I asked the following question here in our forum: (How to calculate this cohomology? )
Proposition (1): Let $\mathcal{F}$ be be coherent sheaf on $\mathbb{P}^{n}$ and let $E$ be a locally free sheaf on $\mathbb{P}^{n}$. If $\mathcal{F}$ is $m$-regular and $E$ is $\ell$-regular, then $\mathcal{F} \otimes E$ is (m + $\ell$)-regular.
Proposition (2): If $E$ is an $m$-regular locally free sheaf, then $\bigwedge^{p}E$ is $(pm)$-regular.
Proposition (3): Let $C \subset \mathbb{P}^{n}$ be an irreducible (but possibly singular) reduced curve of degree $d$. Assume that $C$ is non-degenerate, i.e, that it doesn't lie in any hyperplane. Then $C$ is $(d + 2 - n)$-regular.
Serre Duality : $H^{q}(X, E)^{'} \simeq H^{n-q}(X, E^{*} \otimes \omega_{X})$
Vector bundle isomorphism : $\Omega_{\mathbb{P}^{n}}^{p *} \simeq \Omega_{\mathbb{P}^{n}}^{n-p} \otimes \Omega_{\mathbb{P}^{n}}^{n *}$
Using the above tools together with a projection formula, I think I answered the question. I also think it can be generalized.
Using the answer to this question (mathoverflow.net/questions/66980/) and using the above mentioned tools once again, I get: $$H^{1}(\mathbb{P}^{3}, \Omega_{\mathbb{P}^{3}}^{2} \otimes \mathcal{O}_{\mathbb{P}^{3}}(a) \otimes I_{C}^{\otimes 2})$$
We have:
I) $\Omega_{\mathbb{P}^{3}}^{1}$ is $2$-regular (Bott's Formula). Then, by proposition (2), $\Omega_{\mathbb{P}^{3}}^{2}$ is $4$-regular.
II) $\mathcal{O}_{\mathbb{P}^{3}}(a)$ is $(-a)$-regular
III) By proposition (3), assuming the non-degenerate curve, we have : $I_{C}$ is $(\text{deg}(C) + 2 - 3) = (\text{deg}(C) - 1)$-regular.
My doubt is: Can I use proposition (1) to conclude that: $(\Omega_{\mathbb{P}^{3}}^{2} \otimes \mathcal{O}_{\mathbb{P}^{3}}(a) \otimes I_{C}^{\otimes 2})$ is $(4 + (-a) + 2(\text{deg}(C) - 1))$-regular?
References are:
[1] Robert Lazarsfeld. Positivity in Algebraic Geometry I.
[2] Christian Okonek. Vector Bundles on Complex Projective Space.
Thanks in advance.