# Sheaf of smooth functions and restriction to a divisor

My question is targeted towards a very particular detail in my research that I am trying to understand. I will therefore break it down into some more general questions.

Let $$X$$ be a smooth variety, $$i:D\hookrightarrow X$$ a smooth divisor. Let $$E$$ be a holomorphic vector bundle, $$E|_D =i_*i^* E$$ and $$\mathcal{A}_X$$ the sheaf of smooth functions on $$X$$.

1. The tensor product $$E^{\infty}=E\otimes_{\mathcal{O}_X}\mathcal{A}_X$$ is the sheaf of smooth sections of $$E$$ as a vector bundle. How do I understand the sheaf $$E|_D\otimes_{\mathcal{O}_X}\mathcal{A}_X$$?

2. Let $$\rho: E\to E|_D$$ be the restriction of holomorphic sections. Consider $$\rho\otimes\text{id}:E^{\infty}\to E|_D\otimes_{\mathcal{O}_X}\mathcal{A}_X$$. Is there some way to understand how this map acts on the smooth sections of $$E$$?

Now the actual question that I would like to understand.

1. Let $$T$$ be the tubular neighborhood of $$D$$, $$\phi:E|_D\to E|_D$$ an automorphism of the vector bundle on $$D$$. Let $$\tilde{\phi}: E^\infty|_T\xrightarrow{\sim} E^\infty|_T$$ be a smooth extension of $$\phi$$ to $$T$$ (as $$T$$ is smoothly retractible to $$D$$). Does the following diagram commute?

Clearly, if $$T$$ is a holomorphic tubular neighborhood, then question 3 would have a positive answer. But existence of a such neighborhood is very restrictive. Therefore, I am hoping that understanding the first two questions, will help me better understand the last one. I appreciate any comments, corrections and suggestions.