Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the argument that requires a slightly non-trivial fix. The issue does not really affect the subsequent literature, because the main result has since been reproven and extended by methods that avoid the gap (see in particular this subsequent paper of Tataru), but I have decided to describe the gap and its fix on this blog.
I will assume familiarity with the notation of my paper. In Section 10, some complicated spaces are constructed for each frequency scale , and then a further space is constructed for a given frequency envelope by the formula
where is the Littlewood-Paley projection of to frequency magnitudes . Then, given a spacetime slab , we define the restrictions
where the infimum is taken over all extensions of to the Minkowski spacetime ; similarly one defines
The gap in the paper is as follows: it was implicitly assumed that one could restrict (1) to the slab to obtain the equality
(This equality is implicitly used to establish the bound (36) in the paper.) Unfortunately, (1) only gives the lower bound, not the upper bound, and it is the upper bound which is needed here. The problem is that the extensions of that are optimal for computing are not necessarily the Littlewood-Paley projections of the extensions of that are optimal for computing .
To remedy the problem, one has to prove an upper bound of the form
the extension will then obey (5) (here we use Lemma 9 from my paper), but unfortunately is not guaranteed to obey (4) (the norm does control the norm, but a key point about frequency envelopes for the small energy regularity problem is that the coefficients , while bounded, are not necessarily summable).
This can be fixed as follows. For each we introduce a time cutoff supported on that equals on and obeys the usual derivative estimates in between (the time derivative of size for each ). Later we will prove the truncation estimate
Assuming this estimate, then if we set , then using Lemma 9 in my paper and (6), (7) (and the local stability of frequency envelopes) we have the required property (5). (There is a technical issue arising from the fact that is not necessarily Schwartz due to slow decay at temporal infinity, but by considering partial sums in the summation and taking limits we can check that is the strong limit of Schwartz functions, which suffices here; we omit the details for sake of exposition.) So the only issue is to establish (4), that is to say that
for all .
For this is immediate from (2). Now suppose that for some integer (the case when is treated similarly). Then we can split
The contribution of the term is acceptable by (6) and estimate (82) from my paper. The term sums to which is acceptable by (2). So it remains to control the norm of . By the triangle inequality and the fundamental theorem of calculus, we can bound
By hypothesis, . Using the first term in (79) of my paper and Bernstein’s inequality followed by (6) we have
and then we are done by summing the geometric series in .
It remains to prove the truncation estimate (7). This estimate is similar in spirit to the algebra estimates already in my paper, but unfortunately does not seem to follow immediately from these estimates as written, and so one has to repeat the somewhat lengthy decompositions and case checkings used to prove these estimates. We do this below the fold.
— 1. Proof of truncation estimate —
Firstly, by rescaling (and changing as necessary) we may assume that . By the triangle inequality and time translation invariance, it suffices to show an estimate of the form
where is a smooth time cutoff that equals on and is supported in , and all norms are understood to be on . We may normalise the right-hand side to be , thus is supported in frequencies , and by equation (79) of my paper one has the estimates
for all .
for all integers .
Fix . We can use Littlewood-Paley operators to split , where is supported on time frequencies and is supported on time frequencies . For the contribution of one can replace in (15) by (say) and the claim then follows from (14), the Leibniz rule, and Hölder’s inequality (again ignoring spatial derivatives). For the contribution of , we discard and observe that has an norm of (and its time derivative has a norm of ), so this contribution is then acceptable from (8) and Hölder’s inequality.
Finally we need to show (13). Similarly to before, we split . We also split , leaving us with the task of proving the four estimates
The time cutoff commutes with the spatial Fourier projection and can then be discarded by equation (66) of my paper. This term is thus acceptable thanks to (10).
Now we turn to (17). We can freely insert a factor of in front of . Applying estimate (75) from my paper, it then suffices to show that
From the Fourier support of the expression inside the norm, the left-hand side is bounded by
discarding the cutoff and using (9) we see that this contribution is acceptable.
so it suffices to show that
For (20) we note that has an norm of , while from (9) has an norm of , so the claim follows from Hölder’s inequality. For (21) we can similarly observe that has an norm of while from (8) we see that has an norm of , so the claim again follows from Hölder’s inequality. A similar argument gives (22) (with an additional gain of coming from the second derivative on ).
Finally, for (19), we observe from the Fourier separation between and that we may replace by (in fact one could do a much more drastic replacement if desired). The claim now follows from repeating the proof of (18).