 f(s)^2d [L_s,L_s]\right|\right)^{\alpha/2}, \end{eqnarray*} which leads to bounds depending on the kernel via $$\left(\int_0^1 f(s)^2 d s\right)^{\alpha/2}.$$ Any ideas? Using the compound Poisson representation I can derive estimates like this for a Lévy process with Lévy measure supported on $|x|>1$, and I think something like this should be also doable for the situation above.