Fefferman’s ball multiplier counterexample
In the previous post we saw the connection between the ball multiplier and spherical convergence of Fourier transforms. Recall that the operator is defined in dimensions by the relation
where denotes the -dimensional unit ball. The focus of this post will be to prove the following result
Theorem 1 (Fefferman, 1971) The operator is not bounded on if and .
We will restrict our attention to the -dimensional case. The general case can be obtained by a straightforward modification of these arguments, or by appealing to a theorem of de Leeuw which states that the boundedness of on implies the boundedness of on for . Before turning our attention to the operator we must make one more detour. We will need the following geometric result of Besicovitch.
Theorem 2For there exists a collection of rectangles with the following properties: (1) Each is a rectangle of dimensions oriented such that the long axis of the rectangle makes an angle of with the -axis. (2) The collection of rectangles obtained by translating each units in the direction are disjoint, and (3) .
This theorem is a key tool in Besicovtich’s construction of a compact subset of of measure , which contains a unit line segment in every direction. Such sets are known as Besicovitch (or Kakeya) sets. Theorem 2 can be thought of as a discrete analog of a Besicovitch set. The proof of this theorem, which is based on elementary geometry, can be found in the references listed below.
We now turn our attention to the study of the operator . Let us first try to understand how the operator behaves when applied to specific functions. Arguably the simplest -dimensional function is the characteristic function of a rectangle. The Fourier transform, however, interacts more cleanly with Schwartz functions than discontinuous functions, so we will smooth out the characteristic function of an interval.
Let be a non-negative bump function supported on and equal to on the sub-interval . Moreover, we can take to be nearly supported on the interval . Of course, can’t be made to be completely supported on (by the uncertainty principle) but we can construct it to be rapidly decreasing away from this interval. For we can define which is a non-negative function supported on the rectangle and equal to on the sub-square . The function should be thought of as being localized on the rectangle . We will use the phrase localized on informally to mean supported or nearly supported on. We then have that is localized on the interval . The guiding principle here is that if is localized on some rectangle then one can assume that in most estimates. We now define a function, , localized to the rectangle (this choice for the rectangle is connected to the statement of Theorem 2, which we will eventually apply) by
Moreover, we see that
and hence is localized on the rectangle . So what does the disc multiplier do to the function ? For large values of and we see that is localized on a small rectangle near , which of course is contained inside the unit ball. Thus we expect that . This doesn’t seem too helpful in showing that is an unbounded operator. We’ll need to modify a bit. Consider the modulation of (at frequency ) defined by
Since we see that (moreover, ). The Fourier transform of a modulation is a translation, so is localized to the interval . Thus if either or is large then the interval is far from the unit ball and we expect that . Again, this doesn’t seem very helpful in showing that is unbounded.
In order to see some of the non-trivial behavior of let us localize the Fourier transform of to a rectangle that lies near the edge of the unit ball. To do this take . Now is localized on the rectangle . For large and we see that is a very small rectangle centered at . More specifically, we see that this rectangle has length , and a (even smaller) height of . Since is localized on we have that
Recall that . Thus . In fact, as we increase the size of the approximation tends to the truth (pointwise). In other words, if we define we are saying that for large (and ). This can be made precise without too much difficulty, but we’ll skip the calculation for the sake of clarity. (Heuristically, all we are saying is that, to a very small rectangle, the edge of the unit ball looks like a half-plane. One might compare this to the fact that the surface of the Earth looks like a plane at small scales.) In other words, for large , we can show that
where is the variant of the (-dimensional) Hilbert transform that we introduced in the previous post, applied in the variable . Specifically, we have that is defined (on ) by the relation . The operator is just the operator applied to a function on in the first co-ordinate. One then has that . This gives (1).
Rewriting Lemma 3 from the previous post with the interval (which we rename as to avoid confusion with the rectangle ) taken to be , gives
Lemma 3Consider the intervals and . Then there exists a positive constant such that
for all , where is independent of the choice of .
Putting this lemma together with (1) we see that , where is localized to the rectangle and is defined to be the rectangle . Moreover, using the fact that rotations commute with the Fourier transform, we can obtain a function with the properties described above localized to any rotation of the interval . More specifically, we claim the following theorem
Theorem 4Let be the rectangle rotated so that the longest axis is oriented in the direction , and let be the rectangle obtained by translating units in the direction . Then, for large and , there exists a function supported on such that but for all .
Let be a rectangle as described in the above theorem, and let be the bump function localized to this tile. The theorem then states that . In other words we can, very roughly, think of the operator as an operator that translates the rectangle to . We are now in a position to apply Theorem 2, which roughly states that there exists a family of disjoint rectangles with the property that their translates have a large amount of overlap. We now want to exploit the following heuristic, the quantity should be very small compared to the quantity since the function takes small values (either or ) on a set of large measure, while takes large values on a set of small measure (due to the overlap of the rectangles ). It turns out that this line of reasoning isn’t in itself enough to create a counterexample. However, we can amplify this approach by introducing some randomness via Khinchin’s inequality. We recall Khinchin’s inequality.
Lemma 5(Khinchin) Let be a collection of functions and a collection of random -signs. Then, for any , we have that
Instead of considering the function used in the heuristic reasoning above, we will consider . First we have that
Here the choice of signs was inconsequential. Next, using Khinchin’s inequality we see that
Thus, for some choice of signs we have that . Now let us estimate the right-hand side of this expression. Here we’ll use property (3) from Theorem 2 which states that we can take (for any ). This gives us that the function is supported on a set of measure at most . One now can easily see that is as small as possible when the support is as large as possible (that is ) and for all in the support of this function. Thus we have that
Putting this together with (2) we have that
If , taking arbitrarily close to zero shows that is not a bounded operator on . Notice that the argument breaks down when . This should be reassuring since is bounded in this case. We can, however, extend our counterexample to such that by duality. Let be the conjugate exponent to defined . If then and we have that
which shows that is unbounded on as well.
In preparing these notes, I relied on the following references: Davis and Changs’s Lectures on Bochner-Riesz Means, Fefferman’s paper The Multiplier Problem for the Ball, Stein’s Harmonic Analysis, and Tao’s Math 254B notes.