So this problem comes from the functional analysis course I took last semester. Although simple, I thought was pretty cool. In the actual exercise they had broken it down into three parts, which I did not show in the problem post. Here I will show the steps as claims.
Claim 1: The operator is a bijection
Proof: To show injectivity we show that this operator has trivial kernel. One inclusion is obvious, we just need to show that . To see this we recall from elliptic regularity theory for the Laplacian that if for some then
Now back to our question which can be written as , and so applying the estimate we can see that and so . Applying the estimate one more time shows that . For surjectivity we begin taking some and let be the unique solution to , and by the regularity theory . Then let be the unique solution . It follows that and . (Super cool right?!)
Claim 2: If then
Proof: Let , multiplying by and integrating we get
Since we can integrate by parts once with vanishing boundary terms on the left hand side (just replace with a sequence of compactly supported smooth functions, integrate by parts using the definition of weak derivative passing limits by dominated convergence) to get
Now, (since it is the solution to ) and so we can integrate by parts once more, again with vanishing boundary terms and we get
Now if we try to integrate by parts again we will run into a dead end, try it! Instead we play the same game with to get
Equating (1) and (2) gives the claim.
Now we are ready to finish the problem, which is to prove some regularity result on weak solutions to for ; that is . By surjectivity of there exists some such that . We just need to conclude now that . By claim 2 we have that
and since is bijective we have that this is equivalent to
Setting we get that .