Date: July 24th, 2013.
Notations are lifted from Terry Tao’s note, linked in his recent blog post.
We start with a diﬀerent proof to the 1D local smoothing (Corollary 2.2), which is arguably the shortest one (and doesn’t use Fourier transforms in time or space). Consider a solution to the 1D Schrödinger equation, and deﬁne its odd part: is a solution as well. A straightforward computation yields
which is a form of virial identity (MR2518079 does an equivalent derivation by multiplying the equation by ). Integrating over time and noticing that the space integral over may be seen as a duality bracket on (using ), one gets
Reverting to , using space translation invariance and conservation of the Sobolev norm, we get
If we add that is compactly supported in Fourier space, we can erase derivatives and get back to the statement of Corollary 2.2; moreover, if one plays with the explicit representation of the solution to deal with the time slices and send to , one can recover the identity from MR1101221, e.g. the sign in (6) is actually an equality (with the correct numerical constant).
While short and elegant, this argument does not, unlike the one from Terry’s note, bode well with generalizations (besides the obvious one to the dimensional case, see again MR2518079). The fundamental reason for this is that, while it does not use the Fourier transform, a rigid symetry is the key starting point (and some inspiration from boundary value problems: is a solution to the Schrödinger on the half-line with Dirichlet boundary condition).
However, this little computation was what led to the content of MR2518079, which we now connect to Terry’s note and its argument for bilinear estimates.
Recall we have two solutions to the linear Schrödinger equation (in , but the argument holds equally well in any dimension) and that we assume , , and with . By rotationnal invariance, one may freely set .
The bilinear virial argument from MR2518079 provides two interesting controled quantities (see Theorem 2.3 in the arxiv version. Note that this argument is in eﬀect performing a virial type computation on the tensor product of two solutions and , with a suitable weight ). One quantity is a sum of two positive terms and is similar (up to the choice of weight) to the one that appears in Morawetz interaction estimates, and led to a Radon transform bound which is irrelevant here (though it can be used directly to get the bilinear estimates as well). the other quantity (a rewriting of the previous sum of two positive terms, one of which was usually discarded) turns out to be the most interesting in our context. For the linear equation, we actually have an identity (the usual energy/mass bound for the RHS would be equally good, the identity itself isn’t necessary)
where and . This follows from the bilinear virial computation with choice of weight (this choice being consistent with how we set up the Fourier support of ).
Now, , and if we call , then the support of its Fourier transform is such that , and .
By these support considerations and Plancherel, we have
Then, we notice that , and by Sobolev embedding in the direction (which is really Bernstein inequality here), we have
as is constrained to a ball of radius by the previous support considerations.
Gathering estimates, we obtain (as )
The -dimensional version can be obtained by the same procedure (one loses in the Bernstein inequality for the transverse variables).
One should see that the above argument is using in essence the same line of reasoning as Terry’s note. In principle, it should generalize to variable coeﬃcients setting, and certainly one could recover the bilinear estimates from MR2970710 which were obtained by parametrices methods. We did circulate the argument in casual conversations, and inserted it as a remark in my Bourbaki seminar on the mass critical NLS, but we had no real use for it in the context of boundary value problems, where the direction of the Fourier support of a wave packet may change quite drastically due to reﬂexions.
Eventually, arXiv:1112.0710 came up with an argument that got rid of this ennoying directional requirement. There, one proves the bilinear estimate for the Schrödinger equation in 4D, outside a strictly convex obstacle and with Dirichlet boundary condition. Essentially, one replaces the weight by and later takes a trace over the variable after averaging over . Boundary terms are handled using the local smoothing eﬀect (the exterior of a convex obstable is obviously non trapping, a necessary and suﬃcient condition for local smoothing to hold) and lower order terms are handled using the usual Strichartz estimates (which are known to hold in that setting, see MR2672795).
We now present this derivation in a slightly diﬀerent manner, borrowed from arXiv:1210.4362. There, the bilinear estimates are derived in 3D, for any domain with Dirichlet boundary conditions (the size of the time interval on which they hold is then related to whether the domain is non trapping, in which case they hold as in the whole space, or the domain is trapping, in which case they hold at least on a time interval of size , e.g. they match the known bilinear estimates on a generic boundaryless compact manifold). In principle, this derivation can be adapted to the variable coeﬃcients case, and is very much local in space, while requiring only a spectral localization with respect to the Laplacian operator.
We perform the computation in to avoid clutter. Here, it is important to realize that we require only (e.g., is spectrally localized where ).
Consider the weight deﬁned as follows, where :
The bilinear virial computation yields
The right-hand side is easy, after time integration
We now consider the term in the inside (8) above, and restrict to , where for any , , to get
By Cauchy-Schwarz in ,
Here we use that is a solution, as well as the linear estimate (which we stress can be derived by Morawetz interaction estimates or the bilinear virial computation as well: no need to know the full set of Strichartz estimates here).
Going back to (8) and after time integration, restricting the space integration to , we are left with
Using an elementary version of the trace lemma, applied to as a function of , we have
As ,combining (10) for with (10) where is replaced by (which is also a solution to the Schrödinger equation) with the previous pointwise bound on , we get
By Fourier support considerations, we may then erase the gradient to obtain the desired conclusion,
This generalizes to any dimension, with two caveats: on domains, we need the right substitute for the linear bound we use, which requires a square-function estimate (known to hold) to translate a Sobolev bound on to a Besov bound on , for dimensions . In 2D, we do need the bound, which in turn is known to hold only for the exterior of convex obstacles, as a particular case of the Strichartz estimates.
Obviously these families of estimates will have applications on domains; arXiv:1112.0710 already provides one, to the critical problem outside aconvex obstacle, and arXiv:1210.4362 provides another one, to the global well-posedness for smooth data to the cubic NLS on bounded domains. But they are ﬂexible enough that they should ﬁnd other domains of applicability !
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Université Nice Sophia-Antipolis, LJAD, UMR CNRS 7351, F-06108 Nice Cedex 02, et Institut universitaire de France
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