ENDPOINT LEBESGUE ESTIMATES FOR WEIGHTED AVERAGES ON POLYNOMIAL CURVES

Transcription

1 ENDPOINT LEBESGUE ESTIMATES FOR WEIGHTED AVERAGES ON POLYNOMIAL CURVES MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Abstract. We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying a natural nilpotency hypothesis.. Introduction Let (P, g ) and (P 2, g 2 ) be two smooth Riemannian manifolds of dimension n, n 2. In [27], Tao Wright established near-optimal Lebesgue estimates for local averaging operators of the form T f(x 2 ) = f(γ x2 (t))a(x 2, t) γ x 2 (t) g dt, f C 0 (P ), (.) with a continuous and compactly supported, under the hypothesis that the map (x 2, t) γ x2 (t) P is a smooth submersion on the support of a. Our goal in this article is to sharpen the Tao Wright theorem to obtain optimal Lebesgue space estimates, without the cutoff, under an additional polynomial-like hypothesis on the map γ. We replace the Riemannian arclength with a natural generalization of affine arclength measure; this enables us to prove estimates wherein the Lebesgue exponents are independent of the manifolds and curves involved (provided γ is polynomial-like), and operator norms for a fixed exponent pair and fixed polynomial degree are uniformly bounded. Our results are strongest at the Lebesgue endpoints, where the generalized affine arclength measure is essentially the largest measure for which these estimates can hold and, moreover, the resulting inequalities are invariant under a variety of coordinate changes. By duality, bounding the operator T in (.) is equivalent to bounding the bilinear form B(f, f 2 ) = f (γ x2 (t))f 2 (x 2 )a(x 2, t) γ x 2 (t) g dν 2 (x 2 )dt, M where M := P 2 R. For the remainder of the article, we will focus on the problem of bounding such bilinear forms... The Euclidean case. The Tao Wright theorem, being local, may be equivalently stated in Euclidean coordinates. Though we will obtain more general results on manifolds (and also in Euclidean space) by applying diffeomorphism invariance

2 2 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET of our operator and basic results from Lie group theory, the Euclidean version is, in some sense, our main theorem. Let π, π 2 : R n R n be smooth mappings. Define vector fields X j = (dπj dπ n j ), (.2) where denotes the Riemannian Hodge star operator mapping n forms to vector fields. The geometric significance of the X j is that they are tangent to the fibers of the π j, and their magnitude arises in the coarea formula: Ω = π j(ω) π j (y) where H denotes -dimensional Hausdorff measure. We define a map Ψ : R n R n R n by χ Ω (t) X j (t) dh (t) dy, Ω {X j 0}, (.3) Ψ x (t) := e tnxn e tx (x), (.4) where we are using the cyclic notation X j = X j mod 2, j = 3,..., n. Given a multiindex β, we define b = b(β) := ( + β j, + β j ) (.5) j odd j even ρ β (x) = ( β t det D t Ψ x ) (0) b +b 2 (.6) (p, p 2 ) = (p (b(β)), p 2 (b(β))) := ( b +b 2 ) b, b+b2 b 2. (.7) Our main theorem is the following. Theorem.. Let n 3, let N be a positive integer, and let β be a multiindex. Assume that the maps π j : R n R n and associated vector fields X j, defined in (.2) satisfy the following: (i) The X j generate a nilpotent Lie algebra g of step at most N, and for each X g, the map (t, x) e tx (x) is a polynomial of degree at most N; (ii) For each j =, 2 and a.e. y R n, π j ({y}) is contained in a single integral curve of X j. Then with ρ β satisfying (.6) and p, p 2 as in (.7), f π (x) f 2 π 2 (x) ρ β (x) dx CN f p f 2 p2, (.8) R n for some constant C N depending only on the degree N. No explicit nondegeneracy (i.e. finite type) hypothesis is needed, because the weight ρ β is identically zero in the degenerate case. The weights ρ β were introduced in [25], wherein local, non-endpoint Lebesgue estimates were proved in the C case for a multilinear generalization. In Section 9, we give examples showing that the endpoint estimate (.8) may fail in the multilinear case, and that it may also fail in the bilinear case when Hypothesis (i), Hypothesis (ii), or the dimensional restriction n 3 is omitted. Theorem. uniformizes, makes global, and sharpens to Lebesgue endpoints the Tao Wright theorem for averages along curves, under our additional hypotheses. (As the Tao Wright theorem is stated in terms of the spanning of elements from g, not the non-vanishing of ρ, the relationship between the results will take some explanation, which will be given in Section 3.) Moreover, our result generalizes to the fully translation non-invariant case the results of [8, 0, 5, 2, 23], wherein

3 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 3 endpoint Lebesgue estimates were established for convolution and restricted X-ray transforms along polynomial curves with affine arclength measure..2. Averages on curves in manifolds. Let M be an n-dimensional smooth manifold and let P and P 2 be n -dimensional manifolds. We say that a subset of one of these manifolds has measure zero if it has measure zero with respect to Lebesgue measure in any choice of smooth local coordinates. Let π, π 2 be smooth maps from M to P, P 2, resp., and assume that the π j have full rank a.e. We assume that there exist vector fields X, X 2 on M such that X j (π j ) 0 and that the set where X j = 0 coincides with the set where Dπ j fails to have full rank. We assume moreover that the X j generate a nilpotent Lie algebra g of step at most N, that the flow of each element in g is complete (i.e. e tx (x) is defined for all t, x), and that for a.e. y P j, π j ({y}) lies within a single integral curve of X j. We note that these conditions are invariant under diffeomorphisms of both M and of the P j. As we will see, under these conditions, there exists a covering map Φ : R n M, which is a local diffeomorphism, such that the pullbacks X := Φ X, X g, have polynomial flows. Moreover, the deck transformation group acts transitively on the fibers of Φ, and each deck transformation a diffeomorphism with Jacobian determinant identically equal to. The X j, having polynomial flows, must be divergence-free, and thus we will be able to define measures ν j on the P j with respect to which the co-area formula holds for the maps π j := π j Φ. We define weights ρ β on R n following the algorithm above, and, as we will see, the ρ β dx push forward to measures on M, yielding a natural analogue of (.8) for M and the P j. Without further hypotheses, we do not have the freedom to choose the measures ν j on the P j, but if β is the minimal multiindex for which ρ β 0, or if we content ourselves to local results, we do have this flexibility. In Section 9, we will give a counter-example to show that uniform global results may fail unless carefully chosen measures on the P j are employed. We had assumed above that the flows of the Lie group elements were complete; for local results, this is not needed, as we will see..3. Background and sketch of proof. We turn to an outline of the proof of Theorem., and a discussion of the context in the recent literature. We begin with the proof on a single torsion scale {ρ β 2 m }. By uniformity, it suffices to consider the case when m = 0, and thus the restricted weak type version of (.8) is equivalent to the generalized isoperimetric inequality Ω π (Ω) p π 2 (Ω) p 2, Ω {ρ β }. (.9) By simple arithmetic, this is equivalent to the lower bound α b αb2 2 Ω, α j := Ω π j(ω), (.0) with b = (b, b 2 ) as in (.5). To establish (.0), Tao Wright [27], and later Gressman [3], used a version of the iterative approach from [3]. Roughly speaking, for a typical point x 0 Ω, the measure of the set of times t such that e txj (x 0 ) Ω is α j. Iteratively flowing along the vector fields X, X 2 gives a smooth map, Ψ x0 (recall (.4)), from a measurable subset F R n into Ω. The containment Ψ x0 (F ) Ω must then be translated into a lower bound on the volume of Ω.

4 4 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Tao Wright deduce from linear independence of a fixed n-tuple Y,..., Y n g (the Lie algebra generated by X, X 2 ) a lower bound on some fixed derivative β of the Jacobian determinant det DΨ x0. For typical points t R n, we have a lower bound det DΨ x0 (t) t β β det DΨ x0 (0), and this we should be able to use in estimating the volume of Ω: Ω Ψ x0 (F ) det DΨ x0 (t) dt F β det DΨ x0 (0) max t F tβ α b. F Unfortunately, the failure of Ψ x0 to be polynomial in the Tao Wright case and the fact that F is not simply a product of intervals means that this deduction is not so straightforward; in particular, the inequalities surrounded by quotes in the preceding inequality are false in the general case. More precisely, if Ψ x0 is merely C, we cannot uniformly bound the number of preimages in F of a typical point in Ω (so the first inequality may fail), and even for polynomial Ψ x0, if F is not an axis parallel rectangle, then the inequality det DΨ x0 (t) t β may fail for most t F. In the nonendpoint case of [27], it is enough to prove (.0) with a slightly larger power of α on the left; this facilitates an approximation of F by a small, axis parallel rectangle centered at 0, and (using the approximation of F ) an approximation of Ψ x0 by a polynomial. These approximations are sufficiently strong that Ψ x0 is nearly finite-to-one on F (see also [5]) and det DΨ x0 grows essentially as fast on F as its derivative predicts, giving (.0). In [3], wherein the Lie algebra g is assumed to be nilpotent, the map Ψ x0 is lifted to a polynomial map in a higher dimensional space, abrogating the need for the polynomial approximation. This leaves the challenge of producing a suitable approximation of F as a product of intervals, and Gressman takes a different approach from Tao Wright, which avoids the secondary endpoint loss. In Section 2, we reprove Gressman s single scale restricted weak type inequality. A crucial step is an alternate approach to approximating one-dimensional sets by intervals. This alternative approach gives us somewhat better lower bounds for the integrals of polynomials on these sets, and these improved bounds will be useful later on. An advantage of the positive, iterative approach to bounding generalized Radon transforms has been its flexibility, particularly relative to the much more limited exponent range that seems to be amenable to Fourier transform methods. A disadvantage of this approach is that it seems best suited to proving restricted weak type, not strong type estimates. Let us examine the strong type estimate on torsion scale. By positivity of our bilinear form, it suffices to prove 2 j+k χ E j π (x)χ E k 2 π 2 (x) dx ( 2 jp E j ) p ( 2 kp2 E2 k ) p 2, {ρ β } j k j,k for measurable sets E j, Ek 2 R n, j, k Z. Thus a scenario in which we might expect the strong type inequality to fail is when there is some large set J and some set K such that the 2 j χ E j, j J, evenly share the L p norm of f, the 2 k χ E k 2, k K evenly share the L p2 norm of f 2, and the restricted weak type inequality is essentially an equality χ E j π (x)χ E k 2 π 2 (x) dx E j p E2 k p 2, (.) {ρ β }

5 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 5 for each (j, k) J K. In [4] a technique was developed for proving strong type inequalities by defeating such enemies and this approach was used to reprove Littman s bound [6] for convolution with affine surface measure on the paraboloid. This approach was later used [8, 0,, 5, 2, 23] to prove optimal Lebesgue estimates for translation invariant and semi-invariant averages on various classes of curves with affine arclength measure. Key to these arguments was what was called a trilinear estimate in [4], which we now describe. We lose if one E2 k interacts strongly, in the sense of (.) with many sets E j of widely disparate sizes. Suppose that Ek 2 interacts strongly with two sets E ji, i =, 2. Letting Ω i := π (Eji ) π 2 (Ek 2 ) {ρ β }, our hypothesis (.) and the restricted weak type inequality imply that π 2 (Ω i ) must have large intersection with E2 k for i =, 2; let us suppose that E2 k = π 2 (Ω ) = π 2 (Ω 2 ). Assuming that every π 2 fiber is contained in a single X 2 integral curve, for a typical x 0 Ω i, the set of times t such that e tx2 (x 0 ) Ω i must have measure about α2 i := Ω i E2 k ; thus we have Ψ x 0 (F i ) Ω i for measurable sets F i, which are not well-approximated by products of intervals centered at 0. In all of the above mentioned articles [4, 8, 0,, 5, 2, 23], rather strong pointwise bounds on the Jacobian determinant det DΨ x0 were then used to derive mutually incompatible inequalities relating the volumes of the three sets, E j, Ej2, Ek 2 (whence the descriptor trilinear ). In generalizing this approach, we encounter a number of difficulties. First, we lack explicit lower bounds on the Jacobian determinant. We can try to recover these using our estimate β det DΨ x0 (0), but this is difficult to employ on the sets F i, since it is impossible to approximate these sets using products of intervals centered at 0. Finally, in the translation invariant case, it is natural to decompose the bilinear form in time, B(f, f 2 ) = f (x γ(t))f 2 (x) ρ β (t) dt dx, j R n t I j and, thanks to the geometric inequality of [9], there is a natural choice of I j that makes the trilinear enemies defeatable. It is not clear to the authors that an analogue of this decomposition in the general polynomial-like case is feasible. Our solution is to dispense entirely with the pointwise approach. In Section 4, we prove that if the set Ω nearly saturates the restricted weak type inequality (.9), then Ω can be very well approximated by Carnot Carathéodory balls. Thus, if E and E 2 interact strongly, then E and E 2 can be well-approximated by projections (via π, π 2 ) of Carnot Carathéodory balls. The proof of this inverse result relies on the improved polynomial approximation mentioned above, as well as new information, proved in Section 3, on the structure of Carnot Carathéodory balls generated by nilpotent families of vector fields. In Section 5, we prove that it is not possible for a large number of Carnot Carathéodory balls with widely disparate parameters to have essentially the same projection; thus one set E2 k cannot interact strongly with many E j, and so the strong type bounds on a single torsion scale hold. In Section 6, we sum up the torsion scales. In the non-endpoint case considered in [25], this was simply a matter of summing a geometric series, but here we must control the interaction between torsion scales. The crux of our argument is that

6 6 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET many Carnot Carathéodory balls at different torsion scales cannot have essentially the same projection. Section 7 gives relevant background on nilpotent Lie groups which will be used in deducing from Theorem. more general results, including the above-mentioned global result on manifolds. The results of this section are essentially routine deductions from known results in the theory of nilpotent Lie groups, but the authors could not find elsewhere the precise formulations needed here. In Section 8, we prove extensions of our result to the nilpotent case, including a global result on manifolds, and other generalizations. In Section 9, we give counter-examples to a few natural generalizations of our main theorem, discuss its optimality at Lebesgue endpoints, and recall the impossibility of an optimal weight away from Lebesgue endpoints. The appendix, Section 0, contains various useful lemmas on polynomials of one and several variables. Some of these results are new and may be useful elsewhere. Acknowledgements. The authors would like to thank Melanie Matchett Wood for explaining how one manipulates intersections of sets parametrized by polynomials into varieties, Daniel Erman for suggesting the reference for Theorem 0.8, and Terence Tao for advice and encouragement in the very early stages of this article. Christ was supported by NSF DMS Dendrinos was supported by DFG DE 57/2-. Stovall was supported by NSF DMS and DMS Street was supported by NSF DMS Part of this work was completed at MSRI during the 207 Harmonic Analysis program, which was funded in part by NSF DMS Notation. We will use the standard notation A B to mean that A C N B, where the constant C N depends only on the degree. Since ρ β 0 implies that N is larger than some constant depending on n, our constants implicitly depend on the dimension as well. If A B and B A, then we write A B. The notation A B and A B will also be used; it will be defined later on. 2. The restricted weak type inequality on a single scale This section is devoted to a proof, or, more accurately, a reproof, of the restricted weak type inequality on the region where ρ β. The following result is essentially due to Gressman in [3]. Though uniformity is not explicitly claimed in [3], the methods of that paper may be adapted to establish the following. Proposition 2.. [3] For each pair E, E 2 R n of measurable sets, {ρ β } π (E ) π 2 (E 2) E /p E 2 /p2 (2.) holds uniformly, with definitions and hypotheses as in Theorem.. We give a complete proof of the preceding, using partially alternative methods from those in [3], because our approach will facilitate a resolution, in Section 4, of a related inverse problem, namely, to characterize those pairs (E, E 2 ) for which the inequality in (2.) is reversed. Our proof of Proposition 2. is based on the following lemma. Lemma 2.2. Let S R be a measurable set. For each N, there exists an interval J = J(N, S) with J S N S such that for any polynomial P of degree at most

7 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 7 N, S P dt N N j=0 P (j) ( J ( ε)j S L (J) S ) j+. The key improvement of this lemma over the analogous result in [3] is the gain ( J S )( ε)j in the higher order terms. This gain will allow us to transfer control over P into control over the length of J. S Proof of Lemma 2.2. This follows immediately from Proposition 0.2. Proof of Proposition 2.. We take the now standard approach of the method of refinements. We may assume that E, E 2 are open sets. By the coarea formula and our assumptions on the set U in (.8), there exists an open set Ω U with Ω π (E ) π 2 (E 2) {ρ β }, Ω π (E ) π 2 (E 2) {ρ β }, such that for each y R n, π j ({y}) Ω is contained in a single integral curve of X j. Define α j := Ω E j, j =, 2. Let σ j : π j (Ω) Ω be a measurable section of π j Ω. We may write the coarea formula as Ω = χ Ω (e txj (σ j (y))) dt dy, Ω Ω. π j(ω ) We define our refinements iteratively, starting with Ω n := Ω and j = n. For x Ω j, we may write x = e tj(x)xj (σ j (π j (x))), and define S j (x) := {t : e txj (x) Ω j } = S j (σ j (π j (x))) t j (x) J j (x) := J(N, S j (σ j (π j (x)))) t j (x) Ω j := {x Ω j : S j (x) > C j,n α j, 0 J j (x)}. By the coarea formula, Ω j Ω j. We may assume that each Ω j is open. Indeed, supposing that Ω j is open, we will prove that we can take Ω j to be open. Shrinking Ω j slightly, we may assume that Ω j = {e txj (x) : x x 0 < δ, t < δ}, x 0 A for some finite set A and fixed δ > 0. (We have no control over, nor will we use an upper bound on #A, nor a lower bound on δ.) Thus, after refining Ω j a bit more, we can choose our intervals J j (x) so that their endpoints vary continuously over balls whose size depends on A and δ. Thus we may assume that Ω j is a union of open sets, and hence is open. Let x 0 Ω 0, and for t R n, define Define F := S (x 0 ), and for each j = 2,..., n, Ψ x0 (t) = e tnxn e tx (x 0 ). (2.2) F j := {(t, t j ) R j : t F j, t j S j (Ψ x0 (t, 0))}. Thus for t F j, Ψ x0 (t, 0) Ω j, so 0 J j (Ψ x0 (t, 0)). In particular, Ψ x0 (F n ) Ω, so by Lemma 0.9, Ω Ψ x0 (F n ) det DΨ x0 (t) dt. F n

8 8 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Since 0 J j (Ψ x0 (t, 0)) for each t F j, we compute det DΨ x0 (t) dt = det DΨ x0 (t, t n ) dt n dt F n F n S n(ψ x0 (t,0)) αn βn+ βn t n det DΨ x0 (t, 0) dt F n α βn+ n α β+ β t det DΨ x0 (0) α b αb2 2. After a little arithmetic, we see that (2.) is equivalent to Ω α b proposition is proved. αb2 2 (2.3), so the We have not yet used the gain in Lemma 2.2; we will take advantage of that in Section 4 when we prove a structure theorem for pairs of sets for which the restricted weak type inequality (2.) is nearly reversed. Before we state this structure theorem, it will be useful to understand better the geometry of image under Ψ x0 of axis parallel rectangles. 3. Carnot Carathéodory balls associated to polynomial flows In the previous section, we proved uniform restricted weak type inequalities at a single scale. To improve these to strong type inequalities, we need more, namely, an understanding of those sets for which the inequality (2.) is nearly optimal. In this section, we lay the groundwork for that characterization by establishing a few lemmas on Carnot Carathéodory balls associated to nilpotent vector fields with polynomial flows. Results along similar lines have appeared elsewhere, [6, 7, 26, 27] in particular, but we need more uniformity and a few genuinely new lemmas, and, moreover, our polynomial and nilpotency hypotheses allow for simpler proofs than are available in the general case. We begin by reviewing our hypotheses and defining some new notation. We have vector fields X, X 2 X (R n ) that are assumed to generate a Lie subalgebra g X (R n ) that is nilpotent of step at most N, and such that for each X g, the exponential map (t, x) e tx (x) is a polynomial of degree at most N in t and in x. Lemma 3.. The elements of g are divergence-free. Proof. Let X g. Both det De tx (x) and its multiplicative inverse, which may be written det(de tx )(e tx (x)), are polynomials, so both must be constant in t and x. Evaluating at t = 0, we see that these determinants must equal, so the flow of X is volume-preserving, i.e. X is divergence-free. A word is a finite sequence of s and 2 s, and associated to each word w is a vector field X w, where X (i) = X i, i =, 2, and X (i,w) = [X i, X w ]. We let W denote the set of all words w with X w 0. For I W n, we define λ I := det(x w,..., X wn ), and we define Λ := (λ I ) I W n. We denote by Λ the sup-norm. In this section, we will frequently use c to denote a small constant that depends on N and will change from line to line. Lemma 3.2. Assume that λ I (0) δ Λ(0), for some δ > 0. Then for any w W, for all t < cδ. λ I (e txw (0)) λ I (0), λ I (e txw (0)) δ Λ(e txw (0)),

9 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 9 Proof. By Lemma 3., X w is divergence-free. Thus for any I = (w,..., w n) W n, n X w λ I = λ I i, where I i is obtained from I by replacing the i-th entry with [X w, X w i ]. Thus for each k, dk Λ(e txw (0)) Λ(0) δ λ dt k I (0). As t Λ e txw (0) is a polynomial of bounded degree, this implies that Λ(e txw (0)) Λ(0) for t < c, and that λ I (e txw (0)) λ I (0) for t < cδ. The conclusion of the lemma follows. i= For I = (w,..., w n ) W n, we define a map Φ I x 0 (t,..., t n ) := e tnxwn e txw (x0 ). Lemma 3.3. Let I W n, and assume that λ I (0) δ Λ(0). Then for all t < cδ, and Λ Φ I 0(t) Λ(0). det DΦ I 0(t) λ I Φ I 0(t) λ I (0), Proof. By Lemma 3.2 and a simple induction, we have only to show that det DΦ I 0(t) λ I (0), for all t < cδ. Since the flow of each X w is volume-preserving, det DΦ I 0(t) = det(x w (0), φ t X w X w2 (0),..., φ t X w φ t n X wn X wn (0)), where φ X Y (x) := De X (e X (x))y (e X (x)). Since d dt φ tx Y = φ tx [X, Y ], this gives β t det DΦ I 0(0) Λ(0) δ λ I (0) = δ det DΦ I 0(0), for all multiindices β. This gives us the desired bound on det DΦ 0(t), for t < cδ. Lemma 3.4. Assume that λ I (0) δ Λ(0). Then Φ I 0 is one-to-one on { t < cδ}, and for each w W, the pullback Y w := (Φ I 0) X w satisfies Y w (t) δ on { t < cδ}. Proof. We write DΦ 0 (t) = A(t, Φ 0 (t)), where A is the matrix-valued function given by A(t, x) := (φ t nx wn φ t 2X w2 X w (x),, φ t nx wn X wn (x), X wn (x)). By the nilpotency hypothesis, each column of A is polynomial in t, and thus may be computed by differentiating and evaluating at t = 0. Using the Jacobi identity, iterated Lie brackets of the X wi may be expressed as iterated Lie brackets of the X i, and so φ t nx wn φ t i+x wi+ X wi = X wi + w W p wi,w(t)x w, where each p wi,w is a polynomial in (t i+,..., t n ), with p wi,w(0) = 0. By Cramer s rule, n λ I X w = λ I(i) X wi, where I(i) is obtained from I by replacing X wi with X w. Therefore i= A = (X w,..., X wn )(I n + λ I P ),

10 0 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET where I n is the identity matrix and P is a matrix-valued polynomial with P (0) = 0 and P Φ I 0(t) λ I Φ I 0(t) λ I (0) on { t < cδ}. For w W, we define Y w to be the pullback of X w : Then Y wi (0) = t i, i n. Let Y w (t) := DΦ I 0(t) X w Φ I 0(t), Ỹ w := λ I (0) (det DΦ I 0)Y w. Then Ỹw is a polynomial, and Ỹw(0) = Y w (0). Since t < cδ. Y w (t) = (I n + λ I Φ I 0(t)P Φ I 0(t)) (X w Φ I 0(t),..., X wn Φ I 0(t)) X w Φ I 0(t), we know that Y wi (t) = (I n + λ I Φ I 0(t)P Φ I 0(t)) e i. Therefore Y wi t i on { t < cδ}, which implies that for each w W, Y w δ on this region, since n λ Y w = I(i) Φ I 0 λ I Y Φ I wi, 0 i= and we have bounds on the coefficients of the Y wi in this sum. Since det DΦI 0 λ I (0), we also have that Ỹw i t i on { t < cδ}, and since Ỹw i is polynomial and satisfies Ỹw i (0) = t i, this implies the stronger estimate Ỹw i (t) t i δ t on { t < cδ}. Similarly, det DΦI 0 (t) λ I (0) Therefore δ t, whence Y wi (t) t i δ t as well. D s e snywn e syw (0) In δ t, t < cδ, which, by the contraction mapping proof of the Inverse Function Theorem, implies that (s,..., s n ) e snywn e syw (0) is one-to-one on { t < cδ}. Finally, by naturality of exponentiation, Φ I 0 must also be one-to-one on this region. Lemma 3.5. There exists a constant c = c N > 0 such that the following holds. Let x j R n, j =, 2, and assume that I j W n are such that λ Ij (x j ) δ Λ(x j ), j =, 2. Let 0 < ρ < cδ. If 2 j= ΦIj x j ({ t < cδρ}), then Φ I x ({ t < cδρ}) Φ I2 x 2 ({ t < ρ}). Proof. By assumption, each element of Φ I x ({ t < cδρ}) can be written in the form e t2nxw 2n e t X w (x2 ), with w j W n, j =,..., 2n, and t < 2cδρ. By Lemma 3.4, such points are contained in Φ I2 x 2 ({ t < ρ}). We recall that Ψ x0 = Φ x (,2,,2,...) 0, and we define Ψ x0 := Φ x (2,,2,,...) 0. For β Z n 0 a multiindex, we define J β (x 0 ) := β det DΨ x0 (0), J β (x 0 ) := β det D Ψ x0 (0).

11 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES Lemma 3.6. Λ(0) β J β (0) + J β (0). (3.) Proof. The argument that follows is due to Tao Wright, [27]; we reproduce it to keep better track of constants to preserve the uniformity that we need. Direct computation shows that the J β and J β are linear combinations of determinants λ I, and it immediately follows that the left side of (3.) bounds the right. To bound the left side, it suffices to prove that there exists t such that Λ(0) det DΨ 0 (t) + det D Ψ 0 (t), which is equivalent (via naturality of exponentiation and Lemma 3.4) to finding a point s such that det D s e snyn e sy (0) + det D s e snyn+ e sy2 (0), where the vector fields Y i are those defined in Lemma 3.4, the n-tuple I having been chosen to maximize λ I (0). By Lemma 3.4, Y w C N ({ t <c}), for all w W. By induction, this implies that Y w (0) ( Y (0) + Y 2 (0) ). Since Y wi (0) =, Y (0) + Y 2 (0). Thus (3.2) holds for k =, s = 0. Without loss of generality, we may assume that Y (0). Now we proceed inductively, proving that for each k n, there exists a point (s,..., s k+ ) < c such that e s ky k e sy (0) k e s ky k e sy (0) ; (3.2) the case k =, s = 0 having already been proved. Assume that (3.2) holds for some k < n, s = s 0 < c. Then (s,..., s k ) e s ky k e sy (0), s s 0 < c parametrizes a k-dimensional manifold M. The vector fields Z i (e s ky k e sy (0)) := si e siyi e sy (0) form a basis for the tangent space of M at each point, and if (3.2) fails, then for all s s 0 < c, k Y k+ (e s ky k e sy (0)) = c i (s)z i (e s ky k e sy (0))+Y (e s ky k e sy (0)), i= with c i CN ({ s s 0 <}) and α s Y = Z α Y < c N, for c N as small as we like and all α < N. Since Z k = Y k, det(y w,..., Y wn )(e s0 k Y k e s0 Y (0)) < c N, for a (possibly different but) arbitrarily small constant c N. Thus λ I (e s0 k X k e s0 X (0)) < c N Λ(0), which, by Lemma 3.2, contradicts our assumption that λ I (0) Λ(0). We say that a k-tuple (w,..., w k ) W is minimal if w, w 2 {(), (2)}, and for i 3, w i = (j, w l ) for some j =, 2 and l < i. It will be important later that a minimal n-tuple must contain the indices (), (2), and (, 2). The next few lemmas will involve a small parameter ε > 0. We will use the notation A B to mean that there exists a constant C, depending only on N, such that A Cε C B. We will write A B to mean A B and B A.

12 2 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Lemma 3.7. There exists a minimal n-tuple I 0 W n such that for all ε > 0, {x Ψ0 ({ t < }) : λ I 0(x) Λ(x) } ( ε) Ψ0 ({ t < }). Proof. By Lemma 3.2, Λ(x) Λ(0), for all x Ψ 0 ({ t < }). From standard facts about polynomials, if P is a polynomial of degree at most N and ε > 0, { t < : P (t) < ε P C0 ({ t <})} < Cε /C, for some C = C N,n. Thus it suffices to prove that there exists a minimal n-tuple I 0 such that λ I 0 Ψ 0 C0 ({ t <}) Λ(0). Fix an n-tuple I W n such that λ I (0) Λ(0). By Lemmas 3.4 and 3.6, Ψ 0 ({ t < }) Φ I 0({ t < }) Ψ 0 ({ t < }) Φ I 0({ t < }). Thus it suffices to find a minimal n-tuple I 0 such that λ I 0 Φ I 0 C0 ({ t <}) Λ(0), i.e. such that det(y w 0,..., Y w 0 n ) C 0 ({ t <}), with notations as in Lemma 3.4. We will prove inductively that for each k n, there exists a minimal k-tuple (w, 0..., wk 0) such that Y w 0 Y wk 0 C 0 ({ t <}). The initial step of the induction has been done: As we saw in the proof of Lemma 3.6, we may assume that Y (0). For the induction step, it will be useful to have two constants, c N > 0 and δ N > 0, which will be allowed to change from line to line. Our convention is that c N will be sufficiently small to satisfy the hypotheses of the preceding lemmas, and that we must be able to choose δ N arbitrarily small, so as to derive a contradiction if the induction step fails. By Lemma 3.3, det(y w,..., Y wn ) on { t < c N }. Suppose that for some k < n, we have found a minimal k-tuple (w, 0..., wk 0) and some t0 < c N such that Y w 0 (t 0 ) Y w 0 k (t 0 ). (3.3) Without loss of generality, w 0 = (). We may extend Y w 0,..., Y w 0 k to a frame on { t t 0 < c N } by adding vector fields Y wi ; by reindexing, we may assume that Y w 0,..., Y w 0 k, Y wk+,..., Y wn span at every point of { t t 0 < c N }. Thus failure of the inductive step implies that for each w {(2)} {(i, wj 0 ) : i {, 2}, j k}, we can write k n Y w (t) = a 0 i (t)y w 0 i (t) + a j (t)y wj (t), (3.4) i= j=k+ with a 0 i C N ({ t t 0 <c N }) and a j C N ({ t t 0 <c N }) < δ N. Therefore for i =, 2 and w as above, [Y i, Y w ] = k Y i (a i )Y w 0 i + i= k a 0 i [Y i, Y w 0 i ] + i= n j=k+ Y i (a j )Y wj + n j=k+ a j [Y i, Y wj ], may be written in the same form as (3.4), with the same bounds on the coefficients. By induction, det(y w (t 0 ),..., Y wn (t 0 )) < δ N (because the Y wi must all lie near the span of Y w 0,..., Y w 0 k ), a contradiction, since det(y w,..., Y wn ) on t < c.

13 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 3 Lemma 3.8. There exists a minimal n-tuple I W n such that for all ε > 0, there exists a collection A Ψ 0 ({ t < }), of cardinality #A, such that (i) Ψ 0 ({ t < }) x A σ S n Φ Iσ x ({ t < cε C }) ( ε) Ψ 0 ({ t < }), and, moreover, for all x A, σ, σ S n, and y Φ Iσ x ({ t < cε C }), (ii) Φ I σ y is one-to-one on { t < c ε C }, with Jacobian determinant det DΦ I σ y (t) λ I (y) λ I (x) Λ(x) Λ(Φ I σ y (t)) (iii) Φ Iσ x ({ t < cε C }) Φ I σ y ({ t < c ε C }). Proof. By Lemma 3.7, there exists δ and a minimal I W n such that if G := {x Ψ 0 ({ t < }) : λ I (x) δ Λ(x) }, then G ( ε) Ψ 0 ({ t < }). We will cover G by a controllable number of balls in the collection B whose elements are all of the form B x (ρ) := x ({ t < c 2 δ 2 }), x G. σ S n Φ Iσ Taking cε C := c 2 δ 2 and c ε C := cδ, properties (ii) and (iii) of our lemma follow from Lemmas 3.3, 3.4, and 3.5. We will use the generalized version of the Vitali Covering Lemma in [20]; for this, we need to verify the doubling and engulfing properties. By Lemma 3.5, for all 0 < ρ < cδ, σ S n, and x G, Φ Iσ x ({ t < ρ}) B x (ρ) Φ Iσ x ({ t < cδρ}). (3.5) Hence by Lemma 3.3, B x (ρ) λ I (x) ρ n Λ(0) ρ n. Therefore the balls are indeed doubling. The engulfing property also follows from Lemma 3.5, since B x (cδρ) B x2 (cδρ) implies that B x (cδρ) B x2 (ρ). If we choose A G so that {B x (c 3 δ 3 )} x A is a maximal disjoint subset, then G x A B x(c 2 δ 2 ), and finally, by (3.5), and Lemma 3.6, #A Λ(0) G Ψ 0 ({ t < }) Λ(0). We define a polytope P associated to the vector fields X, X 2 to be the convex hull P := ch deg I + [0, ) 2, (3.6) I W n :λ I 0 and we recall that the Newton polytope associated to X, X 2 at the point x 0 is the smaller convex hull := ch deg I + [0, ) 2. (3.7) P x0 I W n :λ I(x 0) 0 It was proved by Tao Wright in [27] that nontrivial local estimates f π (x)f 2 π 2 (x) a(x) dx f p f 2 p2, (3.8) with a C 0 0(R n ), and π, π 2 smooth submersions (but without the polynomial hypothesis (i)) are only possible if (p, p 2 ) = b b + b 2, (3.9)

14 4 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET for some b {a(x) 0} P x, and that if (3.9) holds for some b int P x0, then (3.8) is possible for some a supported on a neighborhood of x 0. Theorem. sharpens this to Lebesgue endpoints and further sharpens the result by replacing the sharp cutoff a with an essentially optimal weight. In one sense, however, the Tao Wright result is aesthetically preferable, because the determinants λ I are somewhat simpler than the derivatives β DΨ x. Under an additional hypothesis, we are able to phrase our result in terms of these determinants. Theorem 3.9. Under the hypotheses (i) and (ii) of Theorem., if b is a minimal element of the Newton polytope P of X, X 2, under the natural, coordinate-wise, partial order on R 2, and deg I = b for some I W n, then b f π (x) f 2 π 2 (x) λ I (x) +b 2 dx N f p f 2 p2, U 0 where (p, p 2 ) = b b +b 2. Proof. For b N 2 and x 0 U 0, set J b (x 0 ) := β det DΨ x0 (0) + b(β)=b b(β)=b β det D Ψ x0 (0). For convenience, we also set J b = 0 for b / N 2. By Lemma 3.6, for all α (0, ) 2 and x 0 U 0, α b λ I (x 0 ) b α b J b (x 0 ) = b P α b J b (x 0 ). (3.0) By our assumption on b and the definition of P, there exists ν (0, ) 2 such that b ν b ν for all b P. Replacing α = (α, α 2 ) with (δ ν α, δ ν2 α 2 ) in (3.0) and sending δ 0, we see that α b λ I (x 0 ) b F α b J b (x 0 ), (3.) where F := {b P : b ν = b ν}. The face F is a line segment, F = {b 0 + tω : 0 t }, for some vector ω perpendicular to ν. Thus (3.) is equivalent to δ δ0 λ I (x 0 ) i δ θi J b θ i (x 0 ), δ > 0, where b = b 0 + θ 0 ω and F N 2 = {b 0 + θ i ω : i m n }. By Lemma 0.3, λ I (x 0 ) J b (x 0 ) + θ j θ 0 θ (J b θi (x 0 )) j θ i (Jb θ j (x 0 )) θ 0 θ i θ j θ i =: J θ 0 (x 0 ), θ i<θ 0<θ j for all x 0 U 0. By Theorem., complex interpolation, and the triangle inequality, f π (x) f 2 π 2 (x) J θ0 b (x) +b 2 dx f p f 2 p2, U 0 and Theorem 3.9 is proved.

15 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 5 4. Quasiextremal pairs for the restricted weak type inequality The purpose of this section is to prove that pairs E, E 2 that nearly saturate inequality (2.) are well approximated as a bounded union of balls parametrized by maps of the form Φ I x, with I a (reordering of a) minimal n-tuple of words. Results of this type had been previously obtained in [4, 22] for other operators and in [2] for a special case (averaging along (t, t 2,..., t n )) of the class of operators here. We begin with some further notation. Notation. We recall the maps Φ I x 0 (t) := e tnxwn e txw (x0 ), I = (w,..., w n ) W n, Ψ x0 := Φ (,2,,2,...) x 0, Ψx0 (t) := Φ (2,,2,,...) x 0 from the previous section. For α (0, ) 2, we define parallelepipeds Q I α := {t R n : t i < α deg wi, i n}, I W n, Q α := Q α (,2,,2,...), Q α := Q (2,,2,,...) α. These give rise to families of balls, B I (x 0 ; α) := Φ I x 0 (Q I α), B n (x 0 ; α) := Ψ x0 (Q α ) Ψ x0 ( Q α ). For I = (w,..., w n ) an n-tuple of words and σ a permutation of S n, we set I σ := (w σ(),..., w σ(n) ). The results of this section will involve a small parameter ε > 0; we will use the notation A B to mean that A ε C B, where C and the implicit constant depend only on N. We will also write A B to mean that A B and B A. Proposition 4.. Let E, E 2 U be open sets, and let ε > 0. Define If Ω := {ρ β } π (E ) π2 (E 2), α j := Ω E j, j =, 2. Ω ε E p E 2 p 2, (4.) there exist a set A Ω of cardinality #A and a minimal n-tuple I W n such that (i) Ω B Iσ (x; cε C α) Ω, x A σ S n (ii) For every x A, σ, σ S n, and y B Iσ (x; cε C α), Φ I σ y Jacobian determinant α deg I det DΦ I σ y α deg I λ I (x) α b Ω, on Q I σ c ε C α, and, moreover, BIσ (x, cε C α) B I σ (y; c ε C α). is one-to-one with By applying Lemma 3.8 with Cε C α X, Cε C α 2 X 2 in place of X, X 2, it suffices to prove the following. Lemma 4.2. Under the hypotheses of Proposition 4., there exist a set A of cardinality #A such that (i) Ω x A Bn (x; Cε C α) Ω (ii) For every x A and y B n (x; Cε C α), I αdeg I λ I (y) α b.

16 6 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Proof of Lemma 4.2. Inequality (4.) implies, after some arithmetic, that Ω α b. (4.2) Conversely, the conclusion of Proposition 2. is equivalent to Ω α b. We will prove this lemma by essentially repeating the proof of Proposition 2., while keeping in mind the constraint (4.2). In the proof, we will extensively use the notations from the proof of Proposition 2.. In the proof of Proposition 2., we only needed to refine n times, but here it will be useful to refine further. Letting x 0 Ω Ω 0, Ψ x0 (t) Ω n, if t j S j ( Ψ x0 (t,..., t j, 0)), j =,..., n Ψ x0 (t) Ω n, if t j S j (Ψ x0 (t,..., t j, 0)), j =,..., n. Thus exactly the arguments leading up to (2.3) imply that Ω β α b(β ) β det DΨ x0 (0) + α b(β ) β det D Ψ x0 (0). As was observed in (2.3), the right side above is at least α b, and by (4.2), it is at most Cε C α b. Let Ω := Ω. We have just seen that α b(β ) β det DΨ x0 (0) + α b(β ) β det D Ψ x0 (0) α b, x 0 Ω, β so by Lemma 3.6, α deg I λ I (x 0 ) α b, x 0 Ω. (4.3) I Moreover, by the proof of Proposition 2., Ω Ω α b. Thus the proof of our lemma will be complete if we can cover a large portion of Ω using a set A Ω. To simplify the notation, we will give the remainder of the argument under the assumption that (4.3) holds on Ω; the general case follows from the same proof, since (4.) holds with Ω replaced by Ω. Our next task is to obtain better control the sets F j, S j ( ) arising in the proof of Proposition 2.. We begin by bounding the measure of these sets. If S n (x) Cε C α n for all x in some subset Ω Ω with Ω Ω, we could have refined so that Ω n Ω, yielding Ω α βn n (Cε C α n ) C ε C α b αb2 2, F n βn n det D t Ψ x0 (t, 0) dt a contradiction to (4.2) for C sufficiently large. Thus we may assume that S n (x) α n on at least half of Ω, and we may refine so that S n (x) α n throughout Ω n. Similarly, we may refine so that S n (x) α n for each x Ω n 2. Thus, after tweaking our method of refinements, for each j n and each t F j, S j (Ψ x0 (t, 0) = {t j R : (t, t j ) F j } α j. (4.4) We have not yet used the gain coming from Lemma 2.2. We will do so now to control the diameter of our parameter set. The key observation is that we may assume that j odd β j and j even β j are both positive. Indeed, this positivity is

17 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 7 trivial for n 4, because if t j = 0 for any < j < n, then det DΨ x0 (t) = 0. Thus the only way our claim can fail is if n = 3 and β = (0, k, 0), but in this case, β det DΨ x0 (0) = 2 k det D Ψ x0 (0), and we can simply interchange the roles of the indices and 2 throughout the argument. Let j be the maximal odd index with β j > 0. Suppose that on at least half of Ω j, J j (x) Cε C S j (x). Then by adjusting our refinement, we may assume that x Ω j implies that J j (x) Cε C S j (x) ; we note that this implies J j (x) Cε C α j. In view of (4.4), Ω α βn+ n α βj++ j+ n βn F j βj j det DΨ x0 (t, 0) ( J j(ψ x0 (t,0)) S j(ψ x0 (t,0)) )( δ)j S j (Ψ x0 (t, 0)) j+ dt Cε Cj α b αb2 2. For C sufficiently large, this gives a contradiction. Thus on at least half of Ω j, J j (x) α j = α, so we may refine so that for each x Ω j, J j (x) α. Repeating this argument for the maximal even index j with β j > 0, we may ensure that for each x Ω j, J j (x) α 2. Finally, replacing Ω n with Ω min{j,j } and then refining, we can ensure that for x 0 Ω 0, j n, and t F j, J j (Ψ x0 (t, 0)) = J(N, {t j R : (t, t j ) F j }) α j. (4.5) Refining further, we obtain a set Ω n Ω 0, with Ω n Ω, such that for each x 0 Ω n, there exists a parameter set such that F x0 [ Cε C α, Cε C α ] [ Cε C α 2, Cε C α 2 ] Ψ x0 (F n ) Ω 0 B(x 0 ; Cε C α), Ψ x0 (F n ) B n (x 0 ; Cε C α). (4.6) We fix a point x 0 Ω n and a parameter set F x0 as above. We add x 0 to A. If (i) holds, we are done. Otherwise, we apply the preceding to Ω \ B n (x; Cε C α), x A and find another point to add to A. By (4.6) and Ω α b, this process stops while #A. This completes the proof of Lemma 4.2, and thus of Proposition 4. as well. 5. Strong-type bounds on a single scale This section is devoted to a proof of the following. Proposition 5.. f π f 2 π 2 dx f p f 2 p2. {ρ β }

18 8 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Proof of Proposition 5.. It suffices to prove the proposition in the special case f i = k 2 k χ E k i, f i pi, i =, 2, with the E k pairwise disjoint, and likewise, the E2 k. Thus we want to bound 2 j+k Ω j,k, Ω j,k := {ρ β } π (Ej ) π 2 (Ek 2 ). j,k We know from Proposition 2. that For 0 < ε, we define Ω j,k E j /p E k 2 /p2. L(ε) := {(j, k) : 2 ε Ej /p E k 2 /p2 Ω j,k 2ε E j /p E k 2 /p2 }. We additionally define for 0 < η, η 2, L(ε, η, η 2 ) := {(j, k) L(ε) : 2 jp E j η, 2 kp2 E k 2 η 2 }. Let ε, η, η 2 and let (j, k) L(ε, η, η 2 ). Set α j,k = (α j,k, αj,k 2 ) := ( Ω j,k E j, Ωj,k E k 2 ). Proposition 4. guarantees the existence of a minimal I W n and a finite set A j,k Ω j,k such that (i) and (ii) of that proposition (appropriately superscripted) hold. (Since there are a bounded number of minimal n-tuples, we may assume in proving the proposition that all of these minimal n-tuples are the same.) Set Ω j,k := Ω j,k B Iσ (x, cε C α j,k ). (5.) σ S n x A j,k Our main task in this section is to prove the following lemma. Lemma 5.2. Fix ε, η, η 2 and set L := L(ε, η, η 2 ). Then π ( Ω j,k ) log ε E j, j Z (5.2) k:(j,k) L j:(j,k) L π 2 ( Ω j,k ) log ε E k 2, k Z. (5.3) We assume Lemma 5.2 for now and complete the proof of Proposition 5.. It suffices to show that for each ε, η, η 2, if L := L(ε, η, η 2 ), then 2 j+k Ω j,k ε a0 η a ηa2 2, (5.4) (j,k) L with each a i positive. Indeed, once we have proved the preceding inequality, we can just sum on dyadic values of ε, η, η 2. We turn to the proof of (5.4). It is a triviality that #L(ε, η, η 2 ) η η 2, so 2 j+k Ω j,k ε 2 j+k E /p E 2 /p2 Define (j,k) L (j,k) L ε(#l)η /p η /p2 2 εη /p η /p 2 2. q i := (p + p 2 )p i, i =, 2, (5.5)

19 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 9 then since p + p 2 = b + b 2 b + b 2 >, we have q i > p i, i =, 2, and q = q 2. Applying Lemma 5.2, (j,k) L 2 j+k Ω j,k (j,k) L ( (j,k) L 2 j+k Ω j,k (j,k) L 2 jq π ( Ω j,k ) q/p) /q ( 2 j+k π ( Ω j,k ) /p π 2 ( Ω j,k ) /p2 (j,k) L η /p /q η /p2 /q2 2 ( 2 jp π ( Ω j,k ) ) /q ( (j,k) L j 2 kq2 π 2 ( Ω j,k ) q2/p2) /q 2 (j,k) L 2 kp2 π 2 ( Ω j,k ) ) /q 2 ( log ε η /p /q η /p2 /q2 2 2 jp E j ) /q ( 2 kp2 E2 k ) /q 2 log ε η /p /q η /p2 /q2 2. Combining this estimate with (5.5) gives (5.4), completing the proof of Proposition 5., conditional on Lemma 5.2. We turn to the proof of Lemma 5.2. We will only prove (5.2), and we will take care that our argument can be adapted to prove (5.3) by interchanging the indices. (The roles of π and π 2 are not a priori symmetric, because their roles in defining the weight ρ are not symmetric.) The argument is rather long and technical, so we start with a broad overview. Assume that (5.2) fails. By Proposition 4., the Ω j,k can be well approximated as the images of ellipsoids (the Q I α j,k ) under polynomials of bounded degree (the Φ I x j,k ). The definition of L ensures that the α j,k, and hence the radii of these ellipsoids, live at many different dyadic scales (this is where the minimality condition in Proposition 4. will be used). On the other hand, the projections π ( Ω j,k ) must have a large degree of overlap (otherwise, the volume of the union would be the sum of the volumes). In particular, we can find a large number of Ω j,k that all have essentially the same projection. These Ω j,k all lie along a single integral curve of X. The shapes of the Ω j,k are determined by widely disparate parameters, the α j,k, and polynomials, the Φ I x j,k. We can take x j,k = e tj,k X (x 0 ), for a fixed x 0, and we use the condition that the projections are all essentially the same to prove that there exists an associated polynomial γ : R R n that is transverse to its derivative γ more than is allowed by Lemma 0.7. We begin by making precise the assertion that many Ω j,k must have essentially the same projection. The main step is an elementary lemma. Lemma 5.3. Let {E k } be a collection of measurable sets, and define E := k Ek. Then for each integer M, k E k ( ) M E + E M M E k E km M. (5.6) k < <k M k

20 20 MICHAEL CHRIST, SPYRIDON DENDRINOS, BETSY STOVALL, AND BRIAN STREET Proof of Lemma 5.3. We review the argument in the case M = 2, which amounts to a rephrasing of an argument from [4]. By Cauchy Schwarz, E k ( E 2 χ Ek 2) 2 ( = E 2 E k + 2 ) 2 E k E k2 k E k k k <k 2 2 E k + 2 E + 2 E ( ) 2 E k E k2 2 ; k k <k 2 Inequality (5.6) follows by subtracting 2 k Ek from both sides. Now to the case of larger M. By Hölder s inequality and some arithmetic, E k ( M i E M M E k l ) M. (5.7) k i= k <...<k i l= Suppose that (5.6) is proved for 2,..., M. Let < i < M. For fixed k < < k i, k i E k E ki E k E ki + E k E ki M i M i E k E ki + ( k i< <k M E k E km k i< <k M E k E km. Inserting this into (5.7), ( E k E M M E k + ) E k E km M, k k < <k M which implies (5.6). k ) M i Our next goal is to reduce the proof of Lemma 5.2, specifically, the proof of (5.2) to the following. Lemma 5.4. For M > M(N) sufficiently large and each A > 0, there exists B > 0 such that for all 0 < δ ε, if j 0 Z and K Z is a (B log δ )-separated set with cardinality #K M and {j 0 } K L, then π ( Ω j0,k ) < A δ A 2 j0p η. (5.8) k K Proof of Lemma 5.2, conditional on Lemma 5.4. We will only prove inequality (5.2). The obvious analogue of Lemma 5.4, which has the same proof as Lemma 5.4, implies inequality (5.3). Fix M = M(N) sufficiently large to satisfy the hypotheses of Lemma 5.4, then fix A > Mp, then fix B = B(M, N, A) as in the conclusion of Lemma 5.4. Let δ = min{δ 0, ε}, with δ 0 to be determined, and let K 0 Z be a finite (B log δ )- separated set. By Lemma 5.3, Lemma 5.4, then the approximation ( #K 0 ) M (#K 0 ) M and the definition of L, π ( Ω j0,k ) M E j0 + Ej0 M M ( π ( Ω j0k ) ) M k K 0 K K 0;#K=M k K M E j0 + Ej0 M M #K0 (A δ A E j0 ) M.

21 ENDPOINT LEBESGUE ESTIMATES FOR AVERAGES ON CURVES 2 Quasiextremality and the restricted weak type inequality give whence δ E j0 p E k 2 p 2 Ω j0k Ω j0k π ( Ω j0k ) p E k 2 p 2, k K 0 k K 0 π ( Ω j0k ) #K 0 δ p E j0. For δ 0 = δ 0 (p, A, M) sufficiently small, δ p M (A δ A ) M, so we must have π ( Ω j0,k ) M E j0, k K 0 which, since K 0 was arbitrary and p, M, A, B all ultimately depend on N alone, implies (5.2). It remains to prove Lemma 5.4. Lemma 5.5. For M > M(N) sufficiently large and each A > 0, there exists B > 0 such that the following holds for all 0 < δ ε. Fix j 0 Z and let K Z be a (B log δ )-separated set with cardinality #K = M and {j 0 } K L. Let x j0k Ω j0k, k K. Then π ( B Iσ (x, cδ C α j0,k )) < A δ A 2 j0p η. (5.9) σ S n k K Proof of Lemma 5.4, conditional on Lemma 5.5. By definition (5.), each Ω j0k is covered by Cε C balls of the form σ S n B Iσ (x, cε C α j0,k ); in fact, by the proof of Proposition 4., it is also covered by Cδ C balls σ S n B Iσ (x, cδ C α j0,k ), for each 0 < δ ε. Thus k K π ( Ω j0,k ) is covered by (Cδ C ) M M-fold intersections of projections of balls, so (5.9) (with a larger value of A) implies (5.8). The remainder of the section will be devoted to the proof of Lemma 5.5. We will give the proof when δ = ε; since an ε-quasiextremal Ω j0k is also δ-quasiextremal for every 0 < δ < ε, all of our arguments below apply equally well in the case δ < ε. The potential failure of π to be a polynomial presents a technical complication. (Coordinate changes are not an option in the non-minimal case.) By reordering the words in I = (w,..., w n ), we may assume that w n = (). Fix k 0 K, and set x 0 = x j0k0. We define a cylinder C := Φ I x 0 (U), U := {(t, t n ) : (t, 0) Q I cε C α j 0 k 0 }. Set U 0 := {(t, 0) U} and define U + := {t U : t n > 0 and for all 0 < s t n, Φ I x 0 (t, s) / Φ I x 0 (U 0 )}, U := {t U : t n < 0 and for all 0 > s t n, Φ I x 0 (t, s) / Φ I x 0 (U 0 )}, and C 0 := Φ I x 0 (U 0 ), C ± := Φ I x 0 (U ± ). Lemma 5.6. The map Φ I x 0 is nonsingular, with det DΦ I x 0 λ I (x 0 ) (α j0k0 ) b deg I, on U. The sets U ± are open, and Φ I x 0 is one-to-one on each of them. Finally, C C + C C 0 C, where C := Φ I x 0 ( U).