L^{p} improving estimates for averages along curves

Terence Tao (UCLA), Jim Wright (UNSW/U. Edinburgh)

    www.math.ucla.edu/∼tao/preprints (in

preparation)

Abstract

Given any function f on an n-dimensional manifold M, and a family γ_{x} of curves in M smoothly parameterized by some n-dimensional set x of parameters, we can form the associated averaging operator

T f(x) := ∫_{γ_{x}} f.

We obtain nearly-sharp L^{p} improving results for these operators.

Introduction

• Let M, N be smooth n-dimensional manifolds. Suppose that we have a smooth assignment x ↦ γ_{x} mapping points in N to curves in M. We can then form the averaging operator

T f(x) := ∫_{γ_{x}} f.

This operator takes functions on M and returns functions on N. (There is the issue of what measure to put on γ_{x}, but this is usually irrelevant).

• To avoid degenerate situations (e.g. when γ_{t} is constant with respect to t) we assume that the dual curve assignment y ↦ γ_{y}^{∗} defined by

x ∈ γ_{y}^{∗} ⟺ y ∈ γ_{x}

is also a smooth assignment from points to curves.

• The oldest example of such an operator is the Radon transform R on R^{2}. This operator takes a function f on R^{2} and transforms it to a function Rf on the set Gr(2, 1) of lines in R^{2} via the formula

Rf(l) := ∫_{l} f.

In this case M = R^{2}, N = Gr(2, 1), and for each line l ∈ N, the curve γ_{l} is given by l itself, γ_{l} = l. (Dually, for every x ∈ R^{2}, the dual curve γ_{x}^{∗} is the one-dimensional manifold of all l ∈ N which contain x.

• Because of this, these operators are sometimes called generalized Radon transforms.

• The translation-invariant case is already interesting. These averaging operators live on R^{n} and are convolution operators of the form

T f := f ∗ dμ

where dμ is a smooth measure on a one-dimensional curve γ. (In this case, γ_{x} = x − γ, and γ_{y}^{∗} = y + γ). For instance, if γ is the unit circle in R^{2}, this circular average operator appears in the study of the 2D wave equation. A more mysterious operator occurs when γ is a portion of a helix in R^{3}.

Interesting questions:

• Invertibility of averaging operators: given T f, can you reconstruct f? More of an algebraic problem than an analytic one.

• Smoothing properties of averaging operators: It is easy to see that if f ∈ L^{p} (with compact support), then T f is also in L^{p}. But does the averaging give any more smoothing? Several ways to capture smoothing:

• L^{2} smoothing: Does T map L^{2} to L_{α}^{2}? This type of problem can generally be attacked by FIO techniques, and is quite effective as long as the curves do not degenerate in too nasty a way. (Singularity theory becomes an issue).

• L^{p} smoothing: Does T map L^{p} to L_{α}^{p}? This is much more difficult; for instance, for the helix, the optimal L^{p} smoothing result is comparable in difficulty to the local smoothing conjecture for the wave equation.

• L^{p} improving: Does T map L^{p} to L^{q} for some q > p? This question is easier than the smoothing question, because it is purely a geometric question (no oscillations or regularity involved).

• This talk is concerned only with the last question.

5

• The L^{p} improving question can be thought of as a continuous version of the following discrete problem:

• Take A curves of the form γ_{x}, and B points y in M. What bounds does one have on the total number of incidences (i.e. pairs (γ_{x}, y) such that y ∈ γ_{x}?).

• One can make the connection more precise by blurring out the points and curves by some small δ > 0. The above problem then becomes equivalent to the restricted weak-type version of the L^{p} improving question.

• It is now clear that the curvature properties of the γ_{x} will play an important role. For instance, if M = N = R^{n} and γ_{x} is the vertical line through x, there is no gain in L^{p} integrability coming from averaging.

• Note that if one considers maximal averaging operators, so that γ_{x} is allowed to depend on an additional parameter t, then these problems become very difficult, and are related to such problems as the Kakeya conjecture.

• Until recently, the progress on this problem (by Seeger, Greenleaf, Oberlin, and others) used the special properties of L^{p} exponents such as 2 and 4, and as such only obtained sharp results for dimensions n ≤ 4. These techniques had a very Fourier-analytic flavour, and are somewhat similar to the T T^{∗} methods used to prove Strichartz and restriction estimates.

• In 1998, Christ introduced a very different combinatorial approach (more similar in spirit to work on the Kakeya problem) to tackle this problem. In the model case of convolution with a generic polynomial curve in R^{n}, he obtained the sharp (L^{p}, L^{q}) estimates up to endpoints.

If you really must know: L^{1} maps to L^{1}, L^{(n+1)/2} almost maps to L^{n(n+1)/2(n−1)}, and one can use duality and interpolation to get a nearly sharp range of estimates.

• We’ve managed to generalize Christ’s technique to arbitrary smooth families of curves, and in all cases we obtain sharp results except for endpoints.

Push-forwards and pull-backs

• In order to state the results cleanly, we first need to lift the problem to n + 1 dimensions.

• Currently we have two n-dimensional manifolds M, N, and a relationship x ↦ γ_{x} taking points in N to curves in M. From this data we can construct an n + 1-dimensional manifold Σ ⊂ M × N defined by

Σ := {(y, x) ∈ M × N : y ∈ γ_{x}}.

The manifold Σ has the obvious projections π_{M} : Σ → M and π_{N} : Σ → N down to M and N. Our assumptions on x ↦ γ_{x} and y ↦ γ_{y}^{∗} imply that these projections are submersions (the derivative map always has full rank).

• A function on M can be pulled back to Σ by the pullback map π_{M}^{∗}, and conversely a function on Σ can be pushed forward to M by the push-forward operator π_{M,∗}, providing we define some smooth measures on M and Σ.

• The averaging operator T can then be factorized as

T = π_{N,∗}π_{M}^{∗}.

Vector fields

• Now we remove M and N and work exclusively in Σ.

• It is easy to check that the (L^{p}, L^{q}) mapping properties of π_{N,∗}π_{M}^{∗} are the same as those of π_{N}^{∗}π_{N,∗}π_{M}^{∗}π_{M,∗} - which is an operator from functions on Σ to functions on Σ.

• The fibers of π_{M} are one-dimensional curves in Σ. One can then create a non-vanishing vector field X_{1} which is a section of the fiber bundle. (This vector field is only determined up to multiplication by scalar fields, but that is not important for our purposes). The operator π_{M}^{∗}π_{M,∗} can then be written (essentially) as

π_{M}^{∗}π_{M,∗}f(x) = ∫ e^{t_{1}X_{1}}f(x) dt_{1}.

In other words, it is an averaging operator along the vector field X_{1}.

• The fibers of π_{N} similarly generate a vector field X_{2}. The question is then: what is the (L^{p}(Σ), L^{q}(Σ))? properties of the compound averaging operator

Af := ∫ ∫ e^{t_{2}X_{2}}e^{t_{1}X_{1}}f dt_{1}dt_{2}?

• This problem can easily be localized to a small neighbourhood of a point in Σ, call it 0. It is then clear that the Lie algebra generated by X_{1} and X_{2} at 0 will play a crucial role. Indeed, if the Lie algebra at 0 fails to span the entire tangent space, then it is easy to see that there are no L^{p} improving estimates.

Two-parameter Carnot-Caratheodory balls

• On our manifold M there are only two structures, the vector fields X_{1} and X_{2}. These generate a very natural two-parameter metric structure.

• Define the two-parameter Carnot-Caratheodory ball B(x; δ_{1}, δ_{2}) to be the set of all points obtainable from x by applying the flow maps e^{tX_{1}} and e^{tX_{2}}, as long as the total time spent flowing along X_{1} does not exceed δ_{1}, and the total time spent flowing along X_{2} does not exceed δ_{2}. (When δ_{1} = δ_{2} this is essentially the standard Carnot-Cartheodory metric generated by X_{1}, X_{2}).

• These balls provide natural test functions for the averaging operator A. In fact, it is easy to see that if B = B(0; δ_{1}, δ_{2}) is such a ball, then we have the pointwise estimate

Aχ_{B} ≳ δ_{1}δ_{2}χ_{B}.

• Also, the volume of the balls B(0; δ_{1}, δ_{2}) have essentially a polynomial relationship with δ_{1} and δ_{2} as δ_{1}, δ_{2} → 0, with the exact relationship given by the nature of Lie algebra of X_{1} and X_{2} at 0 (or specifically, which combinations of Lie brackets are linearly dependent). The relationship is a bit complicated, but straightforward, and can be proven rigorously using the Baker-Campbell-Hausdorff formula.

• Combining these observations one gets some natural restrictions on the set of all possible p, q for which we have an (L^{p}, L^{q}) mapping property for the averaging operator A - a certain convex polygon determined by the degeneracy of the Lie algebra.

• Theorem [T., Wright]: In the interior of this polygon, the operator A does indeed map L^{p} to L^{q}.

• In other words, up to epsilons, the two-parameter Carnot-Caratheodory balls provide the only obstructions to L^{p} improving estimates.

T T^{∗} methods

• To prove L^{p} → L^{q} mapping properties of the averaging operator A, we follow Christ’s variant of the T T^{∗} method.

• Recall that to prove L^{2} → L^{2} boundedness of a selfadjoint operator T it (morally) suffices to prove this for eigenvectors, i.e. for functions f such that T f = λf. The issue is then to get bounds on λ.

• Even when T is not self-adjoint, there is a related principle, in which one seeks pairs of functions f, g such that T f = λg and T^{∗}g = λf. Again, the issue is to obtain bounds on λ.

• It turns out that there is a similar principle for L^{p} → L^{q} estimates. Very roughly speaking, to prove an operator A with positive kernel is bounded from L^{p} to L^{q} it suffices to verify this for sets E_{1}, E_{2}, λ_{1}, λ_{2} which satisfy the estimates

Aχ_{E_{1}} ≥ λ_{2}χ_{E_{2}}
A^{∗}χ_{E_{2}} ≥ λ_{1}χ_{E_{1}}

and the consistency condition λ_{2}|E_{2}| ∼ λ_{1}|E_{1}| (this reflects the fact that ⟨Aχ_{E_{1}}, χ_{E_{2}}⟩ = ⟨χ_{E_{1}}, A^{∗}χ_{E_{2}}⟩).

• There are combinatorial issues involved in getting the above setup to work correctly (generally, you can get one of the two inequalities to hold 100% of the time, but the other one only works 50% (say) of the time in a certain sense). I’ll ignore this.

• The issue is then to get good bounds in the relationship between λ_{1}, λ_{2}, |E_{1}|, and |E_{2}|. Specifically, to get (L^{p}, L^{q}) boundedness we need λ_{2}|E_{2}|^{1/q} ≲ |E_{1}|^{1/p}.

• When you unravel what the above inequalities tell you about the set E_{1} ⊂ Ω, they are saying (roughly) that when you start in E_{1} and flow randomly along the vector field X_{1}, you will end up in E_{2} with probability about λ_{2}, and if you then flow randomly along X_{2}, you will end up back in E_{1} with probability about λ_{1}.

• Christ’s idea is then to iterate these random flow maps. Start with a generic point in E_{1} (say 0), flow randomly in the X_{1} direction, then the X_{2} direction, and so forth, for m steps; let’s say m is even for sake of notation. With probability λ_{1}λ_{2} … λ_{1}λ_{2}, we stay in E_{1}. This should give a lower bound on E_{1} in terms of λ_{1} and λ_{2}, which when combined with the

consistency condition give (L^{p}, L^{q}) estimates.

• Intuitively one expects to achieve this bound with m = n, because that is how many flows one needs before one spans a full-dimensional subset of Σ with this approach.

• There are two obstructions to this approach, however. Firstly, the iterated flow map need not be oneto-one - many different flows may give the same point in E_{1}. The other obstruction is similar, and has to do with the fact that the Jacobian of the iterated flow map may degenerate to zero, so that the lower bound on the measure of E_{1} may be poor.

• In Christ’s argument, these obstructions are resolved as follows. The Jacobian degeneracy is resolved by the fact that the Jacobian cannot stay close to zero everywhere, and in fact there is some high-order derivative ∂^{α}J of the Jacobian which does not vanish at 0 (exactly which derivative depends on the degeneracy of the Lie algebra). Because of this, one can avoid the low-degeneracy regions of the Jacobian with non-zero probability.

• Model 1D lemma: if J is a monic polynomial of degree n, and E is a subset of [0, 1] of measure λ, then

∫_{E} |J(x)| dx ≳ λλ^{n}.

In the polynomial curve case the Jacobian J factorizes, and these types of calculations become easier (although a fair amount of combinatorics is involved, and one needs to take m ∼ 2n rather than m = n). The bound on the right is sharp, because J can have a zero of order n at some point, and E can concentrate in an interval of length ∼ λ around that point.

• The one-to-one-ness issue is resolved algebraically using Vandermonde determinants. This type of computation restricts Christ to polynomial curves such as (t, t^{2}, …, t^{n}).

17

Widths

• Our main innovation is to do some preparatory reductions on the sets E_{1}, E_{2} before we begin the iterative process. Given a point x ∈ E_{1}, we consider the set

T_{1}(x) = {t : e^{tX_{1}}x ∈ E_{2}}

of times one can flow along X_{1} and end up in E_{2}. We already can control the measure of this set, |T (x)| ≳ λ_{2}. But this does not give any insight as to the shape of this set. To do this we introduce a lemma:

• Lemma. Let T ⊂ [0, 1] be a set of measure λ. Then there exists an interval I ⊂ [0, 1] of some length λ ≤ δ ≤ 1 such that given any two points x, y ∈ T selected in random, one has x, y ∈ I and |x−y| ∼ δ with probability at least ≳ 1/ log(1/λ).

• This lemma is not hard to prove. Basically, one sets σ to be the median separation (or “width”) of two random points in T. (Actually, the mode is a bit better than the median, but you get the idea). The logarithm can be ignored as long as we are not shooting for endpoint results.

• Applying this to our situation, we now obtain two new numbers, δ_{1}, δ_{2}. δ_{1} is the median separation of the flow times from E_{1} to E_{2}, and δ_{2} is the median separation of the flow times from E_{2} to E_{1}.

• From these facts it is fairly easy to conclude that E_{1} (and E_{2}) must concentrate inside a two-parameter ball of the form B(x; δ_{1}, δ_{2}). WLOG we can take x = 0.

Scaling

• The next step is to scale this ball back up to unit size.

• From the Baker-Campbell-Hausdorff formula, the elements of this ball can be written as

exp(δ_{1}t_{1}X_{1} + δ_{2}t_{2}X_{2} + δ_{1}δ_{2}t_{3}[X_{1}, X_{2}] + ...)

where the t_{i} are numbers of size O(1). Only n of the Lie brackets will be relevant, the rest are error terms; the exact choice of these n depends on the relative sizes of δ_{1} and δ_{2}, and the nature of the degeneracy of the Lie algebra at 0.

• One can then parameterize this ball in terms of t_{1}, …, t_{n}, effectively rescaling this ball up to unit size. This scaling will be favorable exactly when (p, q) lies inside the convex polygon determined by the two-parameter Carnot-Caratheodory balls.

• The upshot of this scaling is that one can reduce to the width 1 case, when δ_{1} = δ_{2} = 1.

Avoiding degenerate Jacobians

• The width 1 property is very useful for avoiding regions where the Jacobian vanishes. Recall the estimate

∫_{E} |P (x)| dx ≳ λλ^{n}

when P is a monic polynomial of degree n and |E| = λ? It was sharp because E could concentrate in a λinterval. However, if we assume that E has width 1, then E can no longer concentrate in such a small set, and it turns out that we have a much better bound now:

∫_{E} |P (x)| dx ≳ λ.

The point is that the regions where P are small can be covered by n small intervals, and E cannot concentrate exclusively in those intervals. So at least 50% of E has to live in the set where |P (x)| ≳ 1, hence the claim.

• Using this type of argument, one can use the width 1 property to basically ignore the degenerate regions of the Jacobian, and assume that the iterated flowmap has Jacobian ∼ 1 with large probability. This gives extremely good bounds on the integral - far more than what we need to actually close the theorem, in fact. (If it weren’t for the fact that we had to rescale in the previous slide, we could get better (p, q) estimates than are possible).

One-to-one ness

• The only other thing to check is that the iterated flow map is one-to-one, or at least boundedly manyto-one. Ordinarily, this would be quite difficult and would require sophisticated algebraic geometry.

• However, once we have the Jacobian ∼ 1, the iterated flow map is locally one-to-one simply by virtue of the inverse function theorem. A covering argument then shows that when we restrict ourselves to the set of flow times for which the Jacobian ∼ 1, the map is boundedly many-to-one, and we are done.