Three regularity results in Harmonic Analysis

Terence Tao

A Dissertation Presented to the Faculty of Princeton University

in Candidacy for the Degree of Doctor of Philosophy

Recommended for Acceptance

By the Department of Mathematics

June, 1996 \begin{matrix}c \\ ◯ \end{matrix} Copyright by Terence Tao, 1998.

All Rights Reserved

Abstract

In this dissertation we prove certain regularity properties of three unrelated families of operators arising from separate problems in harmonic analysis. The first result concerns the classical Bochner-Riesz operators S^{δ} on Euclidean spaces R^{n} (as well as more general Riesz means on manifolds). By the work of C. Fefferman, P. Tomas, E. Stein, and M. Christ, one can obtain regularity results on these operators from L^{2} restriction theory. We encompass these results by using the restriction theorem to prove an optimal weak-type estimate for the index δ = \frac{n−1}{2(n+1)}, which is the sharpest possible result one can obtain from L^{2} restriction theory alone.

The second result addresses the question of the pointwise convergence of various wavelet sampling methods. In applications one often samples a function f = ∑_{j,k} a_{j,k}ψ_{j,k} by discarding all but the largest wavelet co-efficients, leaving a reconstructed function of the form ∑_{|a_{j,k}|>λ} a_{j,k}ψ_{j,k}. We show that under general conditions the sampled function converges pointwise almost everywhere to the original function as λ → 0. This is achieved by approximating the sampling operator by a linear sampling method whose convergence was established by Kelly, Kon, and Raphael.

Our final result extends recent work by M. Christ, J. L. Rubio de Francia, and A. Seeger, on weak (1, 1) estimates for rough operators in Euclidean spaces, to more general homogeneous groups. This is still a work in progress by the author, and the results presented here are somewhat partial in nature. We show that a homogeneous singular integral convolution operator is of weak-type (1, 1) if it is bounded on L^{2} and the kernel is L log L on the sphere.

Acknowledgements

The author is deeply indebted to his advisor, Elias Stein, for his constant guidance, advice, and suggestions. The author also wishes to thank Charles Fefferman, Anna Gilbert, Ingrid Daubechies, and Andreas Seeger for several useful discussions. Parts of this research were conducted while the author was supported by a Fulbright Graduate award and a Sloan Graduate Fellowship.

To Garth Gaudry, who set me on the road v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

0 General notation 2

1 Weak bounds for Riesz means 5 1 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . 8 2 Interlude on functions of elliptic operators . . . . . . . . . . . 13 3 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . 16

2 Pointwise convergence of wavelet summation methods 18 1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . 20 2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . 24

3 Weak bounds for homogeneous convolution operators 26 1 Preliminary reductions . . . . . . . . . . . . . . . . . . . . . . 28 2 Interlude on the inverse function theorem . . . . . . . . . . . . 34 3 Conclusion of the proof . . . . . . . . . . . . . . . . . . . . . . 38

Contents

1

Chapter 0

General notation

For further details on some of the terms used below, the reader is referred to [30], [20], or [16].

We shall use the symbol C to denote various large positive constants depending only on the inessential variables, and ε to denote various small constants. In the first chapter the inessential variables are the dimension n, the manifold M, and the pseudo-differential operator P. In the second chapter the only inessential variable is the wavelet ψ. In the third chapter the only inessential variable is the underlying homogeneous group H. We shall write x = O(y) or x ≲ y for the statement that |x| ≤ Cy, and write x ∼ y if x ≲ y and y ≲ x.

If x is a variable of integration, we use dx to denote Lebesgue measure and d#(x) to denote counting measure. When E is a set we use |E| to denote the Lebesgue measure of E, and #E to denote the cardinality of E.

On a Euclidean space R^{n} we use ⟨, ⟩ to denote the inner product, and define the Fourier transform by f̂(ξ) = ∫ e^{−2πi⟨x,ξ⟩}f(x) dx. If ϕ is a function on R^{n}, we define the Fourier multiplier operator ϕ(D) by

{ \atop ϕ(D)f}(ξ) = ϕ(ξ)f̂(ξ).

More generally, we shall use the functional calculus to define operators ϕ(P) whenever P is a suitable operator. For example, if P is self-adjoint and has the spectral resolution P = ∑_{j=1}^{∞} λ_{j}e_{j}, where the λ_{j} are a discrete set of eigenvalues and e_{j} are a mutually orthogonal spanning set of projections, then we can formally define ϕ(P) by ϕ(P) = ∑_{j=1}^{∞} ϕ(λ_{j})e_{j}.

Let T be a linear operator. We use T [x, y] to denote the (distributional) kernel of T; thus

T f(x) = ∫ T [x, y]f(y) dy.

If T [x, y] is supported on the set {(x, y) : d(x, y) ≤ CR} where d is an appropriate metric and R is a positive real, then we say that T is local at the scale of R. Finally, if A and A_{+} are linear operators, we say that A_{+} majorizes A if |A[x, y]| ≤ CA_{+}[x, y] for all x, y, or equivalently that A_{+} is positivity preserving and |Af| ≤ CA_{+}|f| for all functions f.

The following notation will be described for balls in a quasi-metric space, but will also be applied to Euclidean dyadic cubes. Suppose that H is a space with a quasi-distance d(x, y), so that the balls B(x, r) = {y : d(x, y) < r} satisfy the usual Vitali-type doubling and covering properties. If I = B(x, r) is a ball (or cube) and c > 0, we write r(I) for the radius (or sidelength) of I, x_{I} for the center of I, cI for the dilate B(x, cr), and I_{Δ} for an annulus of the form CI − C^{−1}I. If I is dyadic in the sense that r is a power of 2, and both I and i appear in an expression, we shall use the convention that i is the integer such that 2^{i} = r(I) unless i is otherwise defined.

By a wavelet we shall mean a 0-regular orthogonal wavelet ψ on R, in the sense of [20]. In other words, we require ψ to be a function on R that is bounded and rapidly decreasing, satisfies the moment condition ∫ ψ(x) dx = 0, and is such that the family {ψ_{j,k}} = {2^{j/2}ψ(2^{j} · −k)} for j, k ∈ Z form an orthonormal basis of L^{2}(R). Note that no regularity assumptions are assumed on ψ; in particular, our definition of a wavelet will include the Haar wavelet ψ = χ_{[0,1/2)} − χ_{[1/2,1)}.

A homogenous group shall be a canonical Euclidean space R^{n} equipped with a group multiplication xy = x+y+P (x, y) and inverse x^{−1} = −x+Q(x), where P and Q are polynomials consisting only of second-order and higher terms, and a family of dilations

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

for some constants 0 < α_{1} ≤ … ≤ α_{n}, such that x ↦ δ ∘ x is a group automorphism for each δ > 0. It can be shown that Lebesgue measure dx is both left- and right-invariant, and |δ ∘ E| = δ^{N}|E| for all measurable sets E, where N = α_{1} + … + α_{n} is the homogeneous dimension of H. We give H a quasi-norm ρ by declaring ρ(x) = 1 on the Euclidean unit sphere S = {x : |x| = 1}, and ρ(δ ∘ x) = δρ(x) for all δ > 0, x ∈ H. The function d(x, y) = ρ(x^{−1}y) thus becomes a quasi-distance.

Let f be a function on some ball B(x, r) in a homogeneous group. We say that f is an normalized bump function adapted to I if f(xr ∘ y) = φ(y) for some non-negative function φ on the unit ball such that ∥∂^{α}φ∥_{∞} ≤ C_{α} for all multi-indices α. One can of course define analogous notions of bump functions adapted to Euclidean cubes, intervals, annuli, etc.

Weak bounds for Riesz means

Consider the Bochner-Riesz multipliers S^{δ} = (1−|D|^{2})_{+}^{δ} in dimension n ≥ 2. It is conjectured that, for 1 ≤ p ≤ 2n/(n + 1), these operators are of strongtype (p, p) precisely when δ > δ(p) = n(1/p − 1/2) − 1/2. Progress on this conjecture can be found in [12], [4], [2], [33]; it remains open for n > 2 and p close to 2n/(n + 1).

A natural strengthening of the above conjecture is the claim that S^{δ(p)} is of weak-type (p, p) whenever^{1} 1 ≤ p < 2n/(n + 1). This claim has been verified when n = 2 [23], or when the operator is restricted to radial functions [5]. In Christ [8, 7], the claim is proved for p < p_{0}, where p_{0} is such that there is a (p_{0}, 2) restriction theorem for the sphere in R^{n}. The well-known restriction theorem of Tomas and Stein states that such a restriction theorem holds precisely when p_{0} ≤ 2(n + 1)/(n + 3). Thus the conjecture is known when^{2} 1 ≤ p < 2(n + 1)/(n + 3).

Our first result is that the conjecture also holds at the endpoint p = 2(n + 1)/(n + 3).

Theorem 1.1. Suppose there is a (p, 2) restriction theorem for the sphere in R^{n}. Then the operator S^{δ(p)} is of weak-type (p, p).

1The necessity of the condition on p was shown in [13].

2Hardy space analogues of this conjecture for p < 1 are also known; see [31].

In fact, under the restriction theorem hypothesis we have the stronger estimate

∫_{M_{p}f(x)≤α} |S^{δ(p)}f(x)|^{2} dx ≲ α^{2−p}∥f∥_{p}^{p}; (1.1)

here M_{p}f = M(|f|^{p})^{1/p} and M is the Hardy-Littlewood maximal function.

We also show the analogous result for Riesz means on compact manifolds. More precisely, suppose that P is a first-order self-adjoint elliptic pseudodifferential operator on a smooth compact manifold M. We say that there is a discrete (p, 2) restriction theorem for P if ∥χ_{[k,k+1]}(P)∥_{(p,2)} ≲ (1 + k)^{δ(p)} for all k > 0, and consider the Riesz means S_{R}^{δ} = (1 − \frac{P}{R})_{+}^{δ}, defined for R, δ > 0. In Sogge [26] it is shown that, if there is a (p, 2) restriction theorem for P, then S_{R}^{δ} is of strong type (p, p) uniformly in R, for δ > δ(p). Also, it was shown by Christ and Sogge [10] that S_{R}^{δ(1)} is weak (1, 1) uniformly in R for any P. Combining ideas from these results with the methods of [7], it was shown by Seeger [22] that S^{δ(p)} was weak (p, p) uniformly in R for p < p_{0}, if one has a (p_{0}, 2) restriction theorem for P.

By modifying the proof of Theorem 1.1, we have an endpoint version of the result in [22].

Theorem 1.2. Suppose there is a (p, 2) restriction theorem for P. Then the operators S_{R}^{δ(p)} are of weak-type (p, p) uniformly in R.

As in the Euclidean case, we have the stronger estimate

∫_{M_{p}f(x)≤α} |S_{R}^{δ(p)}f(x)|^{2} dx ≲ α^{2−p}∥f∥_{p}^{p} (1.2)

holding uniformly in R.

It is known that the (1, 2) restriction theorem holds for any P. Furthermore, under an additional curvature assumption, one has a (p, 2) restriction theorem for all 1 ≤ p ≤ 2(n + 1)/(n + 3). See [25],[27],[28]. Combining these results with (1.2) and interpolating with the trivial case δ ≥ 0, p = 2, one has a corollary:

Corollary 1.1. Suppose the co-spheres {ξ ∈ T_{x}^{∗}M\0 : p(x, ξ) = 1} have everywhere non-vanishing Gaussian curvature for each x ∈ M. Then the operators S_{R}^{δ} are of weak-type (p, p) uniformly in R when 1 ≤ p ≤ 2 and δ ≥ max(δ(p), \frac{n−1}{2}(\frac{1}{p} − \frac{1}{2})).

The method of proof of Theorems 1.1 and 1.2 is a modification of that used by Christ [7], which involved L^{2} Calderón-Zygmund techniques (as used in [12]), a dyadic decomposition of the Bochner-Riesz multiplier with special properties, and the Cauchy-Schwarz inequality. Our main innovations are the replacement of the dyadic decomposition with a decomposition into two terms (which depend on the level of resolution), and a greater reliance on a certain “locality” principle. In the Euclidean case the principle is the trivial observation that if a multiplier m has its inverse Fourier transform supported on a set of width 2^{i}, then the operator m(D) is local at the scale of 2^{i}. On compact manifolds the corresponding principle is that if a function m on R has its Fourier transform supported on an interval of width 2^{i}/R, then the operator m(P) is - apart from an error with good decay - local at the scale of 2^{i}/R.

1 Proof of Theorem 1.1

Fix n ≥ 2, and assume that 1 ≤ p < 2 is such that a (p, 2) restriction theorem holds for the sphere. Write S^{δ} = m^{δ}(D), where m^{δ}(ξ) = (1 − |ξ|^{2})_{+}^{δ} and δ = δ(p) = n(1/p − 1/2) − 1/2. We have to show that m^{δ}(D) is of weak-type (p, p). Since M_{p} is of weak-type (p, p), it suffices by Tchebyshev’s inequality to prove (1.1). By linearity we may assume that α = C^{−1} for some large C to be determined later.

Fix f, and apply the Calderón-Zygmund decomposition at height C to |f|^{p}. This allows us to write f = g + ∑_{I} b_{I}, where ∥g∥_{∞} ≲ 1, the b_{I} are supported on disjoint dyadic cubes I and satisfy ∥b_{I}∥_{p} ∼ |I|^{1/p}, and the I are such that ∑_{I} |I| ≲ ∥f∥_{p}^{p}. Note that M_{p}f(x) ≳ 1 whenever x ∈ ⋃_{I} CI.

Since m^{δ} is bounded, the contribution of g to (1.1) is acceptable. In fact, as m^{δ} is also compactly supported, we may similarly dispose of the contribution of the small cubes^{3}. Specifically, let g̃ be g + ∑_{i≤0} b_{I} and choose a Gaussian P that majorizes m^{δ}. Then the contribution of g̃ is majorized by ∥P (D)g̃∥_{2}^{2}. But the kernel of P (D) is positive, integrable, and radial decreasing, so we have ∥P (D)g̃∥_{p} ≲ ∥g̃∥_{p} and ∥P (D)g̃∥_{∞} ≲ 1. Thus ∥P (D)g̃∥_{2}^{2} ≲ ∥g̃∥_{p}^{p} ≲ ∥f∥_{p}^{p}, which is acceptable.

Write b_{i} = ∑_{r(I)=2^{i}} b_{I} for each i > 0. From the above reductions, it suffices to show that

∫_{R^{n}−⋃_{I} CI} | ∑_{i=1}^{∞} m^{δ}(D)b_{i}(x)|^{2} dx ≲ ∑_{I} |I| (1.3)

to finish the proof. To estimate this expression we make a decomposition of the multiplier m^{δ} for each i > 0, whose properties are summarized in the following lemma:

Lemma 1.1. For each i > 0, there exists a decomposition m^{δ} = m_{i} + η_{i}n_{i} such that:

3This can be thought of as a dual to the more well-known principle that, if the kernel has compact support, then the large cubes can be safely discarded.

(i) The inverse Fourier transforms of m_{i} and n_{i} are supported in the ball of radius 2^{i} around the origin.

(ii) For all ξ ∈ R^{n}, we have ∑_{i=1}^{∞} |η_{i}(ξ)|^{2} ≲ 1.

(iii) For a certain large N, we have |n_{i}(ξ)| ≲ 2^{−δi}(1 + 2^{i}|1 − |ξ||)^{−N}.

Before we prove this lemma, let us see how it implies (1.3). Write m^{δ}(D)b_{i} as m_{i}(D)b_{i} + η_{i}(D)n_{i}(D)b_{i}. By (i), the operator m_{i}(D) is local at the scale of 2^{i}, so m_{i}(D)b_{i} is supported on ⋃_{r(I)=2^{i}} CI and makes no contribution to (1.3). Thus it suffices to prove ∥ ∑_{i=1}^{∞} η_{i}(D)n_{i}(D)b_{i}∥_{2}^{2} ≲ ∑_{I} |I|. By (ii), Plancherel’s formula, and Cauchy-Schwarz, this expression is O(∑_{i=1}^{∞} ∥n_{i}(D)b_{i}∥_{2}^{2}). So it will suffice to show that

∥n_{i}(D)b_{i}∥_{2}^{2} = ∥ ∑_{r(I)=2^{i}} n_{i}(D)b_{I}∥_{2}^{2} ≲ ∑_{r(I)=2^{i}} |I|

for each i > 0. However, by (i) the operator n_{i}(D) is local at the scale of 2^{i}, so the individual n_{i}(D)b_{I} have almost disjoint support. Thus we only need prove the above estimate for a single cube. But by Plancherel’s formula, (iii), and the restriction theorem, we have

∥n_{i}(D)b_{I}∥_{2}^{2} = ∫_{R_{n}} |n_{i}(ξ)|^{2}|b̂_{I}(ξ)|^{2} dξ
≲ ∫_{0}^{∞} 2^{−2δ(p)i}(1 + 2^{i}|1 − r|)^{−2N} ∫_{S} |b̂_{I}(rω)|^{2} dω r^{n−1}dr
≲ 2^{−2δ(p)i} ∫_{0}^{∞}(1 + 2^{i}|1 − r|)^{−2N}r^{−2n/p′}∥b_{I}∥_{p}^{2} r^{n−1}dr
≲ 2^{−2δ(p)i}2^{−i}|I|^{2/p}
= |I|

as desired, if N is chosen to be sufficiently large.

It remains only to prove Lemma 1.1. Following [7], we start by choosing a smooth radial function ϕ supported on the unit ball and equal to 1 on the half-unit ball, such that ∫_{R_{n}} ϕ̂(ξ) ⟨ξ, e_{1}⟩_{+}^{δ} dξ = 0 for some unit vector e_{1}. Write ϕ_{2^{i}} = ϕ(2^{−i}·), and set m_{i} equal to m^{δ} ∗ ϕ̂_{2^{i}},so that m_{i} obeys (i). Furthermore, since ϕ̂_{2^{i}} is an approximation to the identity of “thickness” 2^{−i}, one has by repeated integration by parts that

|m^{δ}(ξ) − m_{i}(ξ)| ≲ 2^{−δi}(2^{i}|1 − |ξ||)^{−N} (1.4)

for |1 − |ξ|| > 2^{−i} and any large N. When |1 − |ξ|| ≲ 2^{−i}, we claim that the additional estimate

|m^{δ}(ξ) − m_{i}(ξ)| ≲ 2^{−δi}(2^{−i} + 2^{i}|1 − |ξ||)^{ε} (1.5)

holds. Since (1.5) holds for m^{δ}(ξ), it suffices to prove this estimate for m_{i}(ξ). By differentiating under the integral sign we see that the gradient of m_{i}(ξ) is O(2^{i}2^{−δi}) in the region of interest, so it suffices to prove (1.5) for ξ on the unit sphere. Since m_{i} is radial, it suffices to prove it for ξ = e_{1}. But this follows by approximating m^{δ}(e_{1} − ξ) by 2^{δ}⟨ξ, e_{1}⟩_{+}^{δ} and using the assumed moment condition for ϕ̂.

We now construct n_{i} so that (i) holds and

2^{−δi}(1 + 2^{i}|1 − |ξ||)^{−M} ≲ n_{i}(ξ) ≲ 2^{−δi}(1 + 2^{i}|1 − |ξ||)^{−N}

for some large M,N, which we reserve the right to choose later. Once we do this, it can easily be seen from (1.4), (1.5), and the above estimates that the function η_{i} = (m^{δ} − m_{i})/n_{i} satisfies (ii), and so all the required properties will hold.

Let N be a large even integer to be chosen later and consider the onedimensional function ψ(ξ) = (\frac{ξ−sin ξ}{ξ^{3}})^{N/2}, which is comparable to (1 + |ξ|)^{−N} and has a compactly supported inverse Fourier transform. By taking the tensor product of n copies of ψ, averaging over the orthogonal group and then rescaling, we can construct a function ψ_{n} on R^{n} which is positive and comparable to (1 + |ξ|)^{−N−n+1}, and whose inverse Fourier transform is supported on the unit ball. It is then a routine matter to check that the function n_{i} = 2^{−δi}2^{(n−1)i}ψ_{n}(2^{i}·) ∗ dω satisfies all the required properties, with M = N + n − 1; here dω is surface measure on the unit sphere. This completes the proof of Lemma 1.1, and the theorem follows. ■ Remarks 1.1.

1. The method of proof of Theorem 1.1, in particular the construction of m_{i} and the use of the Cauchy-Schwarz inequality, is essentially due to Christ [7]. However, a slightly different decomposition is used in [7]; in our notation, it would be m^{δ} = m_{i} + ∑_{s=0}^{∞}(m_{i+s+1} − m_{i+s}), where the individual m_{i+s+1} −m_{i+s} play much the same role in [7] as the function m^{δ} − m_{i} = η_{i}n_{i} does here.

2. This method can provide alternate proofs of the weak (1, 1) type of various oscillatory convolution operators. For example, the weak boundedness of S^{(n−1)/2} follows immediately from Theorem 1.1, as the (1, 2) restriction theorem is trivial. Another example is the one-dimensional operator f ↦ f ∗ e^{iπx^{2}}/x. The multiplier, which is roughly e^{−iπξ^{2}}/ξ, can be decomposed for each i > 0 as m_{i} + η_{i}n_{i}, where (i) [for m_{i} only] and (ii) of Lemma 1.1 holds and n_{i}(ξ) = 2^{−i}(1 + 2^{−i}|ξ|)^{−1+ε}. We omit the details.

3. By interpolation, (1.1) will hold with δ(p) replaced by δ, for 1 ≤ p ≤ 2 and δ ≥ max(δ(p), \frac{n−1}{2}(\frac{1}{p} − \frac{1}{2})). Simple calculations involving bump functions on small balls or on thin tubes show that this region is best possible. In the interior of this region (1.1) also follows from the weighted inequality

∫_{R_{n}} |S^{δ}g(x)|^{2}ψ(x) dx ≲ ∫_{R_{n}} |g|^{2}M_{r}ψ(x) dx (1.6)

proven by Christ [6]; here r = p/(2 − p) and ψ is any positive function. Indeed, the adjoint of (1.6) is ∫_{R_{n}} \frac{|S^{δ}f(x)|^{2}}{M_{r}ψ(x)} dx ≲ ∫_{R_{n}} \frac{|f|^{2}}{ψ(x)} dx, which yields (1.1) after substituting ψ = |f|^{2−p} and using Tchebyshev’s inequality. Unfortunately (1.6) fails for endpoint values of δ, as the bump function examples will show.

4. We have seen how restriction theorems for the sphere can imply weaktype endpoint bounds for Bochner-Riesz multipliers. A converse to that observation holds: if 1 ≤ p < 2n/(n + 1) is such that S^{δ(p)} is of weak-type (p, p), then there is a strong (p, p) restriction theorem for the sphere. The proof can be sketched^{4} as follows. Assume the weak estimate. It suffices to show that ∥f̂∥_{L^{p}(S)} ≤ C∥f∥_{p} for all f supported in a ball of radius r, where the constant C is independent of r. We may heuristically replace the kernel of the Bochner-Riesz multiplier by |x|^{−n/p}e^{±2πi|x|} away from the origin. This will mean that S^{δ(p)}f(x) behaves like e^{±2πi|x|}|x|^{−n/p}f̂(±x/|x|) when |x| ≫ r^{C}. The claim then follows from observing that the weak-L^{p} quasi-norm of this last expression is comparable to ∥f̂∥_{L^{p}(S)} uniformly in r. In a similar fashion one can recover the (p, 2) restriction theorem from the inequality (1.1).

4A similar argument may be found in [32].

2 Interlude on functions of elliptic operators

Let M be a smooth compact connected manifold without boundary of dimension n > 1, endowed with a smooth positive measure dx. It will be convenient to fix a smooth global Riemannian metric d(x, y) on M.

Let P = P (x, D) be a first-order self-adjoint pseudo-differential operator on M, which we assume to be elliptic in the sense that the principal symbol p(x, ξ) = lim_{λ→∞} λ^{−1}P (x, λξ) of P exists and is positive for non-zero elements ξ of the cotangent space T_{x}^{∗}M. We will assume without loss of generality that the λ_{j} are nonnegative.

If m is a tempered function on R, one can assign an operator m(P) using the functional calculus of P, as described in the chapter on Notation. The purpose of this section is to relate the “locality” (i.e. decay) properties of m(P)[x, y] to those of m̂.

Proposition 1.1. Suppose m is a function of R such that m̂ is supported on [−τ, τ] for some sufficiently small τ. Then

(i) One has |m(P)[x, y]| ≲ ∥m̂∥_{1}τd(x, y)^{−n−1} whenever d(x, y) > Cτ.

(ii) If in addition m(τ·) is in S^{−∞}, then |m(P)[x, y]| ≲ τ^{−n}(1 + \frac{d(x,y)}{τ})^{−n−1} for all x, y.

Part (i) is a slightly stronger version of a result proved in [10]. The key idea is to exploit the fact that e^{itP} is local when t = 0. Further improvements in this lemma are possible but will not be pursued here.

Proof. We first prove (i). By the triangle inequality we may assume that m(P) is of the form e^{itP}. But e^{itP} = I + ∫_{0}^{t} \frac{d}{ds}e^{isP} ds, so it suffices to show that |\frac{d}{dt}e^{itP} [x, y]| ≲ d(x, y)^{−n−1} when d(x, y)/t is large.

To do this we shall apply a well-known approximation of e^{itP} by a Fourier integral operator. Specifically, if ϵ > 0 is sufficiently small, the (distributional) kernel of e^{itP} can be written for |t| ≤ ϵ in the form

e^{itP} [x, y] = η_{ϵ}(x, y) ∫_{T_{y}^{∗}M} e^{iϕ(x,y,ξ)}e^{itp(y,ξ)}q(t, x, y, ξ) dξ + E_{ϵ}(t, x, y), (1.7)

where η_{ϵ}(x, y) is supported on a Cϵ-neighbourhood of the diagonal of M ×M, ϕ(x, y, ξ) is a smooth real function such that |∇_{ξ}ϕ| ≥ C^{−1}d(x, y) and |∇_{ξ}^{k}ϕ| ≤ C_{k}d(x, y)|ξ|^{1−k} for small d(x, y) and k ≥ 1, q(t, x, y, ξ) is an S^{0} symbol in ξ, and E_{ϵ}(t, x, y) is a C^{∞} error. One obtains (1.7) from a construction, originally due to Hörmander, of a local parametrix for the Cauchy problem for P. Details can be found in [18], [28].

If d(x, y)/t is sufficiently large and ϵ is sufficiently small, then the phase Ψ(t, x, y, ξ) = ϕ(x, y, ξ) + tp(y, ξ) will satisfy the same sort of bounds as ϕ on the support of η_{ϵ}. If we then differentiate (1.7), we obtain

\frac{d}{dt}e^{itP} [x, y] = η_{ϵ}(x, y) ∫ e^{iΨ}[\frac{dq}{dt} + pq] dξ + O(1).

Since the integrand is O(1 + |ξ|), the integral on the region {|ξ| ≲ d(x, y)^{−1}} is O(d(x, y)^{−n−1}). Outside of this region, the term in brackets is – after a suitable rescaling by d(x, y) – a symbol of order 1, and so this part of the integral is also O(d(x, y)^{−n−1}), by the standard estimates on non-stationary oscillatory integrals. Thus one has the desired bound on \frac{d}{dt}e^{itP} [x, y], which proves (i).

To show (ii), it suffices by (i) to prove that |m(P)[x, y]| ≲ τ^{−n}. From (1.7) we have

m(P)[x, y] = η_{ϵ}(x, y) ∫ e^{iϕ(x,y,ξ)}( ∫ e^{itp(y,ξ)}q(t, x, y, ξ)m̂(t) dt) dξ + O(1).

But by the usual non-stationary phase estimates and the fact that p(y, ξ) ∼ |ξ|, the inner integral will be O_{N}(1 + τ|ξ|)^{−N} for all N > 0. Taking N > n thus gives the result. ■

We now show that operators with the decay of 1.1(i) are effectively local in a specialized L^{p} sense.

Proposition 1.2. Let {I} be a collection of disjoint “cubes” in M. Suppose that for each I there is a function b_{I} supported on I and a function m_{I} whose Fourier transform is supported on [−2^{i}, 2^{i}] such that ∥b_{I}∥_{1} ≲ |I| and ∥m̂_{I}∥_{1} ≲ 1. Then, for 1 ≤ p < ∞, one has

∥ ∑_{I}(m_{I}(P)b_{I})χ_{CI^{c}}∥_{p}^{p} ≤ C_{p} ∑_{I} |I|.

In particular, if the CI are almost disjoint, then

∥ ∑_{I} m_{I}(P)b_{I}∥_{p}^{p} ≤ C_{p}(∑_{I} ∥m_{I}(P)b_{I}∥_{p}^{p} + ∑_{I} |I|).

Proof. It suffices to prove the first inequality. By Proposition 1.1(i) and elementary estimates, we see that |m_{I}(P)b_{I}(x)χ_{CI^{c}}(x)| ≤ C(Mχ_{I}(x))^{q}, where q = 1 + 1/n > 1. Thus

∥ ∑_{I}(m_{I}(P)b_{I})χ_{CI^{c}}∥_{p}^{p} ≤ C∥(∑_{I}(Mχ_{I})^{q})^{1/q}∥_{pq}^{pq}

≤ C_{p}∥(∑_{I}(χ_{I})^{q})^{1/q}∥_{pq}^{pq}

= C_{p} ∑_{I} |I|

as desired, where we have used the vector-valued maximal theorem of Fefferman and Stein [14], adapted to general manifolds.^{5} ■ Let 1 ≤ p < 2 be such that there is a (p, 2) restriction theorem for P. We wish to show that S_{R}^{δ} is of weak type (p, p) uniformly in R, where δ = δ(p). Since the proof is almost identical to that of Theorem 1.1 except for a rescaling and some minor technical complications, we give only a brief sketch.

5One can also use BMO methods to prove this estimate; cf. the proof of Lemma 3.1.

We first note that, since each projection operator is smoothing, the claim is clearly true when R is bounded. Thus we may assume that R > C for some large C to be determined later.

Secondly, since the eigenvalues of P are nonnegative, we may write S_{R}^{δ} = m_{R}^{δ}(P), where m_{R}^{δ}(λ) = ζ(λ/R)(1 − λ/R)_{+}^{δ} and ζ is smooth, compactly supported, and equal to 1 on [0, 1]. Note that ∥m̂_{R}^{δ}∥_{1} = O(1).

Fix f. It suffices to show that (1.2) holds for α = C^{−1}. We may assume that f is supported on an arbitrarily small subset of M, which we can identify with a subset of R^{n}. We then decompose f = g + ∑_{I} b_{I} as before except that we use a rescaled dyadic mesh, so that the cubes have sidelength 2^{i}/R instead of 2^{i}.

Since m_{R}^{δ} is bounded and supported in [−CR, CR], we may dispose of the contributions of g and the cubes of sidelength at most 1/R. Instead of a Gaussian, one uses an S^{−∞} symbol ψ on R such that ψ̂ is supported on [−C, C] and ψ_{R} = ψ(R^{−1}·) majorizes m_{R}^{δ}. By Proposition 1.1(ii), the kernel of ψ_{R}(P) is majorized by a constant multiple of a positive and “radial decreasing” approximation to the identity of “thickness” 1/R, and one can use the argument in the proof of Theorem 1.1.

The appropriate analogue of Lemma 1.1 is the following decomposition of m_{R}^{δ}:

Lemma 1.2. For each i > 0, there exists a decomposition m_{R}^{δ} = m_{i} + η_{i}n_{i} such that:

(i) The functions m̂_{i} and n̂_{i} are supported on [−2^{i}/R, 2^{i}/R], and each have an L^{1} norm which is O(1).

(ii) For all λ ∈ R, we have ∑_{i=1}^{∞} |η_{i}(λ)|^{2} ≲ 1.

(iii) For a certain large N, we have |n_{i}(λ)| ≲ 2^{−δi}(1 + 2^{i}|1 − λ/R|)^{−N}.

Apart from the last part of (i), Lemma 1.2 is proved in exactly the same way as Lemma 1.1. Since m̂_{i} is by construction the product of m̂_{R}^{δ} and a bounded expression, we see that the L^{1} norm of m̂_{i} is O(1). To prove the corresponding bound for n̂_{i}, use (iii) to observe that n̂_{i} is uniformly bounded by O(2^{−δi}2^{−i}R), and therefore has a L^{1} norm of O(2^{−δi}) = O(1), as desired.

The rest of the argument, deducing (1.2) from the above decomposition, the Cauchy-Schwarz inequality, and the restriction theorem, is virtually unchanged, apart from a rescaling by R. There are only two technical differences: firstly, as the restriction theorem is now discrete, one replaces a certain integral by its Riemann sum instead; and secondly, because the analogous “locality” principle (i.e. Proposition 1.2 for p = 2 combined with Lemma 1.2(i)) is not absolute, one incurs further error terms which are O(∑_{I} |I|), but these errors are acceptable in proving (1.2). We omit the details. ■

Pointwise convergence of wavelet summation methods

Let ψ be a wavelet of the form discussed in the chapter on notation. To each function f ∈ L^{p}(R), 1 < p < ∞, one can associate a wavelet expansion f ∼ ∑_{j,k} a_{j,k}ψ_{j,k}, where a_{j,k} = ⟨f, ψ_{j,k}⟩ for j, k ∈ Z. It is well-known [20] that this series converges unconditionally in L^{p} norm to f, but the pointwise convergence is more delicate, being dependent on the summation method used. In this chapter we show the pointwise convergence of the following three partial summation operators:

the wavelet projections  P_{J}f = ∑_{j<J;k} a_{j,k}ψ_{j,k};
the hard sampling operators T_{λ}f = ∑_{|a_{j,k}|>λ} a_{j,k}ψ_{j,k};
the soft sampling operators T̃_{λ}f = ∑_{|a_{j,k}|>λ} sgn(a_{j,k})(|a_{j,k}| − λ)ψ_{j,k}.

Theorem 2.1. If f ∈ L^{p}(R) for some 1 < p < ∞, then for almost every x one has

lim_{J→∞} P_{J}f(x) = lim_{λ→0} T_{λ}f(x) = lim_{λ→0} T̃_{λ}f(x) = f(x).

The almost everywhere convergence of the wavelet projection operators was already shown (under more general hypotheses) in [19]. The non-linear hard and soft sampling methods are used frequently in applications ([1],[11]) but this seems to be the first pointwise convergence result for these methods.

The proof of the convergence of the hard and soft sampling operators relies on the following observation. At fine scales (that is, when j is large), the narrow support of the ψ_{j,k} ensures that the wavelet coefficients a_{j,k} will be relatively small. Thus, a summation such as T_{λ}f(x) will consist only of coarse wavelet coefficients, and should be well approximated by a wavelet projection P_{J}f(x), where J depends non-linearly on x, λ, and f. One can therefore hope to obtain the convergence of T_{λ} from the corresponding convergence of the linear operator P_{J}.

In the last section we show that there are (somewhat artificial) wavelet summation methods which need not converge pointwise, at least for wavelets with little regularity.

Theorem 2.2. Suppose ψ is the Haar wavelet. Then there exists a function f which is in L^{p} for all 1 < p < ∞ but is such that lim sup_{λ→0} |S_{λ}f(x)| = ∞ for all x ∈ R, where S_{λ} is defined by

S_{λ}f = ∑_{|a_{j,k}|≥2^{−j/2}λ} a_{j,k}ψ_{j,k}.

The proof shall be based on the following key estimate:

(2.1)  sup_{J} |P_{J}f(x)|, sup_{λ} |T_{λ}f(x)|, sup_{λ} |T̃_{λ}f(x)| ≲ Mf(x) a.e.

We first show that (2.1) implies the theorem. Since we only require convergence almost everywhere, we may assume that x is a Lebesgue point^{1} of f and of every ψ_{j,k}. Choose any ε > 0. By the usual method of convolving f with a smooth approximation to the identity and then truncating smoothly, one can find a g in C_{0}^{∞}(R) for which |M(f − g)(x)| ≲ ε. In turn, one can approximate g uniformly almost everywhere by a finite linear combination h of wavelets (for this fact, see [20]). Thus one can find such an h for which |f(x) − h(x)| ≤ |M(f − h)(x)| ≲ ε. Now we use the fact that the wavelet co-efficients of T̃_{λ}f and h + T̃_{λ}(f − h) differ by O(λ) when the wavelet occurs in the expansion of h, and are identical otherwise. This implies that T̃_{λ}f(x) = h(x) + T̃_{λ}(f − h)(x) + O_{h}(λ) = f(x) + O(ε) + O_{h}(λ) as λ → 0. Since ε was arbitrary, we thus have lim_{λ→0} T̃_{λ}f(x) = f(x) as desired^{2}. The argument for P_{J} and T_{λ} is completely analogous.

We now show that (2.1) holds for the P_{J}. We note that this estimate was first proven in [19] under more general hypotheses on ψ and f; we shall give an alternate proof below. The argument relies on two observations. The first is the crude estimate

|ψ_{j,k}(x)\overline{ψ_{j,k}(y)}| ≤ C_{N}2^{−j}(1 + 2^{j}|x − y|)^{−N}(1 + |2^{j}x − k|)^{−N} (2.2)

for all N > 0; this is a simple consequence of the rapid decrease of ψ and the easy inequality 1 + 2^{j}|x − y| ≤ (1 + |2^{j}x − k|)(1 + |2^{j}y − k|). The second is the fact that

1In this chapter we shall not identify functions which agree almost everywhere, so the notion of a Lebesgue point is well defined.

2This argument will in fact give pointwise convergence on the entire Lebesgue set of f, if one assumes some minimal regularity (such as right-continuity) for ψ. See [19].

∑_{j,k} ψ_{j,k}(x)\overline{ψ_{j,k}(y)} = 0 a.e.

The identity follows from the fact that the {ψ_{j,k}}_{j,k∈Z} form an orthonormal basis for L^{2}(R), so that the above summation converges in a distributional sense to δ(x−y). Since the series is absolutely convergent for x ≠ y by (2.2), the two limits must agree almost everywhere, hence the claim.

As a consequence of these two observations, we see that the kernel defined almost everywhere by

K(x, y) = ∑_{j<0;k} ψ_{j,k}(x)\overline{ψ_{j,k}(y)} = − ∑_{j≥0;k} ψ_{j,k}(x)\overline{ψ_{j,k}(y)}

converges absolutely and is bounded and rapidly decreasing away from the diagonal. However, we have from Fubini’s theorem that

P_{J}f(x) = ∑_{j<J;k} ψ_{j,k}(x) ∫_{R} \overline{ψ_{j,k}(y)}f(y) dy

= ∫_{R} ∑_{j<J;k} ψ_{j,k}(x)\overline{ψ_{j,k}(y)}f(y) dy

= ∫_{R} 2^{J}K(2^{−J}x, 2^{−J}y)f(y) dy;

the justification for Fubini’s theorem follow from (2.2), the fact that f ∈ L^{p}, and Hölder’s inequality. Since K(2^{−J}x, 2^{−J}y), as a function of y, is rapidly decreasing away from x, the desired estimate (2.1) now follows from standard maximal function estimates (see e.g. [29]).

We now show that (2.1) holds for the T_{λ}. Fix f ∈ L^{p}(R), λ > 0, and x ∈ R. Since we may assume that Mf(x) is nonzero, we can choose a J for which λ ∼ 2^{−J/2}Mf(x); one can think of J as the finest scale for which significant terms in the summation for T_{λ}f(x) can appear. We will show that |T_{λ}f(x) − P_{J}f(x)| ≲ Mf(x), which will imply (2.1) from our results for the P_{J}. We shall first need some simple estimates. For each j, k ∈ Z, let D(j, k) be the integer part of 2 + |2^{j}x − k|. Note that for each j and for each integer d ≥ 2 there are at most two k for which D(j, k) = d. From the rapid decrease of ψ and the usual estimates on approximations to the identity, we have that, for all N > 0:

|ψ_{j,k}(x)| ≤ C_{N}2^{j/2}D(j, k)^{−N} and
|a_{j,k}| ≤ C2^{−j/2}D(j, k)Mf(x); thus
|a_{j,k}ψ_{j,k}(x)| ≤ C_{N}D(j, k)^{−N}Mf(x).

Note that the second estimate implies that |a_{j,k}| is greater than λ only if j < J + C log D(j, k).

The estimate for |T_{λ}f(x) − P_{J}f(x)| is now straightforward:

|T_{λ}f(x) − P_{J}f(x)| = | ∑_{|a_{j,k}|>λ} a_{j,k}ψ_{j,k}(x) − ∑_{j<J;k} a_{j,k}ψ_{j,k}(x)|

≤ | ∑_{|a_{j,k}|≤λ;j<J} a_{j,k}ψ_{j,k}(x)| + | ∑_{|a_{j,k}|>λ;J≤j} a_{j,k}ψ_{j,k}(x)|

≤ ∑_{j<J;k} λ|ψ_{j,k}(x)| + ∑_{J≤j<J+C log D(j,k)} |a_{j,k}ψ_{j,k}(x)|

≲ ∑_{d=2}^{∞} ∑_{j<J} λ2^{j/2}d^{−N} + ∑_{d=2}^{∞} ∑_{J≤j<J+C log d} d^{−N}Mf(x)

≲ λ2^{J/2} + ∑_{d=2}^{∞} Mf(x)d^{−N} log d

≲ Mf(x) + Mf(x),

as desired, where N is a suitable large number. Note that this argument also shows that the sum defining T_{λ}f(x) is absolutely convergent, since we know the sum for P_{J}f(x) is.

The proof of (2.1) for T̃_{λ} is almost identical to that for T_{λ}, although one must use the estimate

∑_{|a_{j,k}|>λ;j<J} sgn(a_{j,k})(|a_{j,k}| − λ)ψ_{j,k}(x) = ∑_{j<J;k} a_{j,k}ψ_{j,k}(x) + O( ∑_{j<J;k} λ|ψ_{j,k}(x)|)

in the above argument; we omit the details. Thus (2.1) holds for all three operators under consideration, and the theorem is proved. ■ Remarks 2.1.

1. This result can be generalized to to bi-orthogonal wavelets, and to decompositions involving more than one “mother wavelet”. The decay conditions on ψ can also be relaxed; for instance, one can get the pointwise convergence result for the P_{J} alone if one merely assumes that ψ has an integrable radial majorant. The proofs of these assertions are simple applications of the ideas developed in [19].

2. The result is also robust under changes in the thresholding method. For example, if one samples under the threshold |a_{j,k}| ≥ α^{j}λ instead of |a_{j,k}| ≥ λ, then one has almost everywhere convergence as long as α ≠ 2^{−1/2}. When α = 2^{−1/2} one is thresholding according to the “heights” of the summation terms a_{j,k}ψ_{j,k}, and by fine-tuning these heights one can sum a large number of terms of comparable magnitude in any desired order. As we shall see in the next section, this freedom in summation can destroy the cancellative effect of the wavelet series, so that one no longer has pointwise convergence.

We shall construct, for each integer N ≥ 1, a function f_{N} = ∑_{j,k} a_{j,k}^{N}ψ_{j,k} on [0, 1) for which:

(i) All but a finite number of the a_{j,k}^{N} are 0 and those that are not zero satisfy 1 < 2^{j/2}a_{j,k}^{N} < 2;

(ii) ∥f_{N}∥_{p} ≤ C_{p}N^{1/2} for every 1 < p < ∞;

(iii) sup_{λ} |T_{λ}^{α}f_{N}(x)| > N/3 for every x ∈ [0, 1).

Once we do this, one can see that, if one chooses an increasing sequence of natural numbers M_{i} whose successive differences are sufficiently large, the function

F (x) = ∑_{i=1}^{∞} 2^{−i} ∑_{k=0}^{2^{M_{i}}−1} f_{3^{i}}(2^{M_{i}}x − k)

will be in L^{p} for every 1 < p < ∞, but is such that lim sup_{λ} |T_{λ}^{α}F (x)| = ∞ for every x ∈ [0, 1). The function f(x) = ∑_{n=−∞}^{∞} 2^{−|n|}F (x − n) thus satisfies all the required conditions.

We shall define f_{N} explicitly:

f_{N}(x) = ∑_{j=0}^{N−1} ∑_{k=0}^{2^{j}−1}(1 + 2^{−j}(k + \frac{1}{2}))ψ(2^{j}x − k).

The verification of (i) is immediate, and (ii) follows quickly from Khinchin’s inequality^{3}

∥ ∑_{j,k} a_{j,k}ψ_{j,k}∥_{p} ∼ ∥(∑_{j,k} |a_{j,k}|^{2}|ψ_{j,k}|^{2})^{1/2}∥_{p}.

To prove (iii), fix an x ∈ [0, 1) and observe that all but N of the factors ψ(2^{j}x − k) in the sum defining f_{N}(x) are zero, and the remainder are either +1 or −1. Furthermore, this factor is nonnegative if 1 + 2^{−j}(k + \frac{1}{2}) > 1 + x and nonpositive otherwise. Thus, by inspection of the definition of T_{λ}^{α} for λ = 1 + x and λ = 2, we have

3A proof of this inequality can be found in [20].

T_{1+x}^{α}f(x) = ∑_{ψ(2^{j}x−k)=+1}(1 + 2^{−j}(k + \frac{1}{2})), and
T_{1+x}^{α}f(x) − T_{2}^{α}f(x) = ∑_{ψ(2^{j}x−k)=−1}(1 + 2^{−j}(k + \frac{1}{2})); so
2T_{1+x}^{α}f(x) − T_{2}^{α}f(x) = ∑_{ψ(2^{j}x−k)≠0}(1 + 2^{−j}(k + \frac{1}{2})) > N,

which implies (iii). This proves the theorem. ■

Weak bounds for homogeneous convolution operators

Let H be an arbitrary homogeneous group, and suppose that Ω is a function on the unit sphere S of H, which we endow with the natural measure dω induced by the dilation structure. We consider the question of what the minimal conditions on Ω are so that the convolution operator T_{Ω} : f ↦ f ∗K is of weak-type (1, 1), where the convolution kernel K = K_{Ω} is given by K = K_{Ω} defined by K(r ∘ ω) = r^{−N}Ω(ω) and N is the homogeneous dimension of H.

Among the necessary conditions known are that Ω must be in L^{1}(S), and T_{Ω} must be bounded on L^{2}. This in turn necessitates that Ω satisfies the cancellation condition ∫_{S} Ω(ω) dω = 0. When H is an isotropic Euclidean space, the classical theorem of Calderón and Zygmund [3] shows that T is indeed bounded on L^{2} when Ω satisfies the cancellation condition and is either odd and in L^{1}(S), or even and in the Hardy space H^{1}(S). In particular, we have boundedness on L^{2} whenever Ω satisfies the cancellation condition and is in the Orlicz class L log L. However, their argument relies 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[15]).

It is natural to conjecture that analogous L^{2} results hold for arbitrary homogeneous groups. If Ω is odd and in L^{1} one can use the method of rotations and the work of Ricci and Stein [21] on convolution operators on singular sets in homogeneous groups to obtain L^{2} boundedness, but to the best of the author’s knowledge the corresponding question for Ω even and in H^{1} or L log L remains open.

The endpoint question in Euclidean space has been considered by several authors ([8],[9], [17],[24]); recently A. Seeger [24] has shown that T_{Ω} is of weak-type (1, 1) for all Ω in L log L that satisfy the cancellation condition. The corresponding questions for odd L^{1} or even H^{1} kernels remain open.

In this chapter we generalize the result in [24] to arbitrary homogeneous groups. Specifically, we show that if Ω is any function in L log L such that T_{Ω} is bounded on L^{2}, then T_{Ω} is automatically of weak-type (1, 1). If the above conjecture is true, the L^{2} hypothesis on T_{Ω} would just reduce to the cancellation condition.

The methods in [24] 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 [9], in that one considers the expression T_{Ω}f as an operator acting on the kernel K 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 Ω, 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. One can also use the arguments in [24] to weaken the radial regularity on K to a Dini-type continuity condition.

1 Preliminary reductions

Let H be an arbitrary homogeneous group, and let Ω be an L log L function such that T = T_{Ω} is bounded on L^{2}. Our objective is to show that T maps L^{1} to weak-L^{1}. We will assume that all relevant functions are real-valued, since the extension to complex numbers is a trivial matter. Also, we suppose that n = dim(H) ≥ 2, since the n = 1 theory is treatable by the classical theory of the Hilbert transform.

By linearity we may set ∥Ω∥_{L log L} equal to 1. Let f be an arbitrary L^{1} function. By linearity again, the desired estimate reduces to

|{x : |T f(x)| ≳ 1}| ≲ ∥f∥_{1}.

We now decompose f in the standard Calderón-Zygmund fashion as f = g + ∑_{I} b_{I}, where ∥g∥_{2}^{2} ≲ ∥f∥_{1}, and the b_{I} are functions supported on finitely overlapping balls I with dyadic radii 2^{i}, with ∑_{I} |I| ≲ ∥f∥_{1}, such that ∫_{I} b_{I} = 0 and ∫_{I} |b_{I}| ≲ |I|. Since T is bounded on L^{2}, we may dispose of g in the usual manner, and it suffices to prove that

|{x : | ∑_{I} b_{I} ∗ K(x)| ≳ 1}| ≲ ∑_{I} |I|. (3.1)

By rescaling we may assume that ∑_{I} |I| is comparable to 1.

The next step is to break up the kernel K into dyadic pieces, although our decomposition scheme will differ slightly from the standard one. We start with an auxilliary decomposition K = ∑_{i=−∞}^{∞} K_{i} which can be constructed by any of the standard methods; the only assumptions we place on the decomposition are that the K_{i} depend linearly on Ω, are supported on some annulus B(0, 2^{i})_{Δ}, and satisfy the bounds ∥K_{i}∥_{p} ∼ 2^{Ni/p′}∥Ω∥_{p}. Define the pushforward operators δ_{∗}^{t} corresponding to a dilation by t as

δ_{∗}^{t}f(y) = t^{−N}f(t^{−1} ∘ y).

Since K is homogeneous of degree −N, we have δ_{∗}^{t}K = K for all t > 0. Thus, if φ is a suitably normalized bump function adapted to the interval [C^{−1}, C], we can write K = ∑_{i} K̃_{i}, where K̃_{i} = D_{φ}K_{i} is a smoothed out version of K_{i} and D_{φ} = ∫ δ_{∗}^{t} φ(t)\frac{dt}{t} is an averaged dilation operator.

We split the kernel generator Ω into two pieces Ω = Ω^{>,s} + Ω^{≤,s} for each s > 0, where Ω^{>,s} is the portion of Ω on the set {|Ω| > 2^{εs}}. One defines the corresponding functions K̃_{i}^{>,s}, etc. in the obvious manner. The sum in (3.1) can now be rewritten as

∑_{I} b_{I} ∗ K = ∑_{I} ∑_{s<C} b_{I} ∗ K̃_{i+s} + ∑_{s≥C} ∑_{I} b_{I} ∗ K̃_{i+s}^{>,s} + ∑_{s≥C} ∑_{I} b_{I} ∗ K̃_{i+s}^{≤,s}.

The first sum in the right-hand side is supported in the set ⋃_{I} CI, which has an acceptable measure, and so can be discarded. To treat the second sum, we estimate its L^{1} norm by the method of rotations as follows:

∥ ∑_{s≥C} ∑_{I} b_{I} ∗ K̃_{i+s}^{>,s}∥_{1} ≲ ∑_{s≥C} ∑_{I} ∥b_{I}∥_{1}∥K̃_{i+s}^{>,s}∥_{1}

≲ (∑_{I} |I|)(∑_{s≥C} ∥Ω^{>,s}∥_{1})

≤ C_{ε}(∑_{I} |I|)∥Ω∥_{L log L}.

Thus, by Tchebyshev’s inequality, this term is also acceptable.

It remains to treat the last sum, which is the most interesting of the three. By Tchebyshev’s inequality again, it suffices to prove the following L^{p} estimate for some 1 ≤ p ≤ ∞ and all s > 0:

∥ ∑_{I} b_{I} ∗ K̃_{i+s}^{≤,s}∥_{p} ≤ C_{p}2^{−ε_{p}s}.

Suppose that the parameter s > 0, the balls I, and the functions b_{I} are fixed. Rewrite b_{I} ∗ K̃_{i+s}^{≤,s} as ψ_{I}τ_{I}D_{φ}k_{I}, where τ_{I} is the operator corresponding to left-convolution by \frac{b_{I}}{|I|}, ψ_{I} is a normalized bump function equal to 1 on a suitable annulus 2^{s}I_{Δ} and supported on a slightly thickened version of that annulus, and k_{I} = |I|K_{i+s}^{≤,s}. Since ∥k_{I}∥_{2} ≲ 2^{εs}2^{−Ns/2}|I|^{1/2} for some arbitrarily small ε, we see that the desired inequality will be implied by the following proposition.

Proposition 3.1. Let ℐ = {I} be a collection of essentially disjoint balls with dyadic radii such that ∑_{I} |I| ∼ 1, and let b_{I} be functions on I such that ∥b_{I}∥_{1} ≲ |I| and ∫_{I} b_{I} = 0. Then, if k_{I} are an arbitrary family of functions in L^{2}(H), one has the following estimate for all 1 < p < 2 and all s > 0:

∥2^{−Ns} ∑_{I} ψ_{I}τ_{I}D_{φ}k_{I}∥_{p} ≤ C_{p}2^{−ε_{p}s}(2^{−Ns} ∑_{I} ∥k_{I}∥_{2}^{2})^{1/2}. (3.2)

Note that all the support, size, regularity, and interdependence assumptions on k_{I} have been discarded; those aspects of the inequality have either disappeared or been transferred to other components of (3.2). This deliberate forgetting of structure is advantageous because it simplifies the Hilbert space in which the {k_{I}} live in.

By the T T^{∗} method, the estimate (3.2) is equivalent to the claim that ∥A∥_{(p′,p)} ≤ C_{p}2^{−ε_{p}s} for all 1 < p < 2, where A : L^{p′}(H) → L^{p}(H) is the self-adjoint operator

Af = 2^{−Ns} ∑_{I} ψ_{I}τ_{I}D_{φ_{2}}τ_{I}^{∗}ψ_{I}f,

and φ_{2} is the bump function φ_{2}(t) = ∫ s^{N}φ(s)φ(ts) \frac{ds}{s}.

For each I, let ψ̃_{I} be a normalized bump function on 2^{s}I_{Δ} which is equal to 1 on the support of ψ_{I}, and let φ̃_{2} be a similar enlargment of φ_{2} which is self-adjoint in the sense that φ̃_{2}(t) = t^{−N} φ̃_{2}(t^{−1}). Let A^{+} be the self-adjoint operator

A_{+} = 2^{−Ns} ∑_{I} ψ̃_{I}τ_{I}^{+}Dφ̃_{2}τ_{I}^{+∗}ψ̃_{I},

where τ_{I}^{+} is defined as τ_{I} but with b_{I} replaced by |b_{I}|. Our strategy will be to show that (apart from some error terms) A is dominated by 2^{−εs}A_{+} in the sense of the Introduction.

We now prove the “trivial” estimates on A_{+} (and consequently A), which do not use cancellation or regularization properties of the τ and D operators, and so do not achieve the desired exponential gain 2^{−εs}.

Lemma 3.1. For all 1 ≤ p < q ≤ ∞ we have ∥A_{+}∥_{(q,p)} ≤ C_{q,p}.

Proof. Since the τ and D operators are averages of translations and dilations by factors comparable to the identity, they are uniformly bounded on every L^{p} space. Thus, the claim will follow by combining the inequality

(2^{−Ns} ∑_{I} ∥ψ̃_{I}f∥_{r}^{r})^{1/r} ≤ C_{q,r}∥f∥_{q},

which is valid for 1 ≤ r < q ≤ ∞, with its adjoint

∥2^{−Ns} ∑_{I} ψ̃_{I}f_{I}∥_{p} ≤ C_{p′,r′}(2^{−Ns} ∑_{I} ∥f_{I}∥_{r}^{r})^{1/r},

which is valid for 1 ≤ p < r ≤ ∞. To prove this inequality, it suffices by the generalized Hölder inequality to prove the estimate ∥2^{−Ns} ∑_{I} ψ̃_{I}^{a}∥_{b} ≤ C_{a,b} for all 1 ≤ a, b < ∞. Since all the powers of ψ̃_{I} are essentially comparable we shall assume that a = 1. When b = 1 the claim follows immediately from the triangle inequality and the normalization condition ∑_{I} |I| ∼ 1. It therefore suffices to check the endpoint b = BMO; in other words, we will show that \frac{1}{|J|} ∫_{J} |2^{−Ns} ∑_{I} ψ̃_{I} − c_{J}| ≲ 1 for all balls J and some constants c_{J}.

Fix J. We shall estimate the mean oscillation of each summand seperately. When r(2^{s}I) ≲ r(J), there is a non-zero contribution only if I is in CJ, and even then the contribution is majorized by \frac{1}{|J|}C2^{−Ns}|2^{s}I| = C|I|/|J|. Thus the total contribution of the small cubes is at most C|CJ|/|J|, which is acceptable. When r(2^{s}I) ≳ r(J), there are at most C2^{Ns} cubes I at each scale for which there is a non-zero contribution, and an integration by parts argument shows that the maximum contribution from each such cube is O(2^{−Ns} \frac{r(J)}{r(2^{s}I)}). Summing over all large scales thus gives the desired BMO estimate, and we are done. ■

Because of the trivial estimates on A, any bound of the form ∥A^{M}∥_{q,p} ≲ 2^{−εs} for one pair of exponents 1 < p < q < ∞ and a single integer M > 0 will imply the desired estimate ∥A∥_{(p′,p)} ≤ C_{p}2^{−ε_{p}s} for any 1 < p < 2. To see this, note that one can repeatedly compose such an estimate on A^{M} with the trivial estimate and interpolate with further trivial estimates until M is a power of 2 and q is the dual exponent of p. One can then use the T T^{∗} method to obtain a boundedness result (e.g. (2, p) boundedness) on A^{M/2} with exponential decay, which one can interpolate with the trivial estimates to again obtain a (p′, p) boundedness result. Iterating this procedure eventually gives a bound on A, which can be interpolated to give the desired estimate for all 1 < p < 2 (although the decay factor ε would of course have been considerably worsened by this process).

Thus, to finish the proof of the weak (1,1) boundedness of T, we have to find some estimate on a power A^{M} of A which improves on the corresponding trivial estimate on A_{+}^{M}. Since each D_{φ} operator contained in A has a smoothing effect in one “random” direction, one intuitively expects from that n iterations of A should create a smoothing effect in all directions, which would interact with cancellation in the b_{I} to obtain the desired gain. There are two difficulties with this approach, however. Firstly, if one composes parts of A that come from balls I of vastly different sizes, then the cutoff functions ψ_{I} corresponding to the smaller balls can seriously truncate the smoothing effect from the D_{φ} corresponding to the larger balls. Secondly, when H is a general homogeneous group, the smoothing effects of the D_{φ} will tend to be along almost parallel directions, rather than being isotropically dispersed. The combined effect of these two obstacles will be that one might only get smoothing effects along very short, very parallel arcs, which will not give much isotropic regularity. To avoid this problem we shall use a much larger number of iterations, namely M = 2^{2n−3}, which is enough to ensure the existence of at least n untruncated arcs for each M-tuplet of balls I_{1}, …, I_{M}.

Before we continue, however, we shall need a certain amount of machinery related to the inverse function theorem. These rather technical devices unfortunately seem to be necessary when applying iteration methods such as the (T T^{∗})^{M} method; see for example [9], [21].

2 Interlude on the inverse function theorem

As in the previous section, s > 0 is a fixed large number. We shall also set δ > 0 to be an extremely small constant to be chosen later; δ will depend only on the homogeneous group H.

In this section we shall concentrate on the Euclidean structure of H instead of the group structure. Actually, the correct structure to consider should be the vector space structure of the Lie algebra 𝔥, but we shall take advantage of the Euclidean identification between H and 𝔥 and avoid introducing any new spaces.

We will also be using the space ⋀^{k} H of k-forms for all 0 ≤ k ≤ n, which we give a Euclidean structure by declaring the canonical basis e_{K} to be orthonormal, where K runs over all increasing subsequences of 1, …, n of length k. This will induce a metric on the projective space P(⋀^{k} H). We shall use the volume form dx to identify ⋀^{n} H with R in the usual manner.

For technical reasons we shall need a co-ordinate dependent approximate partial ordering ≾ on ⋀^{k}, as well as a rather artificial non-cancellative analogue ♢ of the wedge product. We say that ∑_{K} x_{K}e_{K} ≾ ∑_{K} y_{K}e_{K}, or alternatively ∑_{K} x_{K}e_{K} = Õ(∑_{K} y_{K}e_{K}), whenever |x_{K}| ≲ 2^{Cδs}|y_{K}| for each basis element e_{K}. We use x ≈ y to denote the statement that x ≾ y and y ≾ x.

The non-linear operator ♢, taking j-forms and k-forms to j + k-forms, is defined by

∑_{J} b_{J}e_{J}♢ ∑_{K} c_{K}e_{K} = ∑_{J} ∑_{K} ±|b_{J}c_{K}|e_{J} ∧ e_{K},

where the sign is chosen so that ±e_{J} ∧ e_{K} is a basis (j + k)-form whenever possible. Note that the ♢ operator preserves the ≾ ordering, and that x∧y ≾ x♢y for all forms x, y. A heuristic converse holds: for “most” x and y one should expect that x ∧ y ≈ x♢y. This motivates the following definition: a collection x_{1}, …, x_{k} of vectors in H shall be called good if one has

x_{a_{1}} ∧ … ∧ x_{a_{r}} ≈ x_{a_{1}}♢ … ♢x_{a_{r}}

for all subsequences a_{1}, …, a_{r} of 1, … k of length at most n. Note that the property of being good is invariant under Euclidean dilations of the individual x_{i}, or under a single homogeneous group dilation acting on all the x_{i}. Also, note that the property of being good is equivalent to the existence of vectors y_{1}, …, y_{k} such that x_{a} ≾ y_{a} for all a and

x_{a_{1}} ∧ … ∧ x_{a_{r}} ≿ y_{a_{1}}♢ … ♢y_{a_{r}}

for all subsequences a_{1}, …, a_{r}.

We shall frequently use the element 1 = (1, …, 1) of H, or more generally expressions such as (δ_{1} ∘ 1)♢ … ♢(δ_{k} ∘ 1), to dominate various expressions using the ≾ relation.

Let ϕ be an H-valued function of n real parameters t_{1}, …, t_{n}. For our purposes, there are two ways to define an H-valued partial derivative of ϕ, depending on whether one uses the left action of group multiplication or the vector space structure to identify various tangent spaces of H. The group identification is more natural, but the vector space identification is easier to handle technically. We therefore define the functions \frac{∂ϕ}{∂t_{k}} and ϕ_{,k} for 1 ≤ k ≤ n as follows:

\frac{∂ϕ}{∂t_{k}} (t) = lim_{dt_{k}→0}(dt_{k})^{−1} · (ϕ(t)^{−1}ϕ(t + dt_{k})),

ϕ_{,k}(t) = lim_{dt_{k}→0}(dt_{k})^{−1} · (ϕ(t + dt_{k}) − ϕ(t)).

We shall need the following relationship between the two derivatives: if \frac{∂ϕ}{∂t_{k}} (t) ≾ 2^{i} ∘ 1 and ρ(ϕ(t)) ≤ α2^{i} for some α ≪ 1, then \frac{∂ϕ}{∂t_{k}} (t) − ϕ_{,k}(t) ≾ α^{ε} · (2^{i} ∘ 1). This is easily verified by rescaling to i = 1 and using the fact that group multiplication is approximately equal to vector addition near the origin.

We are now in a position to state a precise version of the following principle: if the partial derivatives \frac{∂ϕ}{∂t_{k}} are good and essentially constant, then the pushforward ϕ_{∗}ψ of bump functions ψ is absolutely continuous and varies in a uniformly continuous manner with respect to uniformly continuous changes in ϕ.

Proposition 3.2. Suppose that B ⊂ R^{n} is a Euclidean ball centered at the origin of radius 2^{−σs} for some δ ≪ σ ≪ 1, i_{1}, …, i_{n} are integers, and ϕ_{0}, ϕ_{1} are continuously differentiable H-valued functions on B such that the following properties hold for j, j′ = 0, 1.

(i) The partial derivatives of ϕ_{j} are good at 0. More precisely, we have \frac{∂ϕ_{j}}{∂t_{k}} ≾ 2^{i_{k}+s} ∘ 1 and

\frac{∂ϕ_{j}}{∂t_{a_{1}}} (0) ∧ … ∧ \frac{∂ϕ_{j}}{∂t_{a_{r}}} (0) ≿ (2^{i_{a_{1}}+s} ∘ 1)♢ … ♢(2^{i_{ar} +s} ∘ 1).

(ii) For all x, x′ in B we have

\frac{∂ϕ_{j}}{∂t_{k}} (x) − \frac{∂ϕ_{j′}}{∂t_{k}} (x′) ≾ 2^{−εσs} · (2^{i_{k}+s} ∘ 1).

(iii) We have d(ϕ_{0}(x), ϕ_{1}(x)) ≤ r for all x ∈ B, where r > 0 is such that r ≲ 2^{−Cσs}2^{i_{k}+s} for all k.

Then the following is true for all integrable functions f and all normalized bump functions ψ, ψ̃ adapted to B such that |ψ(x) − ψ(y)| ≲ 2^{σs}ψ̃(x) for all x, y:

| ∫_{B} f(ϕ_{1}(x))ψ(x) dx − ∫_{B} f(ϕ_{0}(x))ψ(x) dx| ≲ 2^{−εσs} ∫_{B} |f(ϕ_{0}(x))|ψ̃(x) dx.

Proof. By rescaling we can assume that r = 1, so that 2^{i_{k}+s} ≳ 2^{Cσs} for all k. Let Dϕ_{j}(x) be the n × n matrix whose columns are the vectors \frac{∂ϕ_{j}}{∂t_{k}} (x), and let ∇ϕ_{j} be the matrix whose columns are ϕ_{j,k}. By (i), Cramer’s rule, and the bounds on 2^{i_{k}+s}, we see that the components of (Dϕ_{0})^{−1}(x) are O(2^{−Cσs}), so that the ellipsoids

B_{x_{0}} = {x : |Dϕ_{0}(x_{0})(x − x_{0})| ≲ 2^{εσs}}

have diameter at most O(2^{−Cσs}). A covering argument and (ii) then shows that we can restrict our domain from B to a single ellipsoid B_{x_{0}}, and by a partition of unity we can assume that ψ, ψ̃ are now adapted to B_{x_{0}} instead of B and satisfy the Lipschitz condition

|ψ(x_{1}) − ψ(x_{2})| ≲ 2^{−εσs}|Dϕ_{0}(x_{0})(x_{1} − x_{2})|ψ̃(x_{1}).

We may assume that x_{0} and ϕ_{0}(x_{0}) are equal to 0. From (ii) and the bounds on the derivatives of ϕ_{j} we see that ϕ_{j}(x) ≾ 2^{εσs} · 1 for all x ∈ B_{0}, j = 0, 1. Because of these bounds on ϕ_{j}(x), the two derivatives \frac{∂ϕ_{j}}{∂t_{k}} and ϕ_{j,k} are approximately equal, so that ϕ_{j,k} satisfies the same estimates (i), (ii) as \frac{∂ϕ_{j}}{∂t_{k}}. One can then use Cramer’s rule again to apply the argument in the standard proof of the inverse function theorem, which uses the fact that the map G_{j,y}(x) = x − ∇ϕ_{j}(0)^{−1}(ϕ_{j}(x) − y) is a contraction for all y, to show that ϕ_{j} is injective on B_{0} and surjective on {y : y ≾ 2^{εσs} · 1}. Furthermore, one has ϕ_{j}(x) − ϕ_{j}(y) = ∇ϕ_{j}(0)(x − y) + Õ(1) for x, y ∈ B_{0}.

To complete the proof one now makes the change of variables y = ϕ_{1}(x) and y = ϕ_{0}(x) and uses the above estimates, together with the bound

| det(∇ϕ_{j}(x)) − det(∇ϕ_{j′}(x′))| ≲ 2^{−εσs}| det(∇ϕ_{j}(x))|

on the variation in the Jacobian to measure the error between the two integrals. We omit the details. ■

3 Conclusion of the proof

We now conclude the proof of weak (1,1) boundedness of the convolution operator T by dominating most of the operator A^{m} by C2^{−εs} times the corresponding portion of A_{+}^{m}.

One minor difficulty will be that the smoothing effect of the D_{φ} will be along curves instead of straight lines when H is nonisotropic. To avoid this problem we shall make a decomposition which will partition these arcs into a relatively small number of segments which are approximately straight. Choose σ such that δ ≪ σ ≪ 1, and partition the interval [C^{−1}, C] as the finitely overlapping union of a set G of intervals γ of length O(2^{−σs}), where #G ∼ 2^{σs}. Write φ_{2} = ∑_{γ} η_{γ}, φ̃_{2} = ∑_{γ} η̃_{γ}, where η_{γ} and η̃_{γ} are normalized bump functions adapted to γ that satisfy the Lipschitz condition |η_{γ}(x) − η_{γ}(y)| ≲ 2^{σs}|x − y|η̃_{γ}(x) for all x, y.

We can now express A and A_{+} as integral operators which are reminiscent of Markov chain operators. Define a step at I to be any quintuplet Σ = (I, v, w, γ, t), where I is one of the balls in ℐ, v, w ∈ I are points, γ ∈ G is an interval and t ∈ γ is a real number. We say that a point x ∈ H is in range of a step Σ if x is in 2^{s}I_{Δ}. Let X_{I} be the space of all steps at I, endowed with the measure

dΣ_{I} = \frac{b_{I}(v)dv}{|I|} \frac{b_{I}(w)dw}{|I|}t^{N}η_{γ}(t^{−1})\frac{dt}{t}d#(γ),

as well as the non-cancellative analogue

dΣ_{I}^{+} = \frac{|b_{I}(v)|dv}{|I|} \frac{|b_{I}(w)|dw}{|I|}t^{N} η̃_{γ}(t^{−1})\frac{dt}{t}d#(γ).

We use X = ⋃_{I} X_{I} to denote the space of all steps. Finally, we define the propogator P : H × X → H to be the function

P (x, (I, v, w, γ, t)) = v(t ∘ (w^{−1}x)).

The operators A and A_{+} can now be written as

Af(x) = ∑_{I} A_{I}f(x) = 2^{−Ns} ∑_{2^{s}I_{Δ}∋x} ∫_{X_{I}} f(P (x, Σ))ψ_{I}(x)ψ_{I}(P (x, Σ)) dΣ_{I},

A_{+}f(x) = ∑_{I} A_{I}^{+}f(x) = 2^{−Ns} ∑_{2^{s}I_{Δ}∋x} ∫_{X_{I}} f(P (x, Σ))ψ̃_{I}(x)ψ̃_{I}(P (x, Σ)) dΣ_{I}^{+}.

Before we proceed further we shall record in the following proposition some facts on how the propogator reacts to changes in the x, v, and t variables, and how evenly distributed the tangent vectors \frac{∂P}{∂t} are.

Proposition 3.3. Let Σ = (I, v, w, γ, t) be an arbitrary step, x, x′ ∈ H be in range of Σ and such that d(x, x′) ≲ 2^{−εσs}2^{i+s}, and let Σ′ = (I, v′, w, γ, t′) be a step that differs from Σ only in the v and t variables. Then the following statements hold:

P (x, Σ)^{−1}P (x′, Σ) = t ∘ (x^{−1}x).
d(P (x, Σ), P (x′, Σ′)) ≲ 2^{i+(1−εσ)s}.
if t = t′, then d(P (x, Σ), P (x, Σ′)) ≲ 2^{i+(1−ε)s}.
\frac{∂P}{∂t} (x, Σ) ≾ 2^{i+s} ∘ 1.
\frac{∂P}{∂t} (x, Σ) − \frac{∂P}{∂t} (x′, Σ′) ≾ 2^{−εσs} · (2^{i+s} ∘ 1).

Finally, for any set y = {y_{1}, …, y_{k}} of good vectors and any Σ, we have that y_{1}, …, y_{k}, \frac{∂P}{∂t} (x, Σ) are good for most x. More precisely, if one defines E_{y,Σ} to be the bad set

\{x ∈ 2^{s}I_{Δ} : y_{a_{1}} ∧…∧y_{a_{r}} ∧ \frac{∂P}{∂t} (x, Σ) ≿̸ y_{a_{1}}♢ … ♢y_{a_{r}}♢(2^{i+s} ∘1) ∀ a_{1}, …, a_{r}\},

and Ẽ_{y,Σ} to be the slight enlargment E_{y,Σ} = {x : d(x, E_{y,Σ}) ≲ 2^{i+εδs}}, then

|E_{y,Σ}|, |Ẽ_{y,Σ}| ≤ C_{k}2^{−εδs}|2^{s}I_{Δ}|,

where the constant C_{k} is independent of y_{1}, …, y_{k} and Σ.

Proof. All the statements above are invariant under dilations and translations, so we may assume that 2^{i+s} = 1 and x_{I} = 0. The first identity is proven by inspection, and the next four inequalities follow from the polynomial nature of the expressions on the left-hand sides and the bounds on the variation in x and Σ. To prove the final claim, we write \frac{∂P}{∂t} as

\frac{∂P}{∂t} (x, Σ) = t ∘ (t^{−1} · X(wx)),

where X(x) is the tangent vector at the origin to the curve t ↦ x^{−1}(1+t)∘x. The map X is obviously given by polynomials; we shall see shortly that X is invertible and that the inverse map is also polynomial. This implies that, for every hyperplane π, the set {x : ρ(x) ∼ 1, d(X(x), π) ≲ 2^{−εδs}} has measure at most C2^{−εδs}, uniformly in π. The claim now follows by noticing that the property of x being in E_{y,Σ} (or Ẽ_{y,Σ}) is essentially equivalent to the statement that X(wx) avoids a certain finite union of 2^{−εδs}-neighbourhoods of hyperplanes.

It remains to prove that X is invertible and that X^{−1} is given by polynomials. This will come from the fact that H is a nilpotent Lie group, and that all the group operations are “lower diagonal” in some sense. Since the basis vectors e_{1}, …, e_{n} are arranged according to their homogeneous weight, the multiplication and inversion laws must hav the form

(xy)_{j} = x_{j} + y_{j} + P_{j}(x_{1}, …, x_{j−1}, y_{1}, …, y_{j−1}),
(x^{−1})_{j} = −x_{j} + Q_{j}(x_{1}, …, x_{j−1}).

From the definition of X we can therefore write

X(x)_{j} = α_{j}x_{j} + R_{j}(x_{1}, …, x_{j−1}),

where the R_{j} are polynomials, and this map can be inverted explicitly with a polynomial inverse by the usual iterative procedure. ■

Let M = 2^{2n−3}. We write A^{M} as ∑_{I_{1}...I_{M}} A_{I_{1}} … A_{I_{M}}, and use the following combinatorial lemma to extract for each such sequence a monotonic subsequence of length n which avoids the truncation problems mentioned earlier.

Lemma 3.2. Let I_{1}, …, I_{M} be an arbitrary collection of balls, where M = 2^{2n−3}. Then there exists a strictly monotonic sequence k_{1}, …, k_{n} of integers in {1, …, M} such that r(I_{k_{j}}) ≤ r(I_{k}) for all j = 1, …, n − 1, and all k between k_{j} and k_{n} inclusive.

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 inductive procedure. Let S_{1} be the interval {1, …, M}. For each j = 1, …, 2n − 2 in turn, we choose l_{j} ∈ S_{j} so that I_{l_{j}} has minimal radius among all the balls {I_{l} : l ∈ S_{j}}. Removing the element l_{j} from S_{j} divides the remainder into two intervals {l ∈ S_{j} : l < l_{j}} and {l ∈ S_{j} : l > l_{j}}; we choose S_{j+1} to be the larger of the two intervals. One can easily show that |S_{j}| ≥ 2^{2n−2−j} for all j, so that all the l_{j} are well defined. Furthermore, one has r(I_{l_{j}}) ≤ r(I_{l}) for all l between l_{j} and l_{2n−2}. Since one of the sets {l_{i} ≤ l_{2n−2}}, {l_{i} ≥ l_{2n−2}} has a cardinality of at least n, we can extract a monotonic subsequence k_{1}, …, k_{n} with the desired properties. ■

Fix the sequence I_{1}, …, I_{M}, and let k_{1}, …, k_{n} be as in the lemma. Except for an error term which we shall deal with later, we shall majorize A_{I_{1}} … A_{I_{M}} pointwise by 2^{−εs}A_{I_{1}}^{+} … A_{I_{M}}^{+} . By taking adjoints if necessary and using the fact that the A_{I} and A_{I}^{+} operators are self-adjoint, one can assume that the k_{i} are monotonically increasing. We can also truncate the sequence I_{1}, …, I_{M} at both ends, reducing M in the process, so that k_{1} = 1 and k_{n} = M. In fact, we will make the simplifying assumption that M = n and k_{j} = j for j = 1, …, n, so that i_{j} is a monotonically nondecreasing sequence; the general case is completely analogous, but would involve more intricate notation.

We will dominate most of |A_{I_{1}} … A_{I_{n}}f(x)| by 2^{−εs}A_{I_{1}}^{+} … A_{I_{n}}^{+}|f|(x) for all f and x; we may take x = 0 without loss of generality. The expression A_{I_{1}} … A_{I_{n}}f(0) can be written as a repeated integral

∫_{Z} ∫_{I_{1}} ∫_{γ_{1}}⋯∫_{γ_{n}} f(x_{n})Ψ(z, v_{1}, t) ∏_{j=1}^{n}(t^{N}η_{γ_{j}}(t_{j}^{−1})\frac{dt_{j}}{t_{j}} )\frac{b_{I_{1}}(v_{1})dv_{1}}{|I_{1}|} dμ(z),

where Z is the set of (3n − 1)-tuples z = (v_{2}, …, v_{n}, w_{1}, …, w_{n}, γ_{1}, …, γ_{n}), the x_{j} are defined inductively by x_{0} = 0, x_{j} = P (x_{j−1}, (I_{j}, v_{j}, w_{j}, γ_{j}, t_{j})), the cutoff function Ψ is given by Ψ(z, v_{1}, t) = ∏_{j=1}^{n} ψ_{I_{j}}(x_{j−1}), and dμ(z) is some measure whose explicit form is not important. One can of course write a similar integral formula for the A_{I_{1}}^{+} … A_{I_{n}}^{+}|f|(0), with all terms replaced by enlarged, non-cancellative versions of themselves.

We may assume that x_{j−1}, x_{j} are in range of I_{j} for each j. By Proposition 3.3 and the monotonicity of the i_{j}, the maximum variation of each x_{j} as a function of the v_{1} variable is O(2^{i_{j}+(1−ε)s}), and the maximum variation as a function of the t variables is O(2^{i_{j}+(1−εσ)s}). In particular, this means that the Ψ is essentially independent of v_{1} and t, as the variation in Ψ can be dominated by 2^{−εs} times the corresponding function Ψ̃ in the expression for A_{I_{1}}^{+} … A_{I_{n}}^{+}|f|(0). We can therefore treat Ψ as a function of z only, and subsume it into the measure dμ(z).

Let us call a history (x_{0}, Σ_{1}, …, Σ_{M}) of steps good if for each 1 ≤ j ≤ M the sequence \frac{∂x_{j}}{∂t_{1}} , …, \frac{∂x_{j}}{∂t_{j}} is good, where x_{j} is defined in the inductive manner used before. From the estimates in Proposition 3.3, we see that goodness of the history corresponding to z, v_{1}, t is essentially independent of v_{1} and t, and so is only a property of z. Let us discard those z that give rise to bad histories as error terms, to be treated later, and restrict our attention to the good z. By the standard integration by parts argument, to prove the desired majorization result it suffices to show that, for each fixed good z, the v_{1}-variation in the quantity

∫_{γ_{1}}⋯∫_{γ_{n}} f(x_{n}) ∏_{j=1}^{n}(t_{j}^{N}η_{γ_{j}}(t_{j}^{−1})\frac{dt_{j}}{t_{j}} )

is majorized by

2^{−εσs} ∫_{γ_{1}}⋯∫_{γ_{n}} |f(x_{n})| ∏_{j=1}^{n}(t_{j}^{N} η̃_{γ_{j}}(t_{j}^{−1})\frac{dt_{j}}{t_{j}} ).

This however will follow directly from Proposition 3.2, Proposition 3.3, and the assumption that z was good.

It remains to estimate the error terms arising from the bad histories. The total contribution of these terms to the original expression A^{M}f(x), with M = 2^{2n−3}, can be dominated by

∫_{X_{x}^{M}} |f(x_{M})|χ(Γ) dΓ_{+},

where X_{x}^{M} is the space of all histories Γ = (x, Σ_{1}, …, Σ_{M}) endowed with the measure

dΓ_{+} = ∏_{j=1}^{M}(2^{−Ns} ∑_{I_{j}∈ℐ} ψ̃_{j}(x_{j−1})ψ̃_{j}(x_{j})d#(I_{j})dΣ_{I_{j}}^{+}),

and χ is a cutoff function that vanishes on good histories Γ.

We now construct a large set of good histories. Let S = ⋃_{k=1}^{n} P(⋀^{k} H) be the disjoint union of the projective exterior algebras of H, and let N = {ν} be a maximal 2^{−δs}-seperated set of S, so that #N = O(2^{Cδs}). For any ν ∈ N, γ ∈ G, and x ∈ H, define ℐ_{x,γ,ν} to be the set of balls

ℐ_{x,γ,ν} = {I ∈ ℐ : x is in range of I, (t ∘ v) ∧ X(x_{I}^{−1}x) ≿̸ (t ∘ v)♢(2^{i+s} ∘ 1)},

where v is any representative of ν, t is any element of γ, and X is the map in the proof of Proposition 3.3. Define an exceptional set

E = ⋃_{ν∈N,γ∈G}{x ∈ H : |ℐ_{x,γ,ν}| ≥ C2^{(N−εδ)s}};

we shall see presently that E has very small measure if ε is chosen appropriately. We can now construct a large subset Ω_{x}^{M} of X_{x}^{M} by the following inductive procedure. Set Ω_{x}^{1} equal to X_{x}^{1}. For 1 < M ≤ 2^{2n−3}, a history Γ will be in Ω_{x}^{M} if its predecessor (the truncation of Γ of length M − 1) is in Ω_{x}^{M−1}, the point x_{M−1} is not in the set E, and I_{M} avoids a certain finite collection of ℐ_{x_{M−1},γ_{M} ,ν}. More precisely, for any subsequence k_{1}, …, k_{r} of 1, …, M −1, we shall require that I_{M} is not in ℐ_{x_{M−1},γ_{M} ,ν}, whenever ν is a ray in N which is within C2^{−δs} of the projective ray corresponding to \frac{dx_{M}}{dt_{k_{1}}} ∧ … ∧ \frac{dx_{M}}{dt_{kr}}. It is a routine matter to verify inductively that this condition ensures that all elements of Ω_{x}^{M} are good.

Lemma 3.3. |E| ≤ C_{m}2^{−ms} for all m > 0.

Proof. It suffices to show this bound for a single E_{γ,ν} = {x ∈ H : |ℐ_{x,γ,ν}| ≥ C2^{(1−εδ)s}}. Fix γ and ν, and for each I ∈ ℐ let ψ_{I,γ,ν}(x) be a suitable bump function equal to 1 when I ∈ ℐ_{x,γ,ν} and zero on an 2^{i+εδ}-neighbourhood of that set. We need to show that |{x : ∑_{I} ψ_{I,γ,ν}(x) ≥ C2^{(N−εδ)s}}| ≤ C_{m}2^{−ms} for all m > 0. From the John-Nirenberg inequality

|\{x : |f(x)| ≥ α\}| ≲ e^{−Cα/∥f∥_{BMO}} \frac{∥f∥_{1}}{α} ,

it suffices to show the two estimates

∥ ∑_{I} ψ_{I,γ,ν}∥_{1} ≲ 2^{(N−εδ)s}, ∥ ∑_{I} ψ_{I,γ,ν}∥_{BMO} ≲ s2^{(N−εδ)s}.

By Proposition 3.3, each ψ_{I,γ,ν} is supported on a subset of 2^{s}I_{Δ} of measure O(2^{−εδs}|2^{s}I_{Δ}|) and is essentially constant on sets of width O(2^{i}) as measured by the homogeneous norm ρ, with all constants independent of the choice of ν. Furthermore, ψ_{I,γ,ν} commutes with left-translations of I. One now repeats the argument in the proof of Lemma 3.1 to obtain the desired estimates. There will be O(s) exceptional scales for which one cannot take advantage of either the smoothness or support conditions on the ψ_{I,γ,ν}, and which must be treated individually (using the left-invariance of the ψ_{I,γ,ν}); this is the cause of the additional s factor in the BMO estimate. We omit the details. ■

The complement of Ω_{x}^{M} consists of those histories Γ which are such that there exists a 1 ≤ j < M such that either x_{j} is in E, or that x_{j} is not in E but I_{j+1} is on an exceptional set ℐ_{Γ} consisting of a bounded number of the ℐ_{x_{j},γ_{j+1},ν} as detailed above, so that #ℐ_{Γ} = O(2^{(N−εδ)s}). We shall prove the estimate

∥ ∫_{X_{·}^{M} −Ω_{·}^{M}} |f(x_{M})|dΓ_{+}∥_{p} ≤ C_{p,q}2^{−ε_{p,q}s}∥f∥_{q}

for all 1 < p < q < ∞, which will finish the proof of weak (1,1) boundedness. This however will be immediate from a finite number of applications of the following lemma.

Lemma 3.4. Let M be a fixed integer, and suppose that χ(Γ) is a bounded function defined on ⋃_{x∈H} X_{x}^{M}. Then, if χ satisfies either of the two conditions stated below, one has

∥ ∫_{X_{·}^{M}} |f(x_{M})|χ(Γ)dΓ_{+}∥_{p} ≤ C_{p,q}2^{−ε_{p,q}δs}∥f∥_{q}

for any 1 < p < q < ∞ and f ∈ L^{q}. (I) There exists 0 ≤ k ≤ M and a set E of measure O(2^{−εs}) such that χ(Γ) vanishes whenever x_{k} is outside E. OR (II) There exists 0 ≤ k < M and sets ℐ_{Γ} of cardinality O(2^{(N−εδ)s}) that depend only on x_{0}, Σ_{0}, …, Σ_{k}, γ_{k+1}, such that χ(Γ) vanishes whenever I_{k+1} is not in ℐ_{Γ}.

Proof. Assume first that Condition I holds. The expression whose norm is to be bounded is then majorized pointwise by C(A_{+})^{k}χ_{E}(A_{+})^{M−k}|f|, and so the desired estimate follows immediately from the trivial bounds on A_{+} and the fact that ∥χ_{E}∥_{(q,p)} ≤ C_{q,p}2^{−ε_{q,p}s} for all 1 < p < q < ∞. Now assume that Condition II holds. In this case it will suffice to find (L^{q}, L^{p}) bounds with exponential decay on the operator B defined by

Bf(x) = ∫_{X_{x}^{k}} 2^{−Ns} ∑_{γ∈G} ∑_{I∈ℐ_{Γ}} ψ̃_{I}τ_{I}^{+}Dη̃_{γ}τ_{I}^{+∗}ψ̃_{I}f(x_{k}) dΓ_{+},

since the desired expression is majorized by B(A_{+})^{M−k−1}|f|. Since B is majorized by (A_{+})^{k+1}, it suffices by interpolation to prove a bound of the form ∥B∥_{(∞,p)} ≤ C_{p}2^{−ε_{p}} for some 1 < p < ∞. However, as the operators ψ̃_{I}, τ_{I}^{+}, |G|Dη̃_{γ}, and τ_{I}^{+∗} are all uniformly bounded in L^{∞}, we have

|Bf(x)| ≲ ∫_{X_{x}^{k}} 2^{−Ns}|ℐ_{Γ}|∥f∥_{∞} dΓ_{+} ≲ 2^{−εδs}∥f∥_{∞}(A_{+})^{k}1(x),

and so the estimate follows from the trivial (∞, p) bounds on (A_{+})^{k}. ■

Bibliography

[1] A. Averbuch, G. Beylkin, R. Coifman, and M. Israeli, Fast Adaptive Algorithm for the Elliptic equations with periodic boundary conditions, preprint.

[2] J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147–187.

[3] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309.

[4] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972): 287–299.

[5] S. Chanillo, B. Muckenhoupt, Weak type estimates for Bochner-Riesz spherical summation multipliers, Trans. Amer. Math. Soc. 294 (1986): 693–703.

[6] M. Christ, On almost-everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985): 16–20.

[7] M. Christ, Weak type endpoint bounds for Bochner-Riesz multipliers, Revista Mat. Iberoamericana 3 (1987), 25–31.

[8] M. Christ, Weak type (1,1) bounds for rough operators, Annals of Mathematics, 128 (1988): 19–42.

[9] M. Christ and J. L. Rubio de Francia, Weak-type (1,1) bounds for rough operators II, Invent. Math. 93 (1988): 225–237

[10] M. Christ and C. D. Sogge, The weak-type L^{1} convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math. 94 (1988): 421–453.

[11] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81 (1994): 425–455.

[12] C. Feffermann, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.

[13] C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971): 330–336.

[14] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971): 107–115.

[15] R. Fefferman, A theory of entropy in Fourier Analysis, Adv. in Math. 30 (1978): 171–201.

[16] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Math Notes #28, Princeton Univ. Press.

[17] S. Hofmann, Weak (1, 1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988): 260–264.

[18] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.

[19] S. Kelly, M. A. Kon, and L. A. Raphael, Local convergence for wavelet expansions, J. Funct. Anal. 126 (1994), 102-138.

[20] Y. Meyer, Ondelettes, Hermann, Paris, 1990.

[21] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals II: Singular kernels supported on submanifolds, J. Funct. Anal. 78:56–84.

[22] A. Seeger, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Math. J. 40 (1991): 471-533.

[23] A. Seeger, Endpoint inequalities for Bochner-Riesz multipliers in the plane, to appear, Pacific J. Math.

[24] A. Seeger, Singular integral operators with rough convolution kernels, preprint.

[25] A. Seeger and C. D. Sogge, On the boundedness of functions of (pseudo)-differential operators on compact manifolds, Duke Math. J. 59 (1989): 709–736.

[26] C. D. Sogge, On the convergence of Riesz means on compact manifolds, Annals of Math. 126 (1987), 439–447.

[27] C. D. Sogge, Concerning the L^{p} norm of spectral clusters for secondorder elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988): 123–138.

[28] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Math. # 105, Cambridge Univ. Press, 1993.

[29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[30] E. M. Stein, Harmonic Analysis, Princeton University Press, 1993.

[31] E. M. Stein, M. H. Taibleson, and G. Weiss, Weak-type estimates for maximal operators on certain H^{p} spaces, Rend. Circ. Mat. Palermo suppl. 1 (1981): 81–97.

[32] T. H. Wolff, Recent Work on Sharp Estimates in Second-Order Elliptic Unique Continuation Problems, J. Geom. Anal. 3 (1993): 621–650.

[33] T. H. Wolff, An improved bound for Kakeya type maximal functions, to appear in Revista Mat. Iberoamericana.