@PedroLauridsenRibeiro Thank you for your comment, I had assumed it held true on the basis of the Laplacian, but this seems to be much more interesting than I had thought.

I will do that, thank you for all your help!

I have no particular differential operator in mind. I came across the topic through the Laplacian, and thought it was neat how the inverse could map a function from $C^k$ to $C^{k+2}$. Thus for now I think I will start with the inverse of the Laplacian and elliptic operators in general, so something along the lines of the pseudo-differential operators you mentioned.

Thank you, this is exactly the kind of information I was looking for. Do you have any recommended textbooks or readings in this direction?

@JochenGlueck Thank you for your comment. This is the kind of information I am hoping to learn about, do you know of any textbooks that cover this (especially for pseudo-differential operators)?

@AlexandreEremenko Thank you, do you know which volume of the book or even which chapter/section goes over that?

Of course, sorry about not doing so earlier.

Thank you, I'll give it a try then.

That first book seems to be exactly what I need, thank you! If you have happened to have read through it before, how is it? Is it well suited for self-study and with only a minimal background in PDE?

Thank you for your suggestions. These all seem like wonderful books, but I am looking for something more on the PDE side. For example, existence, uniqueness, use of Sobolev spaces, weak solutions...etc.

@AbdelmalekAbdesselam Thank you, I will start with their paper.

@user1504 I will take your advice. I was able to get through Folland's book and develop somewhat of an intuition for the subject, but the books written by and for physicists have been a different story. Thanks again for all your help.