L^{p} improving estimates for averages along curves

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

 www.math.ucla.edu/∼tao/preprints (in

preparation)

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.

• To avoid degenerate situations 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

Rf(l) := ∫_{l} f

where l ranges over lines in R^{2}. Other examples include convolution with the circle, helix, or other curves.

Interesting questions:

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

• L^{2} smoothing: Does T map L^{2} to L_{α}^{2}? FIO techniques, singularity theory.

• L^{p} smoothing: Does T map L^{p} to L_{α}^{p}? Very difficult in general; related to local smoothing for the wave equation and similar hard problems.

• L^{p} improving: Does T map L^{p} to L^{q} for some q > p? Easier than the smoothing question, because no derivatives are involved.

• This talk is concerned only with the last question.

• Work by Seeger, Greenleaf, Oberlin, and others on the L^{p} improving problem obtained sharp results in low dimensions n ≤ 4 and partial results in higher dimensions. Techniques: oscillatory integrals, T T^{∗} method, special properties of L^{2} and L^{4}.

• 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.

• 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

• A fundamental role is played by the n+1-dimensional manifold

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 hypotheses imply that these projections are submersions.

• 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.

• 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. We can essentially write

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.

• 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.

Idea 1: Iteration

• The first ingredient in the proof is an iteration idea of Mike Christ. Instead of proving bounds for the averaging operator A, we study instead a large power of A. After some standard arguments, the problem is now as follows.

• We need two parameters 0 < λ_{1}, λ_{2} < 1. We adopt the notation that X_{i} = X_{1} for i odd and X_{i} = X_{2} for i even, and similarly for λ_{i}.

• Start with a point x_{0} in Σ and Let E_{0} := {x_{0}}, and

write

E_{1} = {e^{tX_{1}}x_{0} : t ∈ T_{1}(x_{0})}

where T_{1}(x_{0}) is a set of times of measure λ_{1}. Then write

E_{2} = {e^{tX_{2}}x_{1} : x_{1} ∈ E_{1}; tinT_{2}(x_{1})}

where T_{2}(x_{1}) is a set of times of measure λ_{2} for each x_{1}. And so forth.

• The set E_{i} lives in an i-dimensional set for each i. Thus one expects E_{n} to be full dimensional. The problem of obtaining L^{p} estimates for A turns out to be equivalent (up to endpoints) to obtaining lower bounds for |E_{n}| in terms of λ_{1} and λ_{2}.

• Thus everything depends on the behaviour of the map

Φ : (t_{1}, …, t_{n}) ↦ e^{t_{n}X_{n}} … e^{t_{1}X_{1}}x_{0}.

Ideally we want Φ to be one-to-one and have Jacobian J(Φ) bounded away from zero.

• Unfortunately, J(Φ) degenerates at places. However, 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).

Widths

• Our main innovation over Christ’s argument is to do some preparatory reductions on the sets T_{j}(x_{j−1}). We need to control not just the measure of this set, but its shape. The key lemma is

• 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.

• This allows one to localize the T_{j} to a fixed interval, which basically localizes the iterated sets E_{1}, E_{2}, … to a two-parameter Carnot-Caratheodory ball. We then rescale the ball to unit size and apply Christ’s argument to obtain a near-optimal bound.

• The large median separation of times allows one to avoid the bad set where J(Φ) is small with very high probability.