Exercise 1.6 from Stein and Shakarchi’s Fourier Analysis
The problem is as follows: Prove that if 
is a twice continuously differentiable function on 
which is a solution of the equation 
, then there exist constants 
and 
such that 
.
Localizations of a ring at maximal ideals correspond to stalks of a sheaf
We have previously observed that localizing a finitely generated algebra at a specific element yields a local ring which is exactly the set of regular functions on the distinguished open set 
, or in other words, the sections of the sheaf of regular functions on . That is to sayーwe have the concept of a (pre)sheaf 
, which ‘collects together’ regular functions that are defined on the open sets of a topological space 
, and we have shown that if we look at the functions that are assigned to special types of open sets (distinguished open sets ), they correspond to the localizations of a finitely generated algebra 
at specific elements of that algebra. (Notably, this means they have global representations as polynomial quotients over each .) However, what happens if we want to look at the behavior of these functionsーsections of the sheaf ーnear a specific point in the topological space?
What are Hilbert and Banach spaces?
It is dangerous to live without knowing what Hilbert spaces are. One might be spontaneously quizzed on one’s recollection, and it would be very embarrassing to not know the answer! Thankfully, you need only remember that Hilbert spaces are the objects which allow you to generalize linear algebra and apply it to analytic settings.
Localizations of a ring at specific elements correspond to sections of a sheaf
One fundamental observation in algebraic geometry is that there are a number of powerful correspondences between ring localizations and algebraic sheaves. Although elementary, this post will outline one such correspondence, which is that the localizations of a ring to a specific element can be exactly thought of as the sections of the sheaf of regular functions over the distinguished open set of . (Consider this post to be a self-directed refresher! It has been a long time since I learned this material.)