It got a tiny bit colder today, mostly because of the rain.

I was originally planning on going out running (first time, like, ever; I decided that it would be a good way to work up my endurance). The thunderstorm made me cancel that plan. My other plan of going food-shopping also went down the drain.

After "waking up" (i.e., a shower and some miscellaneous reading to rev'up the brain), I spent the afternoon working, still, on this paper by Markus Keel and Terence Tao on endpoint Strichartz estimates. The non-endpoint portion was fairly easy, once I remembered stuff like Hardy-Littlewood-Sobolev or Young inequality, and once I looked up bilinear interpolation methods--and so I decided, since there are quite a bit of details (which would be useful for beginning grad students like I am) omitted, I would type up a set of notes, "annotating" the Keel-Tao paper.

Typing/rewriting proofs actually help my understanding a lot. It turns out that quite often I don't actually understand what I thought I understood. Rewriting helps me pick out those spots and make me work at figuring out exactly why it is so. It is a nice way to learn for me. So today I starting rewriting the proof for the endpoint estimate. I got stuck on a triviality (in hindsight) for 2 hours or so, mostly because I still are not familiar enough with harmonic analysis to realize that localization of interaction can usually be generalized to an almost-orthogonality condition, which, in discrete cases, becomes very useful. Case in point:

F, G are functions in L²(R). T is a real-valued bilinear operator defined on L² functions that has only near-interactions, i.e., if the supports of F and G doesn't overlap, then T(F,G) = 0. Suppose, further, that we know T is bounded when the supports of F and G are each contained in an interval of length 1. Show that T is bounded from (L²)² to R.

Without the near-interaction condition, the statement is not true. With the near-interaction, we can say

Write F and G as sums over functions defined on intervals. In particular F can be represented as a sum over F_n, with the support for F_n contained in the interval [n,n+1). Similarly for G. The near-interaction then becomes the orthogonality condition T(F_n,G_m) = 0 whenever n ≠ m. Then, instead of summing T(F_n,G_m), we only need to sum T(F_n,G_n) ≤ |F_n|₂|G_n|₂. Since F and G are L² functions, the sum can be taken via Cauchy-Schwarz to be less than or equal to l² sums of the component-norms, which sums to the L² norm of the function! Without the orthogonality condition, we can not simplify the sum to a form analogous to the l² inner-product between two sequences, and so we must sum in l¹, whose boundedness, in our case, is not guaranteed.

(Yes, I know it is confusing. Once I finish rewriting the paper, the latex'ed version would be much clearer.)

In any case, I should get some dinner.

Also, the weather forecast says it would be 18 degrees warmer tomorrow, shooting into the 70s, before dropping back down to the 50s on Tuesday. Weird.