Abstract. We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L log L on the unit annulus is bounded on L^{p} for 1 < p < ∞ and is of weak-type (1, 1), generalizing the result of Seeger [9]. The proof is in a similar spirit to that of Christ and Rubio de Francia [3].

THE WEAK-TYPE (1,1) OF L log L HOMOGENEOUS CONVOLUTION OPERATORS

TERENCE TAO

1. Introduction

Let K be a homogeneous convolution kernel on a homogeneous group H, so that so that

K(t ∘ x) = t^{−N}K(x)

for all x ∈ H, t > 0, where t ∘ x is the dilation operation on H and t^{N} is the Jacobian of x ↦ t ∘ x. Let K_{0} be the restriction of K to the unit annulus A_{0} = {x ∈ H : 1 ≤ ρ(x) ≤ 2}, where ρ is a norm associated to the dilation structure.

We consider the question of what the minimal conditions on K_{0} are so that the convolution operator T : f ↦ f ∗ K is of weak-type (1, 1). Here of course

f ∗ K(x) = ∫ f(y)K(y^{−1}x) dy.

Among the necessary conditions known are that K_{0} must be in L^{1}(A_{0}), and T must be bounded on L^{2}. This in turn necessitates that K_{0} must have mean zero. When H is an isotropic Euclidean space, the classical theorem of Calderón and Zygmund [1] shows that T is indeed bounded on L^{2} when K_{0} has mean zero and is either odd and in L^{1}(A_{0}), or even and in the Orlicz space L log L(A_{0}). In particular, we have boundedness on L^{2} whenever K_{0} has mean zero and in L log L. This last condition has been relaxed to H^{1} and beyond; see [6]. Unfortunately, these argumenst rely on the method of rotations and therefore cannot be applied directly to the question of weak (1,1) boundedness (cf. the discussion by R. Fefferman[5]).

It is natural to conjecture that analogous L^{2} results hold for arbitrary homogeneous groups. If K_{0} is odd and in L^{1} one can use the method of rotations and the work of Ricci and Stein [8] on convolution operators on singular sets in homogeneous groups to obtain L^{2} boundedness. In case when K_{0} is merely in L log L we shall use a variant of Littlewood-Paley theory and an iterated T T^{∗} method to answer this conjecture affirmatively:

Theorem 1.1. If K_{0} is in L log L and has mean zero, then T is bounded on L^{2}.

1991 Mathematics Subject Classification. 42B20.

The weak-type (1, 1) question in Euclidean space has been considered by several authors ([2],[3], [7],[9]); recently A. Seeger [9] has shown that T is of weak-type (1, 1) on Euclidean space whenever K_{0} is in L log L and has mean zero. The corresponding questions for odd L^{1} or even H^{1} kernels remain open. We remark that the corresponding (H^{1}, L^{1}) conjecture is false by an example of Mike Christ. For this and further discussion, see the survey in [6].

In this paper we generalize the result in [9] to arbitrary homogeneous groups. Specifically, we show that

Theorem 1.2. If K_{0} is in L log L and T is bounded on L^{2}, then T is of weak-type (1, 1).

Combining the two results and using duality we thus have

Corollary 1.3. If K_{0} is in L log L and has mean zero, then T is bounded on L^{p} for all 1 < p < ∞, and is of weak-type (1, 1).

The methods in [9] rely on the Euclidean Fourier transform and do not appear to be adaptable to non-abelian settings. Our approach is more in the spirit of Christ and Rubio de Francia [3], in that one considers the expression T f as an operator acting on the kernel K_{0} rather than one acting on f. One can then reduce weak (1, 1) boundedness to something resembling a (L^{2}, L^{2}) estimate, which is now amenable to orthogonality techniques such as the T T^{∗} and (T T^{∗})^{M} methods.

The argument can also be used to treat the slightly smoother maximal and square function operators corresponding to L log L generators K_{0}, either by direct modification of the proof, or by using a Radamacher function argument based on the fact that ∑_{i} r_{i}(t)f ∗ K_{i} is of weak-type L^{1} uniformly in t. For these operators the mean zero condition is not required. One can also use the arguments in [9] to weaken the radial regularity on K to a Dini-type continuity condition. We will not pursue these matters here.

2. Notation

We will work exclusively with real-valued functions; none of our functions will be complex-valued.

The letters C, (resp. c, ϵ) will always be used to denote large (resp. small) positive constants htat depend only on the homogeneous group H and any other specified quantities. The values of these constants will change from line to line. We use A ≲ B to denote the statement that A ≤ CB, and A ∼ B to denote the statement that A ≲ B and B ≲ A.

We define a homogeneous group to be a nilpotent Lie group H = R^{n} with multiplication, inverse, dilation, and norm structures

(x, y) ↦ xy, x ↦ x^{−1}, (t, x) ↦ t ∘ x, x ↦ ρ(x)

for x, y ∈ H, t > 0, where the multiplication and inverse operations are polynomial and form a group with identity 0, the dilation structure preserves the group operations and is given in co-ordinates by

t ∘ (x_{1}, …, x_{n}) = (t^{α_{1}}x_{1}, …, t^{α_{n}}x_{n}) (1)

for some constants 0 < α_{1} ≤ α_{2} ≤ … ≤ α_{n}, and ρ(x) equals one on the Euclidean unit sphere, and satisfies ρ(t ∘ x) = tρ(x). It can be shown that Lebesgue measure dx is a Haar measure for this group, and that ρ(x) ∼ ρ(x^{−1}). For further properties of homogeneous groups see e.g. [10].

We call n the Euclidean dimension of H, and the quantity N = α_{1} + … + α_{n} the homogeneous dimension of H.

We will always assume H to be a homogeneous group with Euclidean dimension n > 1; the case n = 1 can of course be treated by classical methods. In addition to the homogeneous group structures mentioned above, we shall also exploit the corresponding Euclidean structures

(x, y) ↦ x + y, x ↦ −x, (t, x) ↦ tx, x ↦ |x|,

together with the Euclidean inner product (x, y) ↦ x · y.

We shall also use the Euclidean structure of the exterior algebra Λ of R^{n}. Recall that Λ is spanned by basis elements of the form

e_{P} = e_{p_{1}} ∧ … ∧ e_{p_{r}}

where 0 ≤ r ≤ n and P = (p_{1}, …, p_{r}) is an increasing subsequence of 1, …, n. We give Λ the usual inner product and norm structure

(∑_{P} a_{P} e_{P}) · (∑_{Q} b_{Q}e_{Q}) = ∑_{P} a_{P} b_{P}

and

| ∑_{P} a_{P} e_{P} | = (∑_{P} |a_{P} |^{2})^{1/2}.

Later on we shall define some further structures on Λ which are more compatible with the non-isotropic dilation (1).

We use ∏_{i=1}^{k} x_{i} to denote the product x_{1} … x_{k}, and ∏_{i=k}^{1} x_{i} to denote the product x_{k} … x_{1}.

We define a left-invariant quasi-distance d on H by d(x, y) = ρ(x^{−1}y). A ball J = B(x_{J}, 2^{j}) with center x_{J} and radius 2^{j} is defined to be any set of the form

J = {x : d(x, x_{J}) < 2^{j}}

for some x_{J} ∈ H and j ∈ Z. If J appears in an expression, then x_{J}, j are always understood to be defined as above. If C > 0, then CJ denotes the ball with the same center as J but C times the radius. We use J_{Δ} to denote the annulus CJ\C^{−1}J.

If E is a finite set, we use #E to denote the cardinality of E; if E is a measurable set, we use |E| to denote the Lebesgue measure of E. Note that |t ∘ E| = t^{N}|E| for all t > 0 and E ⊂ H.

For each t define the scaling map Δ[t] by

Δ[t]f(y) = t^{−N}f(t^{−1} ∘ y);

note that these operators are an isometry on L^{1}.

3. Left-invariant differentiation structures

Let f(t) be a smooth function from R to H. The Euclidean derivative ∂_{t}f(t) can of course be defined by Newton’s approximation

f(t + ε) = f(t) + ε∂_{t}f(t) + ε^{2}O(1)

for ε small. We shall also need a left-invariant derivative ∂_{t}^{L}f(t) defined by

f(t + ε) = f(t)(ε∂_{t}^{L}f(t)) + ε^{2}O(1).

If f(t) is bounded, then the operation of left multiplication of f(t) is bilipschitz, and so we have

(2) |∂_{t}f(t)| ∼ |∂_{t}^{L}f(t)| whenever |f(t)| ≲ 1.

We observe the product rule

∂_{t}^{L}(f(t)g(t)) = ∂_{t}^{L}g(t) + C[g(t)]∂_{t}^{L}f(t) (3)

where the linear transformation C[x] : R^{n} → R^{n} is the derivative of the conjugation map y ↦ x^{−1}yx at the origin. In other words, for all x ∈ H, v ∈ R^{n} we have

x^{−1}(εv)x = εC[x]v + ε^{2}O(1).

The rule (3) is easily verified by expanding f(t + ε)g(t + ε) to first order in two different ways. We note the identities

C[t ∘ x](t ∘ v) = t ∘ (C[x]v), C[x]^{−1} = C[x^{−1}]. (4)

Since C[x] and its inverse are both polynomial in x, we have

(5) |C[x]v| ∼ |v| whenever |x| ≲ 1.

Now suppose F (x) is a smooth function from R^{n} to H. We define the leftinvariant derivative D_{x}^{L}F (x) to be the matrix with columns given by

D_{x}^{L}F (x) = (∂_{x_{1}}^{L}F (x), …, ∂_{x_{n}}^{L}F (x)).

In other words, we have the Newton approximation

F (x + εv) = F (x)(εD_{x}^{L}F (x)v) + ε^{2}O(1).

Since dx is a Haar measure, we see that the determinant of D_{x}^{L}F (x) is equal to the Jacobian of F at x with respect to Lebesgue measure. We note that

det D_{x}^{L}F (x)(e_{1} ∧ … ∧ e_{n}) = ∂_{x_{1}}^{L}F (x) ∧ … ∧ ∂_{x_{n}}^{L}F (x). (6)

The vector field

X(x) = ∂_{t}^{L}(t ∘ x)|_{t=1}.

shall be crucial in our arguments. An equivalent definition is

(1 + ε) ∘ x = x(εX(x)) + ε^{2}O(1). (7)

Note that X commutes with dilation:

X(t ∘ x) = t ∘ X(x). (8)

Since X depends polynomially on x, we therefore have

ρ(X(x)) ≲ ρ(x) (9)

For comparison, we also observe the bound

ρ(∂_{t}(t ∘ x)) ∼ ρ(t ∘ x) (10)

which follows immediately from (1).

The left-invariant derivative ∂_{t}^{L} interacts with dilations via the formula

∂_{t}^{L}(s(t) ∘ f(t)) = s(t) ∘ ∂_{t}^{L}f(t) + \frac{s′(t)}{s(t)}(s(t) ∘ X[f(t)]) (11)

which is verified by expanding s(t + ε) ∘ f(t + ε) to first order in two different ways. Finally, we have

Lemma 3.1. The map X : R^{n} → R^{n} is a polynomial diffeomorphism with Jacobian comparable to 1.

Proof From the monotonicity assumptions on α_{i} and the assumption that dilations preserve the multiplication structure, it is easy to see that the multiplication law (x, y) ↦ xy on H must have the upper diagonal form

\begin{aligned}(xy)_{1} & = x_{1} + y_{1} \\ (xy)_{2} & = x_{2} + y_{2} + P_{2}(x_{1}, y_{1}) \\ (xy)_{3} & = x_{3} + y_{3} + P_{3}(x_{1}, x_{2}, y_{1}, y_{2}) \\ & … \\ (xy)_{n} & = x_{n} + y_{n} + P_{n}(x_{1}, …, x_{n−1}, y_{1}, …, y_{n−1}) \end{aligned} (12)

where P_{2}, …, P_{n} are polynomials.

From (1) and (7) we have

ε(α_{1}x_{1}, …, α_{n}x_{n}) = x(εX(x)) + ε^{2}O(1).

Inserting this into (12) and solving recursively for the components of X(x) we see that

X(x)_{1} = α_{1}x_{1}
X(x)_{2} = α_{2}x_{2} + Q_{2}(x_{1})
…
X(x)_{n} = α_{n}x_{n} + Q_{n}(x_{1}, …, x_{n−1})

for some polynomials Q_{2}, …, Q_{n} which depend on the α_{i}. The claim follows.

4. Proof of Theorem 1.1. Kernel truncation and frequency localization.

We now begin the proof of Theorem 1.1. The heart of the argument is an iterated T T^{∗} method, in the spirit of Christ and Rubio de Francia [3]. We may normalize ∥K_{0}∥_{L log L} = 1.

We first partition the kernel K dyadically. From the identity

K = \frac{1}{ln 2} ∫ Δ[t]K_{0} \frac{dt}{t}

we have the decomposition K = ∑_{j} S_{j}K_{0}, where S_{j} is the operator

S_{j}F = 2^{−j} ∫ φ(2^{−j}t)Δ[t]F dt, (13)

and φ is a bump function adapted to \{t ∼ 1\} such that ∑_{j} 2^{−j}tφ(2^{−j}t) = \frac{1}{ln 2}. Note that

∥S_{j}F ∥_{1} ≲ ∥F ∥_{1} (14)

uniformly in j. From the a priori assumptions on K_{0} we see that the S_{j}K_{0} are all C_{0}^{∞} functions.

We need to show that

∥f ∗ ∑_{j} S_{j}K_{0}∥_{2} ≲ ∥f∥_{2}.

Write K_{0} = ∑_{s≥0} K_{0}^{s}, where

K_{0}^{s} = K̂_{0}^{s} − \frac{χ_{A_{0}}}{|A_{0}|} ∫_{A_{0}} K̂_{0}^{s}

and K̂_{0}^{s} is the portion of K_{0} on the set

2^{2^{s}} ≤ 1 + |K_{0}| < 2^{2^{s+1}}.

Note that each K_{0}^{s} has mean zero. By the triangle inequality and the computation

∑_{s≥0} 2^{s}∥K_{0}^{s}∥_{1} ≲ ∑_{s≥0} 2^{s}∥K̂_{0}^{s}∥_{1} ≲ ∥K_{0}∥_{L log L} = 1

it thus suffices to show that

∥f ∗ ∑_{j} S_{j}K_{0}^{s}∥_{2} ≲ ∥f∥_{2}(2^{s}∥K_{0}^{s}∥_{1} + 2^{s}2^{−ε2^{s}})

for all s ≥ 0.

Fix s. For each integer k, let T_{k} denote the operator

T_{k}f = f ∗ ∑_{j=k2^{s}}^{(k+1)2^{s}−1} S_{j}K_{0}^{s}.

Our task is to show the operator norm estimate

∥ ∑_{k} T_{k}∥ ≲ 2^{s}∥K_{0}^{s}∥_{1} + 2^{s}2^{−ε2^{s}}.

From Young’s inequality and (14) we have

∥T_{k}f∥_{2} ≲ ∥f∥_{2}2^{s}∥K_{0}^{s}∥_{1}

for all integers k. In particular, we have the operator norm estimates

∥T_{k}T_{k′}^{∗}∥, ∥T_{k}^{∗}T_{k′}∥ ≲ (2^{s}∥K_{0}^{s}∥_{1})^{2}

for all k, k′. We will shortly show that

∥T_{k}T_{k′}^{∗}∥, ∥T_{k}^{∗}T_{k′}∥ ≲ 2^{2s}2^{−ε2^{s}|k−k′|} (15)

for |k − k′| ≥ C, where C is a large constant to be determined later. From these estimates the desired bound on ∥ ∑_{k} T_{k}∥ follows from the Cotlar-Knapp-Stein lemma (see e.g. [10]).

It remains to prove (15). We prove only the first estimate, as the second is analogous. We rewrite this as

∥ ∑_{j=k2^{s}}^{(k+1)2^{s}−1} ∑_{j′=k′2^{s}}^{(k′+1)2^{s}−1} f ∗ S_{j}K_{0}^{s} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{2s}2^{−ε2^{s}|k−k′|}∥f∥_{2}

where F̃ denotes the function F̃(x) = F (x^{−1}).

By the triangle inequality it suffices to show that

∥f ∗ S_{j}K_{0}^{s} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{−ε|j−j′|}∥f∥_{2} (16)

for all integers j, j′ for which |j − j′| > C2^{s}.

The next step is to introduce a form of Littlewood-Paley theory, although we shall avoid any explicit use of the Fourier transform. Fix a function ϕ on the unit ball with ∥ϕ∥_{C^{1}} ≲ 1 which has unit mass. We may also assume that ϕ = ϕ̃. For each integer k, write

Ψ_{k} = Δ[2^{k−1}]ϕ − Δ[2^{k}]ϕ.

Note that Ψ_{k} is supported on the ball of radius C2^{k}, has mean zero, and Ψ̃_{k} = Ψ_{k}. Since 1 = ∑_{k} Ψ_{k}, we may write

f ∗ S_{j}K_{0}^{s} ∗ S_{0}K̃_{0}^{s} = ∑_{k} ∑_{k′} f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k} ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}.

Suppose for the moment that we could prove

Proposition 4.1. For any integers j, k, and any L^{∞} function K_{0} on the unit annulus with mean zero, we have

∥f ∗ S_{j}K_{0} ∗ Ψ_{k}∥_{2} ≲ 2^{−ε|j−k|}∥f∥_{2}∥K_{0}∥_{∞}.

Then we would have the estimates

∥f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k}∥_{2} ≲ 2^{2^{s+1}}2^{−ε|j−k|}∥g∥_{2}.

and (by duality)

∥g ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{2^{s+1}}2^{−ε|k′−j′|}∥g∥_{2}.

Combining these two estimates we see that

∥f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k} ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{2^{s+2}}2^{−ε|j−k|}2^{−ε|k′−j′|}∥f∥_{2}. (17)

On the other hand, by Young’s inequality we also have the estimate

∥f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k} ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ ∥f∥_{2}∥S_{j}K_{0}^{s}∥_{1}∥Ψ_{k} ∗ Ψ_{k′}∥_{1}∥S_{0}K̃_{0}^{s}∥_{1}.

Since K_{0} is in L log L, K_{0}^{s} is in L^{1}, and so by (14) we have

∥S_{j}K_{0}^{s}∥_{1}, ∥S_{j′} K̃_{0}^{s}∥_{1} ≲ 1.

Also, from the smoothness and mean zero conditions on Ψ_{k}, Ψ_{k′} we have

∥Ψ_{k} ∗ Ψ_{k′}∥_{1} ≲ 2^{−ε|k−k′|}.

Thus we obtain the bound of

∥f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k} ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{−ε|k−k′|}∥f∥_{2}.

Taking the geometric mean of this with (17) we obtain

∥f ∗ S_{j}K_{0}^{s} ∗ Ψ_{k} ∗ Ψ_{k′} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{2^{s+1}}2^{−ε|j−k|/2}2^{−ε|k−k′|/2}2^{−ε|k′−j′|/2}∥f∥_{2}.

If we then sum this in k and k′ we obtain

∥f ∗ S_{j}K_{0}^{s} ∗ S_{j′} K̃_{0}^{s}∥_{2} ≲ 2^{2^{s+1}}|j − j′|2^{−ε|j−j′|/2},

which gives (16) for some ε > 0, if |j − j′| > C2^{s} for a sufficiently large C.

5. Proof of Theorem 1.1 continued. Iterated T T^{∗} methods.

It thus remains to prove Proposition 4.1. We may normalize so that ∥K_{0}∥_{∞} = 1; by scale invariance we may assume that j = 0.

If k ≥ −C then from the mean zero condition on K_{0} and the smoothness of Ψ_{k} we have

∥S_{0}K_{0} ∗ Ψ_{k}∥_{1} ≲ 2^{−εk}

and the desired bound thus follows from Young’s inequality. We may therefore assume that k < −C.

Fix k = −s for some s > C. Our task is now to show

∥f ∗ S_{0}K_{0} ∗ Ψ_{−s}∥_{2} ≲ 2^{−εs}∥f∥_{2}.

It is possible to use Fourier techniques, taking advantage of the microlocal regularity properties of S_{0}K_{0} to handle the estimate. However, we shall pursue a different approach based on the iterated T^{∗}T method, as we shall need these techniques later on for the (more difficult) weak (1,1) estimate.

Roughly speaking, the idea is as follows. The kernel S_{0}K_{0} is smooth along the “radial” direction, but is otherwise rough. Thus there is no obvious way to exploit the cancellation properties of Ψ_{−s}. However, if we convolve S_{0}K_{0} with itself n times then one should obtain a kernel which is smooth in n separate directions at any given point. Assuming that these directions are linearly independent, the iterated kernel thus has isotropic regularity, and one will pick up the desired 2^{εs} gain by exploiting the moment conditions of Ψ_{−s}. Of course, there will be an exceptional portion of the convolution in which the directions of smoothing are not independent. For this portion one cannot exploit cancellation and one must instead replace everything by absolute values.

We now turn to the details. By the T^{∗}T method, it suffices to show that

∥f ∗ Ψ_{−s} ∗ S_{0}K̃_{0} ∗ S_{0}K_{0} ∗ Ψ_{−s}∥_{2} ≲ 2^{−εs}∥f∥_{2}.

From the operator norm identity ∥T^{∗}T ∥ = ∥(T^{∗}T)^{n}∥^{1/n}, it thus suffices to show that

∥f ∗ Ψ_{−s} ∗ S_{0}K̃_{0} ∗ S_{0}K_{0} ∗ Ψ_{−s} ∗ … ∗ Ψ_{−s} ∗ S_{0}K̃_{0} ∗ S_{0}K_{0} ∗ Ψ_{−s}∥_{2} ≲ 2^{−εs}∥f∥_{2}

for a slightly different value of ε > 0, where the convolution is iterated n = dim H times. By Young’s inequality it suffices to show that

∥Ψ_{−s} ∗ S_{0}K̃_{0} ∗ S_{0}K_{0} ∗ Ψ_{−s} ∗ … ∗ Ψ_{−s} ∗ S_{0}K̃_{0} ∗ S_{0}K_{0} ∗ Ψ_{−s}∥_{1} ≲ 2^{−εs}.

The function Ψ_{−s} ∗ S_{0}K̃_{0} is bounded in L^{1}, and is therefore an average of delta functions in B(0, C). From Minkowski’s inequality, it therefore suffices to show that

∥δ_{w_{1}} ∗ S_{0}K_{0} ∗ δ_{w_{2}} ∗ S_{0}K_{0} ∗ … ∗ δ_{w_{n}} ∗ S_{0}K_{0} ∗ Ψ_{−s}∥_{1} ≲ 2^{−εs}

uniformly for all w_{1}, …, w_{n} ∈ B(0, C).

Fix w = (w_{1}, …, w_{n}). It suffices to show that

|⟨δ_{w_{1}} ∗ S_{0}K_{0} ∗ δ_{w_{2}} ∗ S_{0}K_{0} ∗ … ∗ δ_{w_{n}} ∗ S_{0}K_{0} ∗ Ψ_{−s}, g⟩| ≲ 2^{−εs}

for all test functions g which are normalized in L^{∞}.

Fix g. We write the left-hand side as

| ∫ ∫ ∫ Ψ_{k}(x)g(Φ_{y}(t)x) ∏_{q=1}^{n} K_{0}(y_{q})φ(t_{q}) dydtdx|

where t = (t_{1}, …, t_{n}) ∈ [C^{−1}, C]^{n}, y = (y_{1}, …, y_{n}) ∈ A_{0}^{n}, and

Φ_{y}(t) = ∏_{q=1}^{n} w_{q}(t_{q} ∘ y_{q}). (18)

The treatment of this integral depends on whether the map Φ_{y} : R^{n} → H is degenerate or not. This degeneracy is measured by the Jacobian det D_{t}^{L}(Φ_{y}(t)). Accordingly, we split our estimates into

| ∫ ∫ ∫ Ψ_{−s}(x)η(2^{nεs} det D_{t}^{L}(Φ_{y})(t))g(Φ_{y}(t)x) ∏_{q=1}^{n} K_{0}(y_{q})φ(t_{q}) dydtdx| ≲ 2^{−εs} (19)

and

| ∫ ∫ ∫ Ψ_{−s}(x)[1 − η(2^{nεs} det D_{t}^{L}(Φ_{y})(t))]g(Φ_{y}(t)x) ∏_{q=1}^{n} K_{0}(y_{q})φ(t_{q}) dydtdx| ≲ 2^{−εs}, (20)

where η is a smooth non-negative bump function which equals 1 near 1.

6. Proof of Theorem 1.1 continued. The degenerate portion of the integral.

To show (19) we simply replace everything by absolute values, and use the bounds on K_{0}, g, and φ to reduce to

∫ ∫_{[C^{−1},C]^{n}} ∫_{A_{0}^{N}} |Ψ_{−s}(x)|η(2^{nεs} det D_{t}^{L}(Φ_{y})(t)) dydtdx ≲ 2^{−εs}.

Performing the x integration and taking supremums in the t integral, we reduce to

sup_{t∈[C^{−1},C]^{n}} ∫_{y∈A_{0}^{n}:| det D_{t}^{L}(Φ_{y})(t)|≲2^{−nεs}} dy ≲ 2^{−εs}.

Fix t ∈ [C^{−1}, C]^{n}. From (6) we have

| det D_{t}^{L}(Φ_{y})| = |∂_{t_{1}}^{L}Φ_{y} ∧ … ∧ ∂_{t_{n}}^{L}Φ_{y}|.

From (18), (3) and (11) we have

∂_{t_{q}}^{L}Φ_{y} = t_{q}^{−1}C[Q_{q}](t_{q} ∘ X(y_{q})) (21)

for all q, where Q_{q} is the quantity

Q_{q} = ∏_{j=q+1}^{n} w_{j}(t_{j} ∘ y_{j}).

Since |Q_{q}| ≲ 1, t_{q} ∼ 1, and |y_{q}| ∼ 1, we see from (5) that

|∂_{t_{q}}^{L}Φ_{y}| ∼ 1.

We therefore have

∫_{y∈A_{0}^{n}:| det D_{t}(Φ_{y})(t)|≲2^{−nεs}} dy_{1} … dy_{n} ≤ ∑_{q=1}^{n−1} |E_{q}|, (22)

where E_{q} is the set.

E_{q} = {y ∈ A_{0}^{n} : |∂_{t_{q}}^{L}Φ_{y} ∧ … ∧ ∂_{t_{n}}^{L}Φ_{y}| < 2^{−εs}|∂_{t_{q+1}}^{L}Φ_{y} ∧ … ∧ ∂_{t_{n}}^{L}Φ_{y}|}.

It thus suffices to show that |E_{q}| ≲ 2^{−εs} for each q.

Fix q, and freeze all the y_{j} variables except for y_{q}. From (21) and (5), we see that in order for y to be in E_{q}, X(y_{q}) must live in a boundedly finite union of 2^{−εs}- neighbourhoods of planes. These planes depend only on Q_{q} and ∂_{t_{q+1}}Φ_{y}∧…∧∂_{t_{n}}Φ_{y}, and so are independent of y_{q}. Since y_{q} is bounded, we thus see from Lemma 3.1 that y_{q} lives in a finite union of C2^{−εs}-neighbourhoods of compact hypersurfaces. In particular, the variable y_{q} must range in a set of measure O(2^{−εs}). The desired bound on E_{q} follows by unfreezing the remaining y variables. This concludes the proof of (19).

7. Proof of Theorem 1.1 continued. The non-degenerate portion of the integral.

To finish the proof of Theorem 1.1 we must show (20). Since K_{0} is in L^{∞}(A_{0}), it is in L^{1}, and it suffices to show

| ∫ ∫ Ψ_{−s}(x)g(Φ_{y}(t)x)[1 − η(2^{nεs} det D_{T}^{L}(Φ_{y})(t))] ∏_{i=1}^{n} φ(t_{i}) dtdx| ≲ 2^{−εs}

uniformly in y.

Fix y. We now utilize the moment conditions in the Ψ_{−s} by rewriting Ψ_{−s} as a (Euclidean) divergence of a function which is small in L^{1}. More precisely, we shall use

Lemma 7.1. Let f be a function on B(0, C) with mean zero and ∥f∥_{1} ≲ 1. Then there exists functions f_{1}, …, f_{n} supported on a slightly larger ball B(0, C) with ∥f_{i}∥_{1} ≲ 1 and

f(x) = ∑_{i} ∂_{x_{i}}f_{i}(x).

Proof Without loss of generality we may assume that f is supported on the unit cube [0, 1]^{n}.

When n = 1 the lemma is clear. For n > 1 we write

f(x_{1}, … x_{n}) = ∂_{x_{n}}f_{n}(x_{n}) + F_{x_{n}}(x_{1}, …, x_{n−1})

for all x ∈ [0, 1]^{n}, where

f_{n}(x_{n}) = ∫_{x′_{n}≤x_{n}} f(x′) dx′

and

F_{x_{n}}(x_{1}, …, x_{n−1}) = f(x_{1}, …, x_{n}) − ∫_{x′_{n}=x_{n}} f(x′) dx.

Clearly f_{n}, F_{x_{n}} have bounded L^{1} norm on the unit cube, and F_{x_{n}} has mean zero for each x_{n}. The lemma then follows from induction.

Applying this lemma to f = Ψ_{0} and then rescaling, we may write

Ψ_{−s}(x) = ∑_{i=1}^{n} ∂_{x_{i}}f_{i}(x)

where the functions f_{i} are supported on B(0, C2^{−s}) and satisfy

∥f_{i}∥_{1} ≲ 2^{−α_{i}s}. (23)

We thus need to show that

| ∫ ∫ ∂_{x_{i}}f_{i}(x)g(Φ_{y}(t)x)a(t) dtdx| ≲ 2^{−εs} (24) for all i = 1, 2, …, n, where

a(t) = [1 − η(2^{nεs} det D_{T}^{L}(Φ_{y})(t))] ∏_{i=1}^{n} φ(t_{i}).

Fix i. The idea is to use integration by parts to somehow move the derivative ∂_{x_{i}} onto the smooth function a, so that one can exploit (23) and the L^{∞} control on g.

If we integrate by parts in the x_{i} variable, the left-hand side of (24) becomes

| ∫ ∫ f_{i}(x)∂_{x_{i}}g(Φ_{y}(t)x)a(t) dtdx|.

From (23), it thus suffices to show that

| ∫ ∂_{x_{i}}g(Φ_{y}(t)x)a(t) dt| ≲ 2^{Cεs}, (25)

for all x ∈ B(0, C), if ε is chosen sufficiently small.

Fix x. We now apply the following application of the chain rule, which allows one to convert a derivative of one variable to a derivative on another variable, provided that a certain Jacobian is non-zero.

Lemma 7.2. Let f : R × R^{n} → H and F : H → R be smooth functions. Then

∂_{s}F (f(s, t)) = ∇_{t}F (f(s, t)) · (D_{t}^{L}f(s, t))^{−1}∂_{s}^{L}f(s, t) (26)

whenever det D_{t}^{L}f(s, t) is non-zero.

Proof For any small ε, we have the Newton approximations

F (f(s + ε, t)) = F (f(s, t)) + ε∂_{s}F (f(s, t)) + ε^{2}O(1)
f(s + ε, t) = f(s, t)(ε∂_{s}^{L}f(s, t)) + ε^{2}O(1)
f(s, t + εv) = f(s, t)(εD_{t}^{L}f(s, t)v) + ε^{2}O(1)
F (f(s, t + εv)) = F (f(s, t)) + ε∇_{t}F (f(s, t)) · v + ε^{2}O(1).

Combining all these estimates with v = D_{t}^{L}f(x, t)^{−1}∂_{s}^{L}f(s, t) and letting ε → 0 gives the result.

From this lemma, (25) becomes

| ∫ ∇_{t}g(Φ_{y}(t)x) · (D_{t}^{L}(Φ_{y}(t)x))^{−1}∂_{x_{i}}^{L}(Φ_{y}(t)x)a(t) dt| ≲ 2^{Cεs}.

By another integration by parts and the fact that g ∈ L^{∞}, it suffices to show the uniform estimate

|∇_{t} · [(D_{t}^{L}(Φ_{y}(t)x))^{−1}∂_{x_{i}}^{L}(Φ_{y}(t)x)a(t)]| ≲ 2^{Cεs}.

But this is easily verified, since all variables are compactly supported and all functions are smooth, with norms at most O(2^{Cεs}). The (D_{t}^{L}(Φ_{y}(t)x))^{−1} term is wellbehaved since

| det D_{t}^{L}(Φ_{y}(t)x)| ∼ | det D_{t}^{L}(Φ_{y}(t))| ≳ 2^{−nεs}

on the support of a(t). This completes the proof of Proposition 4.1 and thus Theorem 1.1.

8. Proof of Theorem 1.2. Truncation of the kernel and strong-type estimates.

We now begin the proof of Theorem 1.2. The arguments will be similar in flavor to the ones used to prove Theorem 1.1, but with two major differences. Firstly, because the function f is now only controlled in L^{1}, one is forced (as in [3]) to perform the T T^{∗} method with respect to K_{0} rather than f. Secondly, the Littlewood-Paley operators are not particularly useful in the L^{1} setting, and we cannot reduce to an estimate on a single scale such as Proposition 4.1. Instead, we are forced to consider the interactions between several scales. This will cause an increase in complexity in our arguments. We remark that if one were to treat the maximal function or square function instead of the singular integral, then one could again localize to a single scale; cf. the arguments in [3].

Let K be as in the statement of the theorem. We wish to show that

{|f ∗ K| ≳ α} ≲ α^{−1}∥f∥_{1}∥K_{0}∥_{L log L}.

We may assume that f is a C_{0}^{∞} function. By linearity we may assume that α = 1 and ∥K_{0}∥_{L log L} = 1.

We perform the standard Calderón-Zygmund decomposition of f at height 1 to obtain f = g + ∑_{J} b_{J}, where ∥g∥_{1} ≲ ∥f∥_{1}, ∥g∥_{∞} ≲ 1, the J range over a collection of disjoint balls with ∑_{J} |J| ≲ ∥f∥_{1}, and for each J the functions b_{J} are supported on CJ with

∥b_{J}∥_{L^{1}(CJ)} ≲ |J|, ∫ b_{J} = 0. (27)

Since f is smooth, the collection of J is finite. We may arrange matters so that the b_{J} are smooth.

We now proceed with the standard reduction argument as employed in [3], [9]. As in the proof of Theorem 1.1, we decompose K = ∑_{j} S_{j}K_{0}. We need to

estimate the set where

f ∗ K = g ∗ K + ∑_{s≤C} ∑_{J} b_{J} ∗ S_{j+s}K_{0} + ∑_{s>C} ∑_{J} b_{J} ∗ S_{j+s}K_{0}

is essentially greater than 1; here and in the sequel, j = j(J) is the integer such that J has side-length 2^{j}.

The first term can be handled by the L^{2} boundedness hypothesis and Chebyshev’s inequality because g is in L^{2} with norm O(∥f∥_{1}^{1/2}). The second term is supported in ⋃_{J} CJ, and so that contribution is acceptable since ∑_{J} |CJ| ≲ ∥f∥_{1}.

To handle the remaining term it suffices to show that

|{∑_{s>C} | ∑_{J} b_{J} ∗ S_{j+s}K_{0}| ≳ 1}| ≲ ∑_{J} |J|.

We introduce a cutoff to emphasize the fact that b_{J} ∗ S_{j+s}K_{0} is supported on the annulus (2^{s}J)_{Δ}. Namely, we rewrite the above as

|{∑_{s>C} | ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K_{0})| ≳ 1}| ≲ ∑_{J} |J| (28)

where ψ_{J}(x) = ψ(2^{−j} ∘ (x_{J}^{−1}x)) and ψ is a suitable cutoff function supported on (a slight thickening of) the unit annulus A_{0}. We now claim that (28) will follow from

Proposition 8.1. Let s ≥ C, J be a non-empty finite collection of disjoint balls such that

∑_{J} |J| ≲ 1, (29)

and b_{J} be a collection of smooth functions satisfying (27). Let ψ_{J} be defined as above. Let 1 < p < 2 be an exponent. Then there exists an exceptional set E = E_{s} such that |E| ≲ 2^{−εs} and

∥ ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}F_{J})∥_{L^{p}(E^{c})} ≲ 2^{−εs}(∑_{J} |J|∥F_{J}∥_{2}^{2})^{1/2} (30)

for all functions F_{J} in L^{2}(H).

We will prove this proposition in later sections. For now, we show why Proposition 8.1 implies (28). It suffices by dilation invariance to verify (28) in the case when ∑_{J} |J| ∼ 1. In particular, we may assume that (29) holds.

For each s > C we decompose K_{0} as K_{0} = K^{≤s} +K^{>s}, where K^{≤s} is the portion of K_{0} supported on the set |K_{0}| ≲ 2^{εs/2}. We have to show that

|{| ∑_{s>C} ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K^{>s})| ≳ 1}| ≲ 1 (31)

and

|{| ∑_{s>C} ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K^{≤s})| ≳ 1}| ≲ 1. (32)

To show (31) it suffices by Chebyshev’s inequality to show that

∥ ∑_{s>C} ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K^{>s})∥_{1} ≲ 1.

But from (14), (27), and Young’s inequality one sees that

∥ψ_{J}(b_{J} ∗ S_{j+s}K^{>s})∥_{1} ≲ |J|∥K^{>s}∥_{1},

and the desired estimate follows from (29) and the observation that

∑_{s>C} ∥K^{>s}∥_{1} ≲ ∥K_{0}∥_{L log L} = 1.

To show (32) it suffices by Chebyshev and the observation | ⋃_{s>C} E_{s}| ≲ 1 to show that

∥ ∑_{s>C} ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K^{≤s})∥_{L^{p}((⋃_{s>C} E_{s})^{c})} ≲ 1.

By the triangle inequality it suffices to show

∥ ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}K^{≤s})∥_{L^{p}(E_{s}^{c})} ≲ 2^{−εs/2}

for each s. But this follows from (30) with F_{J} = K^{≤s} for all J, since

(∑_{J} J∥K^{≤s}∥_{2}^{2})^{1/2} ≲ ∥K^{≤s}∥_{2} ≲ ∥K^{≤s}∥_{∞} ≲ 2^{εs/2}.

This completes the derivation of (28) from Proposition 30.

9. Proof of Theorem 1.2 continued. Bounded overlap of dilated balls.

In the remainder of the argument, s > C and 1 < p < 2 will be fixed.

To prove Proposition 8.1, we first prove it under a natural multiplicity assumption on the overlap of the sets 2^{s}J. More precisely, we will show in later sections that

Proposition 9.1. Let J be a non-empty finite collection of disjoint balls such that (29) and

∥ ∑_{J} χ_{C2^{s}J}∥_{∞} ≲ 2^{Ns} (33)

hold. Let b_{J} be a collection of smooth functions satisfying (27), and let ψ_{J} be defined as above. Then we have

∥ ∑_{J} ψ_{J}(b_{J} ∗ S_{j+s}F_{J})∥_{p} ≲ 2^{−εs}(∑_{J} |J|∥F_{J}∥_{2}^{2})^{1/2} (34)

for all functions F_{J} in L^{2}(H).

In this section we show how Proposition 9.1 can be used to imply Proposition 8.1.

Suppose that we are in the situation of Proposition 8.1. We first observe a useful lemma which will also be needed much later in this argument.

Lemma 9.2. Let B ⊂ B(0, C) be any Euclidean ball of side-length at least 2^{−εs}, and define the functions ψ_{J,B} by

ψ_{J,B}(x) = ψ_{B}(2^{−j−s} ∘ (x_{J}^{−1}x))

where ψ_{B} is any bump function which is adapted to B. Then we have

|{∑_{J} ψ_{J,B}(x) > s^{3}2^{Ns}|B|}| ≲ 2^{−εs^{2}}.

Proof It suffices to show the two estimates

∥ ∑_{J} ψ_{J,B}∥_{1} ≲ 2^{Ns}|B| (35)

and

∥ ∑_{J} ψ_{J,B}∥_{BMO} ≲ s2^{Ns}|B| (36)

since the desired distributional estimate then follows from the inequality

|\{|f| ≥ α\} ≲ e^{−Cα/∥f∥_{BMO}} \frac{∥f∥_{1}}{α}

which follows immediately from the John-Nirenberg inequality and the Calderón-Zygmund decomposition.

The estimate (35) follows trivially from the triangle inequality and (29). To show (36), It suffices to show that

∑_{J} osc_{I}ψ_{J,B} ≲ 2^{Ns}|B|

for all balls I, where osc_{I}f = \frac{1}{|I|} ∫_{I} |f − f_{I}| and f_{I} is the mean of f on I.

Fix I, and suppose that I has radius 2^{i}. We divide into three cases, depending on the relative sizes of I, J and 2^{s}J.

We first consider the case where I is larger than 2^{s}J. In this case osc_{I}ψ_{J,B} vanishes unless J is in CI, in which case the oscillation is O(2^{Ns}|J||B|/|I|). Since the J are disjoint and live in CI the total contribution from these balls is acceptable.

Next, we consider the case where I has size between J and 2^{s}J inclusive. For each scale j, there are at most O(2^{Ns}|B|) balls J of size 2^{j} which give a non-zero contribution. Since each ball contributes at most O(1), we are done.

Finally, we consider the case where I has size smaller than J. For each scale j, there are at most O(2^{Ns}|B|) balls J which contribute. But from the smoothness of ψ_{J,B} we see that each ball gives a contribution of O(2^{−ε(j+s−i)}) for some ε > 0. Summing in j we see that this contribution is also acceptable.

By applying this lemma with a ball of size roughly 1 and a non-negative cutoff, we obtain

|{∑_{J} χ_{C2^{s}J} ≳ s^{3}2^{Ns}}| ≲ 2^{−εs^{2}}. (37)

To pass from this to (33) we shall use a sieving argument of Córdoba [4].

For any ball J ∈ J, define the height h(J) to be the number

h(J) = #{J′ ∈ J : 2J ⊂ 2J′}.

We first deal with the contribution of those balls in J with height at least s^{3}2^{Ns}. Clearly, the counting function ∑_{J∈J} χ_{C2^{s}J} is at least s^{3}2^{Ns} on these balls. By the above lemma, the total measure of these balls is O(2^{−εs^{2}}). This implies that the contribution of these balls to (8.1) is supported on a set of measure O(2^{Ns}2^{−εs^{2}}), which can safely be placed in the exceptional set E.

We now consider for each a = 0, 1, …, s^{3} − 1 the contribution of those balls in J of height between a2^{Ns} and (a + 1)2^{Ns}. If we denote this collection of balls by J_{a}, then we claim that J_{a} obeys (33). The estimate (30) would then follow from s^{3} applications of (34) and the triangle inequality.

It remains to verify (33). Let x be an arbitrary point and let J^{x} be the set of all J ∈ J_{a} for which x ∈ 2^{s}J. We wish to show that #J^{x} ≲ 2^{Ns}.

We may of course assume that J^{x} is non-empty. Let J_{0}, J_{1} be elements of J^{x} with minimal and maximal radius 2^{j_{0}} and 2^{j_{1}} respectively. We observe that there are at most O(2^{Ns}) balls in J^{x} of radius comparable to 2^{j_{0}} since the balls are disjoint. Similarly there are at most O(2^{Ns}) balls in J^{x} of radius comparable to 2^{j_{1}}. So it only remains to show that there are at most O(2^{Ns}) balls in J^{x} of radius much larger than 2^{j_{0}} and much smaller than 2^{j_{1}}. But each such ball makes a contribution of 1 to h(j_{1}) − h(j_{0}), which is O(2^{Ns}) by assumption. This completes the derivation of Proposition 8.1 from Proposition 9.1.

10. Proof of Theorem 1.2 continued. Iterated T T^{∗} methods.

Let J, b_{J} satisfy the conditions of Proposition 9.1. To finish the proof of Theorem 1.2 it suffices to show (34). By duality, it suffices to show

(∑_{J} |J|^{−1}∥S_{j+s}^{∗}(b̃_{J} ∗ (ψ_{J}F))∥_{2}^{2})^{1/2} ≲ 2^{−εs}∥F ∥_{p′}

for all test functions F on H, where b̃_{J}(x) = b_{J}(x^{−1}). By the T T^{∗} method, it therefore suffices to show that

∥ ∑_{J} |J|^{−1}ψ_{J}(b_{J} ∗ S_{j+s}S_{j+s}^{∗}(b̃_{J} ∗ (ψ_{J}F)))∥_{p′} ≲ 2^{−εs}∥F ∥_{p′}.

Since S_{j+s}S_{j+s}^{∗} = 2^{−Ns}|J|^{−1}S_{0}S_{0}^{∗}, we may rewrite this as

∥T F ∥_{p} ≲ 2^{−εs}∥F ∥_{p′}, (38)

where

T = 2^{−Ns} ∑_{J} ψ_{J}T_{J}ψ_{J}

and T_{J} is the self-adjoint operator

T_{J}F = \frac{b_{J}}{|J|} ∗ S_{0}S_{0}^{∗}(\frac{b̃_{J}}{|J|} ∗ F)).

Define the smooth functions c_{J} supported on the ball B(0, C) by

c_{J}(v) = |J|^{−1}b_{J}(d_{J}(v))

where d_{J} : B(0, C) → CJ is the map

d_{J}(v) = x_{J}(2^{j} ∘ v); (39)

from (27) we see that

∥c_{J}∥_{L^{1}(B(0,C))} ≲ 1, ∫_{B(0,C)} c_{J} = 0. (40)

Also, note that

S_{0}S_{0}^{∗}F (x) = ∫ φ̃(t)F (t ∘ x) dt

where φ̃ is a bump function adapted to {t ∼ 1}. We may rewrite T_{J} as

T_{J}F (x) = ∫ ∫ ∫ c_{J}(v)φ̃(t)c_{J}(w)F (d_{J}(w)t ∘ (d_{J}(v)^{−1}x)) dwdtdv. (41)

We now define a slightly larger, and non-cancellative, version of T_{J}. For each J, let ψ_{J}^{+} be a slight enlargement of ψ_{J} which is positive on the support of ψ_{J}. Also, apply Lemma 7.1 to find functions c_{J}^{1}, …, c_{J}^{n} supported on B(0, C) such that

c_{J} = ∑_{i=1}^{n} ∂_{x_{i}}c_{J}^{i} (42)

and ∥c_{J}^{i}∥_{1} ≲ 1. If one then defines

c_{J}^{+} = |c_{J}| + ∑_{i=1}^{n} |c_{J}^{i}|,

then one sees that c_{J}^{+} is a non-negative function on B(0, C) with

∥c_{J}^{+}∥_{1} ≲ 1. (43)

Finally, we choose φ^{+} to be any enlargement of φ̃ which is strictly positive on the support of φ̃, and obeys the condition φ^{+}(t) = t^{−N}φ^{+}(t^{−1}). We then define the self-adjoint operator T_{J}^{+} by

T_{J}^{+}F (x) = ∫ ∫ ∫ c_{J}^{+}(v)φ^{+}(t)c_{J}^{+}(w)F (d_{J}(w)t ∘ (d_{J}(v)^{−1}x)) dwdtdv,

and

T^{+} = 2^{−Ns} ∑_{J} ψ_{J}^{+}T_{J}^{+}ψ_{J}^{+}.

Clearly we have the pointwise bounds T_{J}F (x) ≤ T_{J}^{+}F (x) and T F (x) ≤ T^{+}F (x) for all J and non-negative F.

The operator

F (·) ↦ F (d_{J}(w)t ∘ (d_{J}(v)^{−1}·))

is bounded on every L^{p} uniformly in all variables. From this, (41), (43) and Minkowski’s inequality we therefore have ∥T_{J}^{+}F ∥_{p} ≲ ∥F ∥_{p} uniformly in p and J.

We now show

∥T^{+}F ∥_{p} ≲ ∥F ∥_{q} (44)

for all 1 ≤ p ≤ q ≤ ∞; note that this would imply (38) were it not for the 2^{−2εs} factor. By interpolation and duality it suffices to verify this for q = ∞. Since T_{J}^{+} is positivity preserving it suffices to verify this when F is identically 1, i.e. we need to show that

∥2^{−Ns} ∑_{J} ψ_{J}T_{J}^{+}ψ_{J}∥_{p} ≲ 1.

By (33) and Hölder’s inequality we have the pointwise estimate

2^{−Ns} ∑_{J} ψ_{J}T_{J}^{+}ψ_{J} ≲ 2^{−Ns/p}(∑_{J}(T_{J}^{+}ψ_{J})^{p})^{1/p}.

Thus to show (44) it suffices to show

2^{−Ns/p}(∑_{J} ∥T_{J}^{+}ψ_{J}∥_{p}^{p})^{1/p} ≲ 1.

But this follows from the L^{p} boundedness of T_{J}^{+} and (29).

To finish the proof of Theorem 1.2 we must obtain the gain of 2^{−2εs} for (38). To obtain this gain, we will iterate T m times as in the proof of Theorem 1.1, until the kernel is smooth enough to profitably interact with the derivatives in (42). As long as there is enough isotropic smoothing, we hope to bound (in an appropriate sense) T^{m} by 2^{−εs}(T^{+})^{m}. As before, there will be an exceptional portion of T^{m}, but we hope to also show this part is also very small.

Naively, one expects this to work with m = n. However, one runs into two difficulties with this choice. Firstly, if one composes the operator T_{I} with T_{J}, where I is much larger than J, the cutoff functions ψ_{2^{s}J} corresponding to J can seriously truncate the smoothing effect from T_{I}. Secondly, when H is a general homogeneous group, the smoothing effects of the T_{J} will tend to be along almost parallel directions, rather than being isotropically dispersed. Although each of these obstacles is individually tractable, the combined effect of these two obstacles may restrict the smoothing effects discussed above to very short, very parallel arcs, which will not give much isotropic regularity^{1}. To avoid this problem we shall iterate T by considerably more than n times to ensure the existence of at least n untruncated arcs. In fact we shall iterate m = 2^{2n−3} times.

We now turn to the details. Let m be a large number to be chosen later. To show (38) it suffices to show that

∥T^{m}F ∥_{p} ≲ 2^{−εs}∥F ∥_{p′} (45)

for some ε > 0. To see this, observe from the T T^{∗} method and the self-adjointness of T that (45) implies

∥T^{m/2}F ∥_{p} ≲ 2^{−εs}∥F ∥_{2}

with a slightly worse value of ε. On the other hand, from many applications of (44) we have

∥T^{m/2}F ∥_{p} ≲ ∥F ∥_{q}

for all q ≥ p. By interpolation we thus obtain

∥T^{m/2}F ∥_{p} ≲ 2^{−εs}∥F ∥_{p′}

for an even worse value of ε. By iterating this argument 2n−3 times we thus obtain (38) (for a very small value of ε).

1These obstructions also show that in the non-Euclidean case the smoothing effect is global rather than local, and one cannot exhibit this effect by naive microlocal methods (as used in the standard proof of the averaging lemma); this is in contrast with the Fourier transform analysis of [9] in the Euclidean case.

It remains to show (45). Since A^{m} is bounded on L^{2} by (44), it suffices by interpolation to prove this for p = 1. By expanding T^{m}, we thus reduce to showing that

∥2^{−Nms} ∑_{J_{1}...J_{m}∈J} (∏_{i=1}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F ∥_{1} ≲ 2^{−εs}∥F ∥_{∞}.

We use 2^{j_{i}} to denote the radius of J_{i}.

The balls J_{i} may be radically different sizes, and need not be arranged in any sort of monotone order. Nevertheless, we can still extract a subsequence of n balls whose sizes do increase monotonically, and have no smaller balls between elements of the sequence. More precisely, we have

Definition 10.1. Let k = (k_{1}, …, k_{n}) be a strictly increasing n-tuple of integers in {1, …, m}. We say that an m-tuple J = (J_{1}, …, J_{m}) of balls is ascending with respect to k if

j_{k_{q}} ≤ j_{l} for all k_{q} ≤ l ≤ k_{n},

and write this as J ↗ k. Similarly, we say that J is descending with respect to k if

j_{k_{q}} ≤ j_{l} for all k_{1} ≤ l ≤ k_{q},

and write this as J ↘ k.

Lemma 10.2. If m ≥ 2^{2n−3} and J ∈ J^{m}, then there exists a sequence k such that either J ↗ k or J ↘ k.

Proof We first construct an auxilliary sequence l_{1}, …, l_{2n−2} of integers and a sequence S_{1}, … S_{2n−2} of intervals of integers by the following iterative procedure.

Let S_{1} be the interval {1, …, m}. For each p = 1, …, 2n − 2 in turn, we choose l_{p} ∈ S_{p} so that J_{l_{q}} has minimal radius among all the balls {J_{l} : l ∈ S_{p}}. Removing the element l_{p} from S_{p} divides the remainder into two intervals {l ∈ S_{p} : l < l_{p}} and {l ∈ S_{p} : l > l_{p}}; we choose S_{p+1} to be the larger of the two intervals. We then increment p and iterate the above construction.

One can easily show inductively that |S_{p}| ≥ 2^{2n−2−p} for all s, so that all the l_{p} are well defined. Furthermore, one has j_{l_{q}} ≤ j_{l} for all l between l_{q} and l_{2n−2}. One of the sets {p : l_{p} ≤ l_{2n−2}}, {p : l_{p} ≥ l_{2n−2}} has a cardinality of at least n. If the former set is larger, we choose k to be the first n elements of this set and observe that J ↗ k. Otherwise we choose k to be the first n elements of the latter set and observe that J ↘ k.

Temporarily set m = 2^{2n−3}. Order the sequences k lexicographically, so in particular we have k < k′ whenever k_{1} < k′_{1}. For all J ∈ J^{m}, let k_{max}(J) be the largest sequence with respect to this ordering so that either J ↗ k or J ↘ k. From the above lemma we see that k_{max}(J) is well-defined. Since the number of sequences is finite, it suffices to show that

∥2^{−Nms} ∑_{J_{1},... ,J_{m}∈J:k_{max}(J_{1},... ,J_{m})=k}(∏_{i=1}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F ∥_{1} ≲ 2^{−εs}∥F ∥_{∞}

for each k.

Fix k. The purpose of the following (somewhat technical) discussion is to enable us to reduce to the case when k_{1} = 1 and k_{n} = m.

We observe from the lexicographical ordering that the property that k_{max}(J_{1}, …, J_{m}) = k is independent of the choices of J_{i} for 1 ≤ i < k_{1}. We thus abuse notation and write

k_{max}(J_{k_{1}}, …, J_{m}) = k instead of k_{max}(J_{1}, …, J_{m}) = k.

The desired estimate can then be factored as

∥2^{−N(m−k_{1}+1)s} ∑_{J_{k_{1}} ,... ,J_{m}∈J:k_{max}(J_{k_{1}} ,... ,J_{m})=k}( ∏_{i=k_{1}}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})T^{k_{1}−1}F ∥_{1} ≲ 2^{−εs}∥F ∥_{∞}.

By (44), T is bounded on L^{∞}, and it suffices to show that

2^{−N(m−k_{1}+1)s}∥ ∑_{J_{k_{1}} ,... ,J_{m}∈J:k_{max}(J_{k_{1}} ,... ,J_{m})=k}( ∏_{i=k_{1}}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F ∥_{1} ≲ 2^{−εs}∥F ∥_{∞}.

The left-hand side is majorized by

2^{−N(m−k_{1}+1)s}∥ ∑_{J_{k_{1}} ,... ,J_{m}∈J:J↗k or J↘k} |( ∏_{i=k_{1}}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F |∥_{1} (46)

where J = (J_{1}, …, J_{m}), and the choices of J_{1}, …, J_{k_{1}−1} are irrelevant. Since the properties J ↗ k, J ↘ k do not depend on J_{k_{n}+1}, …, J_{m}, we may estimate (46) crudely by

2^{−N(k_{n}−k_{1}+1)s}∥(T^{+})^{m−k_{n}} ∑_{J_{k_{1}} ,... ,J_{kn} ∈J:J↗k or J↘k} |( ∏_{i=k_{1}}^{k_{n}} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F |∥_{1}

By (44) we may discard the (T^{+})^{m−k_{n}} operator. By a re-labelling of J and k, and reducing m to k_{n} − k_{1} + 1, it thus suffices to show that

2^{−Nms}∥ ∑_{J_{1},... ,J_{m}∈J:J↗k or J↘k} |(∏_{i=1}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F |∥_{1} ≲ 2^{−εs}∥F ∥_{∞}

for all m ≤ 2^{2n−3} and all k such that k_{1} = 1, k_{n} = m.

Fix m, k. By duality it suffices to show that

2^{−Nms} ∑_{J_{1},... ,J_{m}∈J:J↗k or J↘k} |⟨(∏_{i=1}^{m} ψ_{J_{i}}T_{J_{i}}ψ_{J_{i}})F_{J}, G_{J}⟩| ≲ 2^{−εs} (47)

for all functions F_{J}, G_{J} in the unit ball of L^{∞}. It suffices to consider the contribution of J ↗ k, since the other contribution then follows by self-adjointness.

For each J ↗ k, we expand the inner product in (47) as

∫ ∫ ∫ ∫ G_{J}(x_{0})F_{J}(x_{m})(∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i})) dx_{0}dwdtdv (48)

where v = (v_{1}, …, v_{n}), w = (w_{1}, …, w_{n}) range over B(0, C)^{n}, t = (t_{1}, …, t_{n}) ranges over [C^{−1}, C]^{n}, dw = ∏_{i=1}^{n} dw_{i}, dv = ∏_{i=1}^{n} dv_{i}, x_{0} ranges over H, and x_{1}, …, x_{m} are defined recursively by

x_{i} = F (d_{J_{i}}(w_{i})t_{i} ∘ (d_{J_{i}}(v_{i})^{−1}x_{i−1})) for i = 1, …, m. (49)

Note that each x_{i} is a function of x_{0} and J_{l}, v_{l}, t_{l}, w_{l} for all l = 1, …, i. We call the variables x_{0} and v_{l}, t_{l}, w_{l} for l = 1, …, m integration variables.

There are many variables of integration here, but the only ones that we shall actively use are the dilation parameters t_{k_{1}}, …, t_{k_{n}} and the translation parameter v_{1}. Accordingly, we define new variables τ = (τ_{1}, …, τ_{n}), y = (y_{1}, …, y_{n}) ∈ R^{n} by τ_{q} = t_{k_{q}} and y = v_{1}.

Each τ_{q} integration is smoothing in one direction. The combined smoothing effect of all the τ variables shall be beneficial provided that the Jacobian

det D_{τ}^{L}(x_{m})

is sufficiently large.

As will become clear later, the natural size for det D_{τ}^{L} is 2^{M_{n}}, where the quantities M_{0}, …, M_{n} are defined by

M_{q} = ∑_{i=1}^{q} α_{i}(j_{k_{i}} + s). (50)

Accordingly, we shall decompose the J ↗ k portion of (47) into

\begin{matrix}∑_{J∈J^{m}:J↗k} | & ∫ ∫ ∫ ∫ G_{J}(x_{0})F_{J}(x_{m})(∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i})) \\ η(2^{δs}2^{−M_{n}} det D_{τ}^{L}(x_{m})) dx_{0}dwdtdv| ≲ 2^{−εs}2^{Nms} \end{matrix} (51)

and

\begin{matrix}∑_{J∈J^{m}:J↗k} | & ∫ ∫ ∫ ∫ G_{J}(x_{0})F_{J}(x_{m})(∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i})) \\ (1 − η(2^{δs}2^{−M_{n}} det D_{τ}^{L}(x_{m}))) dx_{0}dwdtdv| ≲ 2^{−εs}2^{Nms}. \end{matrix} (52)

Here δ > 0 is a small number to be chosen later, and η is a bump function which equals 1 near 1.

We shall prove these two estimates in later sections. But first we must introduce some preliminaries to treat the Jacobian det D_{τ}^{L}(x_{m}), which is the wedge product of n vectors of vastly different sizes.

HOMOGENEOUS CONVOLUTION OPERATORS 23

11. The exterior algebra and non-isotropic scaling It shall be necessary to define some artificial structures on the exterior algebra

Λ of R^{n}. Define a quasi-order ≾ on Λ by

∑_{P} a_{P} e_{P} ≾ ∑_{P} b_{P} e_{P} ⟺ |a_{P} | ≲ b_{P} for all P,

and write w ≈ w′ if w ≾ w′ and w′ ≾ w. We define an absolute value by

∥ ∑_{P} a_{P} e_{P} ∥ = ∑_{P} |a_{P} |e_{P} .

Of course, ∥a∥ = |a| for scalars a. We let 1 denote the vector 1 = (1, …, 1).

We define a non-cancellative analogue ⋄ of the wedge product by

(∑_{P} a_{P} e_{P}) ⋄ (∑_{Q} b_{Q}e_{Q}) = ∑_{P} ∑_{Q} a_{P} b_{Q}∥e_{P} ∧ e_{Q}∥.

Note that ⋄ is bilinear and associative. This operation dominates the wedge product in the following sense: if ω_{1} ≾ a_{1} and ω_{2} ≾ a_{2} then ∥ω_{1} ∧ ω_{2}∥ ≾ a_{1} ⋄ a_{2}, |ω_{1} · ω_{2}| ≲ a_{1} · a_{2}. (53) Finally, we observe that if 1 ≤ r ≤ n and i_{1}, …, i_{r} is any non-decreasing sequence of integers, then

(2^{i_{1}} ∘ 1) ⋄ … ⋄ (2^{i_{r}} ∘ 1) ≈ ∑_{1≤p_{1}<...<p_{r}≤n} 2^{α_{p_{1}} i_{1}+...+α_{pr} i_{r}}e_{p_{1}} ∧ … ∧ e_{p_{r}}. (54)

In particular, from (50) and the hypothesis J ↗ k we have (2^{j_{k_{1}} +s} ∘ 1) ⋄ … ⋄ (2^{j_{kn} +s} ∘ 1) ≈ 2^{M_{n}}e_{1} ∧ … ∧ e_{n}. (55)

If F (x) is a form-valued function of x and C > 0, define

∥(1 + C∇_{x})F (x)∥ = ∥F (x)∥ + C ∑_{i=1}^{n} ∥∂_{x_{i}}F (x)∥.

More generally, we define

∥(A+B∇_{x}+C∇_{y})F (x, y)∥ = A∥F (x, y)∥+B ∑_{i=1}^{n} ∥∂_{x_{i}}F (x, y)∥+C ∑_{q=1}^{n} ∥∂_{y_{q}}F (x, y)∥.

From the product rule and (53) we observe that ∥(1 + C∇_{x})(F · G)∥ ≲ ∥(1 + C∇_{x})F ∥ · ∥(1 + C∇_{x})G∥ (56) and ∥(1 + C∇_{x})(F ∧ G)∥ ≾ ∥(1 + C∇_{x})F ∥ ⋄ ∥(1 + C∇_{x})G∥. (57)

We record the following estimates on the size and derivatives of x_{l}, and det D_{T}^{L}(x_{m}).

Lemma 11.1. If t_{l} ∼ 1 and x_{l} ∈ (2^{s}J_{l})_{Δ} for all 1 ≤ l ≤ m, then

(58)  ∥(1 + ∇_{τ})∂_{y_{i}}^{L}x_{l}∥ ≾ 2^{j_{1}+s−cs} ∘ 1
(59)  ∥(1 + 2^{cs}∇_{y} + ∇_{τ})∂_{τ_{q}}^{L}x_{m}∥ ≾ 2^{j_{kq} +s} ∘ 1
(60)  ∥(1 + 2^{cs}∇_{y} + ∇_{τ}) det D_{τ}^{L}(x_{m})∥ ≲ 2^{M_{n}}
(61)  ∥(1 + 2^{cs}∇_{y} + ∇_{τ})ψ_{J_{l}}(x_{l′})∥ ≲ ψ_{J_{l}}^{+}(x_{l′})

for all i, q, q′ = 1, … n, 1 ≤ l ≤ m, 0 ≤ l′ ≤ l, where c > 0 is a constant independent of ε, δ.

Proof From (49), (39), (3), and (11) we have

∂_{y_{i}}^{L}x_{l} = (t_{l} … t_{0}) ∘ C[x_{J_{1}}^{−1}x_{0}]∂_{y_{i}}^{L}(2^{j_{1}} ∘ y^{−1}). (62)

By (4) and (11), this is becomes

∂_{y_{i}}^{L}x_{l} = (t_{l} … t_{0}2^{j_{1}+s}) ∘ C[2^{−j_{1}−s} ∘ x_{J_{1}}^{−1}x_{0}](2^{−s} ∘ ∂_{y_{i}}^{L}y^{−1}).

Since x_{0} ∈ (2^{s}J_{1})_{Δ}, 2^{−j_{1}−s} ∘ x_{J_{1}}^{−1}x_{0} is bounded. From this, (5) and the observation that |∂_{y_{i}}^{L}y^{−1}| ≲ 1, we thus have

|C[2^{−j_{1}−s} ∘ x_{J_{1}}^{−1}x_{0}](2^{−s} ∘ ∂_{y_{i}}^{L}y^{−1})| ≲ 2^{−cs}.

Inserting this into the previous estimate we thus obtain

∂_{y_{i}}^{L}x_{l} ≾ 2^{j_{1}+s−cs} ∘ 1

which is the first part of (58). The ∇_{τ} portion of (58) then follows from (62) and (10).

We now turn to (59). From (49), (3), and (11), we have

∂_{τ_{q}}^{L}x_{m} = t_{k_{q}}^{−1}((t_{m} … t_{k_{q}}2^{j_{kq} +s}) ∘ X(u_{q})) (63)

where u_{q} is the quantity

u_{q} = 2^{−j_{kq} −s} ∘ (d_{J_{kq}} (v_{k_{q}})^{−1}x_{k_{q}−1})) = (2^{−j_{kq} −s}τ_{q}^{−1}) ∘ (d_{J_{kq}} (w_{k_{q}})^{−1}x_{k_{q}}). (64)

Since x_{k_{q}−1} ∈ (2^{s}J_{k_{q}})_{Δ}, u_{q} is bounded, and so the first part of (59) obtains. To show the 2^{cs}∇_{y} portion of (59), it suffices from (63) and the chain rule to show that ∥∂_{y_{i}}^{L}u_{q}∥ ≲ 2^{−cs}. But by (64), (3), and (11) we have

∂_{y_{i}}^{L}u_{q} = (τ_{q}^{−1}2^{−j_{kq} −s}) ∘ ∂_{y_{i}}^{L}x_{k_{q}},

and the claim follows from (58) and the inequality j_{k_{1}} ≤ j_{k_{q}} arising from the hypothesis J ↗ k.

We now show the ∇_{τ} portion of (59). We consider the ∂_{τ_{q′}} derivatives for q′ ≥ q and q′ < q separately. If q′ ≥ q, then we see from (63) and (10) that

ρ(∂_{τ_{q′}} ∂_{τ_{q}}^{L}x_{m}) ≲ ρ(∂_{τ_{q}}^{L}x_{m}),

so the claim follows from the first part of (59). If q′ < q, then from (63) we have

∂_{τ_{q′}} ∂_{τ_{q}}^{L}x_{m} = t_{k_{q}}^{−1}(t_{m} … t_{k_{q}}2^{j_{kq} +s}) ∘ ∂_{τ_{q′}} X(u_{q}).

Since X is polynomial and u_{q} is bounded, it thus suffices by (2) to show that

|∂_{τ_{q′}}^{L} u_{q}| ≲ 1.

But from (64), (3), and (11), we have

∂_{τ_{q′}}^{L} u_{q} = 2^{−j_{kq} −s} ∘ ∂_{τ_{q′}}^{L} x_{k_{q}−1},

and the claim follows from the first part of (59).

We now turn to (60). It suffices to show that

|(1 + 2^{cs}∇_{y} + ∇_{τ})(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{n}}^{L}x_{m})| ≾ 2^{M}e_{1} ∧ … ∧ e_{n}.

From (57) and (53) we have

|(1 + 2^{cs}∇_{y} + ∇_{τ})(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{n}}^{L}x_{m})|

≾ |(1 + 2^{cs}∇_{y} + ∇_{τ})∂_{τ_{1}}^{L}x_{m}| ⋄ … ⋄ |(1 + 2^{cs}∇_{y} + ∇_{τ})∂_{τ_{n}}^{L}x_{m}|.

By (59) this is majorized by

(2^{j_{k_{1}} +s} ∘ 1) ⋄ … ⋄ (2^{j_{k_{l}} +s} ∘ 1).

The claim then follows from (55).

Finally, we show (61). We can rewrite the desired estimate as

|(1 + 2^{cs}∇_{y} + ∇_{τ})ψ(2^{−j_{l}−s} ∘ (x_{J_{l}}^{−1}x_{l′}))| ≲ ψ^{+}(2^{−j_{l}−s} ∘ (x_{J_{l}}^{−1}x_{l′})).

From the support assumptions on ψ and ψ^{+} we have |(1 + ∇)ψ| ≲ ψ^{+}. Thus by the chain rule and (2), it suffices to show that

|2^{cs}∂_{y_{i}}^{L}(2^{−j_{l}−s} ∘ (x_{J_{l}}^{−1}x_{l′})| ≲ 1

and

|∂_{τ_{q}}^{L}(2^{−j_{l}−s} ∘ (x_{J_{l}}^{−1}x_{l′})| ≲ 1

for all i, q = 1, …, n. We may of course assume that 1 ≤ l′ and k_{q} ≤ l′ since the claims are trivial otherwise. But these estimates follow from (3), (11), (58), and (59), noting that j_{1}, j_{k_{q}} ≤ j_{l} from the hypothesis J ↗ k.

12. Proof of Theorem 1.2 continued. The degenerate portion of the integral.

We now prove (51). For this estimate we do not exploit any cancellation, and crudely majorize the left-hand side as

∑_{J∈J^{m}:J↗k} ∫_{| det D_{τ}^{L}x_{m}|≲2^{−δs}2^{Mn}} ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1})c_{J_{i}}^{+}(v_{i})φ^{+}(t_{i})c_{J_{i}}^{+}(w_{i})ψ_{J_{i}}^{+}(x_{i}) dx_{0}dwdtdv (65)

We discard the ψ_{J_{i}}^{+}(x_{i}) multiplier. We may freeze the t_{i}, v_{i}, w_{i} variables using (43) and reduce ourselves to showing

∑_{J∈J^{m}:J↗k} ∫_{|∂_{τ_{1}}^{L} x_{m}∧...∧∂_{τn}^{L} x_{m}|≲2^{−δs}2^{Mn}} ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1}) dx_{0} ≲ 2^{−εs}2^{Nms} (66)

uniformly over all choices of t_{i} ∼ 1, v_{i} = v_{i}(J_{i}) ∈ B(0, C), w_{i} = w_{i}(J_{i}) ∈ B(0, C). Fix t, w, v. To show (66), we first exclude an exceptional set of x’s.

Definition 12.1. For each x in H, define the set S(x) by

S(x) = {2^{−j−s} ∘ (x_{J}^{−1}x) : J ∈ J , x ∈ (2^{s}J)_{Δ}}.

A point x is said to be good if one has

#S(x) ∩ B ≲ s^{3}2^{Ns}|B| (67)

for all balls of side-length 2^{−εs}.

From (33) we see that S(x) is supported in the unit annulus A_{0} and #S(x) ≲ 2^{Ns} for all x. The property (67) can thus be thought of as a statement about the uniform distribution of S(x).

Let E denote the set of all points in x which are not good. Fortunately, E is very small:

Lemma 12.2. If ε is sufficiently small, we have

|E| ≲ 2^{−εs^{2}}. (68)

Proof Since there are at most O(2^{Cs}) finitely overlapping B which need to be considered for (67), it suffices to show that

|{x ∈ H : #S(x) ∩ B ≳ s^{3}2^{Ns}|B|}| ≲ 2^{−εs^{2}}

for each cube B. But this follows from Lemma 9.2 after some re-arranging.

For each i = 1, …, m, the contribution to (65) of the case when x_{i} ∈ E is bounded by

∑_{J∈J^{m}} ∫ ∏_{l=1}^{n} ψ_{J_{l}}^{+}(x_{l−1})χ_{E}(x_{l}) dx_{0}.

By (33) this is bounded by

2^{Nms} ∫ χ_{E}(x_{l}) dx_{0}.

Thus the contribution to (65) is definitely acceptable by (68) and the observation that x_{0} ↦ x_{l} is a diffeomorphism with Jacobian

det D_{x_{0}}(x_{l}) = (t_{1} … t_{l})^{N} ∼ 1 (69)

Thus it remains only to show that

∑_{J∈J^{m}:J↗k} ∫_{|∂_{τ_{1}}^{L} x_{m}∧...∧∂_{τn}^{L} x_{m}|≲2^{−δs}2^{Mn}} ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1})χ_{E^{c}}(x_{i−1}) dx_{0} ≲ 2^{−εs}2^{Nms}.

For each q = 0, …, n, define P_{q} to be the property that

2^{−qδs/n}2^{M_{q}} ≲ |∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{q}}^{L}x_{m} · e_{1} ∧ … ∧ e_{q}|,

HOMOGENEOUS CONVOLUTION OPERATORS 27

where M_{q} was defined in (??). The desired estimate can thus be rewritten as

∑_{J∈J^{m}:J↗k} ∫_{P_{n} fails} ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1})χ_{E^{c}}(x_{i−1}) dx_{0} ≲ 2^{−εs}2^{Nms}.

Since P_{0} is vacuously true, it thus suffices to show

(70) ∑_{J∈J^{m}:J↗k} ∫_{P_{q−1} holds,P_{q} fails} ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1})χ_{E^{c}}(x_{i−1}) dx_{0} ≲ 2^{−εs}2^{Nms}

for all q = 1, …, n (cf. (22)).

Fix 1 ≤ q ≤ n. We now make the key observation

Proposition 12.3. If we fix x_{0} and all the J_{i} except for J_{k_{q}}, then we have

(71) #{J_{k_{q}} : J ∈ J_{k}^{m}, P_{q−1} holds, P_{q} fails} ≲ 2^{−εs}2^{Ns} provided that x_{k_{q}−1} is good. Proof Suppose J_{k_{q}} is in the set in (71). Since P_{q} fails, we have

|(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{q}}^{L}x_{m}) · (e_{1} ∧ … ∧ e_{q})| ≲ 2^{−qδs/n}2^{M_{q}}.

We rewrite this as |(2^{−j_{kq} −s} ∘ ∂_{τ_{q}}^{L}x_{m}) · a| ≲ 2^{−qδs/n} (72) where the vector a = a_{1}e_{1} + … + a_{q}e_{q} is defined by

a_{l} = 2^{α_{l}(j_{kq} +s)}2^{−M_{q}}(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{q−1}}^{L}x_{m} ∧ e_{l}) · (e_{1} ∧ … ∧ e_{q}).

Since M_{q} = M_{q−1} + α_{q}(j_{k_{q}} + s), we may rewrite this as

a_{l} = ±2^{−(α_{q}−α_{l})(j_{kq} +s)}2^{−M_{q−1}}(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{τ_{q−1}}^{L}x_{m}) · (e_{1} ∧ … ê_{l} … ∧ e_{q}) (73)

where the ê_{l} denotes that the e_{l} term is missing from the wedge product. Since P_{q−1} holds, we thus see that |a_{q}| ≳ 2^{−(q−1)δs/n}. (74)

Also, from (59) we see that |a_{l}| ≲ 2^{−(α_{q}−α_{l})(j_{kq} +s)}2^{−M_{q−1}}((2^{j_{k_{1}} +s} ∘ 1) ⋄ … ⋄ (2^{j_{k_{q−1}} +s} ∘ 1)) · (e_{1} ∧ … ê_{l} … ∧ e_{q}). By (54), we thus have

|a_{l}| ≲ 2^{−(α_{q}−α_{l})(j_{kq} +s)}2^{−M_{q−1}}( ∏_{l′=1}^{l−1} 2^{α_{l′} (j_{k_{l′}} +s)})(∏_{l′=l}^{q−1} 2^{α_{l′+1}(j_{k_{l′}} +s)}).

By (??), this simplifies to

|a_{l}| ≲ 2^{−(α_{q}−α_{l})(j_{kq} +s)} ∏_{l′=l}^{q−1} 2^{(α_{l′+1}−α_{l′})(j_{k_{l′}} +s)}).

Since J ↗ k, we have j_{k_{l′}} + s ≤ j_{k_{q−1}} + s. Applying this inequality, we obtain a telescoping product which simplifies to

a_{l} ≲ 2^{−(α_{q}−α_{l})(j_{kq} −j_{k_{q−1}} )} (75)

From (73) we see that a_{l} is independent of j_{k_{q}} if α_{l} = α_{q}. If α_{l} < α_{q}, then a_{l} can vary with j_{k_{q}}. However, from (75) we see that a_{l} = O(2^{−Cs}) unless j_{k_{q}} = j_{k_{q−1}} + O(s).

In both cases we thus conclude that, up to an error of 2^{−Cs}, the quantities a_{l} can each take at most O(s) values. From this, (74), and (72), we see that 2^{−j_{kq} −s}∘∂_{τ_{q}}^{L}x_{m} lies in a union of O(s^{C}) O(2^{−δs/n})-neighbourhoods of hyperplanes.

From (63) and the fact that the frozen quantities t_{i} are comparable to 1, we thus see that X(u_{q}) also lives in a union of O(s^{C}) O(2^{−δs/n})-neighbourhoods of hyperplanes. From Lemma 3.1 and the boundedness of u_{q}, we thus see that u_{q} lives in a union of O(s^{C}) O(2^{−δs/n})-neighbourhoods of compact hypersurfaces. From (64) we have

u_{q} = (2^{−s} ∘ v_{k_{q}}^{−1})2^{−j_{kq} −s} ∘ (x_{J_{kq}}^{−1} x_{k_{q}−1}),

and so 2^{−j_{kq} −s}∘(x_{J_{kq}}^{−1} x_{k_{q}−1}) also lives in the union of O(s^{C}) O(2^{−δs/n})-neighbourhoods of compact hypersurfaces. The desired cardinality bound on the possible J_{k_{q}} then follows from (67) and a covering argument.

From this proposition, we may estimate the left-hand side of (70) as

∑_{(J_{i})_{i≠kq} ∈J^{m−1}} ∫ 2^{−εs}2^{Ns} ∏_{i≠k_{q}} ψ_{J_{i}}^{+}(x_{i−1}) dx_{0}.

Choose an i_{0} ∈ {1, …, n} not equal to k_{q}. By applying (33) to all J_{i} other than J_{i_{0}}, we estimate this by

2^{−εs}2^{N(m−1)s} ∑_{J_{i_{0}} ∈J} ∫ ψ_{J_{i_{0}}}^{+} (x_{i_{0}−1}) dx_{0}.

This estimate (70) then follows from (29) and (69). This concludes the proof of (51).

13. Proof of Theorem 1.2 continued. The non-degenerate portion of the integral.

It remains to show (52). By (44), it suffices to show that

∑_{J∈J_{k}^{m}} | ∫ ∫ ∫ ∫ G_{J}(x_{0})F_{J}(x_{m}) ∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i}))
η(2^{δs}2^{−M} det D_{τ} (x_{m})) dx_{0}dwdtdv| ≲ 2^{−εs}⟨(T^{+})^{m}1, 1⟩.

By expanding out T^{+}, we see that it suffices to show that

| ∫ ∫ ∫ ∫ G_{J}(x_{0})F_{J}(x_{m}) ∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i}))

η(2^{δs}2^{−M} det D_{τ} (x_{m})) dx_{0}dwdtdv|

≲ 2^{−εs} ∫ ∫ ∫ ∫ ∏_{i=1}^{m}(ψ_{J_{i}}^{+}(x_{i−1})c_{J_{i}}^{+}(v_{i})φ^{+}(t_{i})c_{J_{i}}^{+}(w_{i})ψ_{J_{i}}^{+}(x_{i})) dx_{0}dwdtdv

for all J ↗ k.

Fix J ↗ k. We freeze all the integration variables except for τ_{1}, …, τ_{n}, and y. It thus suffices to show that

| ∫ ∫ G_{J}(x_{0}) ∏_{i=1}^{m} ψ_{J_{i}}(x_{i−1})c_{J_{i}}(v_{i})φ̃(t_{i})c_{J_{i}}(w_{i})ψ_{J_{i}}(x_{i}))F_{J}(x_{m})

η(2^{δs}2^{−M} det D_{τ} (x_{m}))dydτ|

≲ 2^{−εs} ∫ ∫ ∏_{i=1}^{m} ψ_{J_{i}}^{+}(x_{i−1})c_{J_{i}}^{+}(v_{i})φ^{+}(t_{i})c_{J_{i}}^{+}(w_{i})ψ_{J_{i}}^{+}(x_{i})) dydτ

uniformly in the frozen variables.

Fix all the frozen variables. Throwing out all the factors in the above expression which do not depend on y or τ, we reduce to

| ∫ ∫ c_{J_{1}}(y)F_{J}(x_{m})a(y, τ) dydτ| ≲ 2^{−εs} ∫ ∫ c_{J_{1}}^{+}(y)a^{+}(y, τ) dydτ

where

a(y, τ) = (∏_{l=1}^{m} ψ_{J_{l}}(x_{l−1})ψ_{J_{l}}(x_{l}))η(2^{δs}2^{−M} det D_{τ} (x_{m})) ∏_{q=1}^{n} φ̃(τ_{q}) (76)

and

a^{+}(y, τ) = (∏_{l=1}^{m} ψ_{J_{l}}^{+}(x_{l−1})ψ_{J_{l}}^{+}(x_{l})) ∏_{q=1}^{n} φ^{+}(τ_{q}). (77)

We now repeat the argument used to treat (20). By (42) it suffices to show that

| ∫ ∫ ∂_{y_{i}}c_{J_{1}}^{i}(y)F_{J}(x_{m})a(y, τ) dydτ| ≲ 2^{−εs} ∫ ∫ c_{J_{1}}^{+}(y)a^{+}(y, τ) dydτ (78)

for all i = 1, …, n.

Fix i. By an integration by parts, the left-hand side of (78) is majorized by

| ∫ ∫ c_{J_{1}}^{i}(y)F_{J}(x_{m})∂_{y_{i}}a(y, τ) dydτ| + | ∫ ∫ c_{J_{1}}^{i}(y)(∂_{y_{i}}F_{J}(x_{m}))a(y, τ) dydτ|. (79)

We now apply

30 TERENCE TAO

Lemma 13.1. We have the pointwise estimate |(1 + 2^{εs}∇_{y} + ∇_{τ})a(y, τ)| ≲ a^{+}(y, τ). (80) Proof From (56), (76), (77), it suffices to verify

|(1 + 2^{εs}∇_{y} + ∇_{τ})ψ_{J_{l}}(x_{l−1}))| ≲ ψ_{J_{l}}^{+}(x_{l−1})
|(1 + 2^{εs}∇_{y} + ∇_{τ})ψ_{J_{l}}(x_{l}))| ≲ ψ_{J_{l}}^{+}(x_{l})
|(1 + 2^{εs}∇_{y} + ∇_{τ})φ̃(τ_{q})| ≲ φ^{+}(τ_{q})
|(1 + 2^{εs}∇_{y} + ∇_{τ})η(2^{δs}2^{−M} det D_{τ}^{L}(x_{m}))| ≲ 1.

The first two estimates follow from (61), while the third is trivial. The fourth estimate follows from the chain rule and (60) providing that δ ≥ ε.

From this lemma we see that the first term of (79) is acceptable. To treat the second term, it suffices to show that | ∫ (∂_{y_{i}}F_{J}(x_{m}))a(y, τ) dτ| ≲ 2^{−εs} ∫ a^{+}(y, τ) dτ (81) uniformly in y.

Fix y. By Lemma 7.2, we can rewrite the left-hand side as

| ∫ ∇_{τ} F_{J}(x_{m}) · (D_{τ}^{L}x_{m})^{−1}∂_{y_{i}}^{L}x_{m}a(y, τ) dτ|.

Integrating by parts, we see that this is equal to

| ∫ F_{J}(x_{m})∇_{τ} · ((D_{τ}^{L}x_{m})^{−1}∂_{y_{i}}^{L}x_{m}a(y, τ)) dτ|.

Thus to show (81), it suffices to verify the pointwise estimate

∥(1 + ∇_{τ})((D_{τ}^{L}x_{m})^{−1}∂_{y_{i}}^{L}x_{m}a(y, τ))∥ ≲ 2^{−εs}a^{+}(y, τ).

We may of course assume that (y, τ) is in the support of a, so that | det D_{τ}^{L}x_{m}| ≳ 2^{−δs}2^{M}. (82) By (80) and (56), the left-hand side is majorized by

a^{+}(y, τ)|(1 + ∇_{τ})((D_{τ}^{L}x_{m})^{−1}∂_{y_{i}}^{L}x_{m})|

and so it suffices to show that

∥(1 + ∇_{τ})((D_{τ}^{L}x_{m})^{−1}∂_{y_{i}}^{L}x_{m})∥ ≲ 2^{−εs}.

By Cramer’s rule, it suffices to show that

∥(1 + ∇_{τ})\frac{∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{y_{i}}^{L}x_{m} ∧ … ∂_{τ_{n}}^{L}x_{m}}{det D_{τ}^{L}x_{m}} ∥ ≲ 2^{−εs}

for all q, where the numerator is the wedge product of all the ∂_{τ_{q′}}^{L} x_{m}, q′ = 1, …, n, but with the q^{th} term ∂_{τ_{q}}^{L}x_{m} replaced by ∂_{y_{i}}^{L}x_{m}.

Fix q. From the quotient rule, (60) and (82) it suffices (if ε and δ are sufficiently small) to show that

∥(1 + ∇_{τ})(∂_{τ_{1}}^{L}x_{m} ∧ … ∧ ∂_{y_{i}}^{L}x_{m} ∧ … ∂_{τ_{n}}^{L}x_{m})∥ ≲ 2^{−cs}2^{M}

for some constant c > 0. On the other hand, from (58) and the inequality j_{1} ≤ j_{k_{q}} arising from the hypothesis J ↗ k, we see that

∥(1 + ∇_{τ})∂_{y_{i}}^{L}x_{m}∥ ≾ 2^{−cs}(2^{j_{kq} +s} ∘ 1).

Meanwhile, from (59) we have

∥(1 + ∇_{τ})∂_{τ_{q′}}^{L} x_{m}∥ ≾ (2^{j_{k_{q′}} +s} ∘ 1).

The desired estimate thus follows from (57) and (55). This concludes the proof of (52) and thus of Theorem 1.2.

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Department of Mathematics, UCLA, Los Angeles, CA 90024

E-mail address: tao@math.ucla.edu