THE CALDERÓN COMMUTATOR ALONG A PARABOLA

ANTHONY CARBERY, STEVE HOFMANN, AND JAMES WRIGHT

Abstract. We introduce an analogue of Calderón’s first commutator along a parabola, and establish its L^{2} boundedness under essentially sharp hypotheses.

1. Introduction

During the past 30 years, many authors have investigated the L^{p} mapping properties of singular integral operators whose “kernels” are actually singular measures supported on lower dimensional subvarieties. One may consult the survey article of Wainger [19] for a history of the subject up to the mid 1980’s. The interested reader may also find a synopsis of more recent developments, along with numerous references, in the monograph of Stein [18, Chapter XI, sections 4.5, 4.7, 4.17]. The prototypical example of of such operators is the “Hilbert transform along a curve”:

H_{γ}f(x) ≡ p.v. ∫_{ℝ} f(x − γ(t))\frac{dt}{t}, (1.1)

where x ∈ ℝ^{n} and γ : ℝ → ℝ^{n} is the parametrization of a smooth curve in ℝ^{n}. This sort of operator was first introduced by Fabes [8] who established L^{2} boundedness of H_{γ} in the special case n = 2, γ(t) = (t, (sgnt)t^{2}). The original motivation for studying operators of this sort is that they arise when one tries to develop a “method of rotations” (see [2] in the classical elliptic case) for parabolic singular integrals of convolution type. In turn, the method of rotations enables one to significantly relax the regularity hypotheses on kernels of singular integral operators.

In the present paper, motivated in part by formal analogy to Fabes’ goal of extending the method of rotations to the parabolic case, and in part by recent developments in the theory and application of parabolic singular integrals which are not of convolution type [16], [12], [13], [15], we introduce a certain bilinear analogue of (1.1): the “Calderón commutator along a parabola”. To describe this operator, suppose that A : ℝ^{2} → ℝ satisfies the Lip_{1, \frac{1}{2}} condition

|A(x + h) − A(x)| ≤ B_{0}r, (1.2)

for some constant B_{0}, whenever h ≡ (h_{1}, h_{2}) satisfies |h_{1}| ≤ r, |h_{2}| ≤ r^{2}. We remark that the results of the present paper may be readily extended to ℝ^{n}, n > 2, by the same arguments which we shall give below, but to minimize technicalities we shall restrict our attention to the case n = 2. Since the story here is not yet complete (we do not yet know how to treat the higher order commutators, for example), it does not seem crucial at this point that we state our results in the greatest generality.

The authors were supported in part by EPSRC. Hofmann supported in part by an NSF grant.

Research at MSRI supported in part by NSF grant DMS-9701755.

For A as in (1.2) and for γ(t) ≡ (t, t^{2}), we define

T_{A}^{γ}f(x) ≡ p.v. ∫_{ℝ}[A(x) − A(x − γ(t))]f(x − γ(t))\frac{dt}{t^{2}}. (1.3)

As mentioned above, the operator T_{A} has a connection with the results of [16], [12], [13], and [15], which may be understood as follows. Let K : ℝ^{2}\{0} → ℝ satisfy the parabolic homogeneity property

K(ρx_{1}, ρ^{2}x_{2}) ≡ ρ^{−4}K(x), (1.4)

and further suppose that K ∈ L^{1}(S^{1}), and that K(x_{1}, x_{2}) is even in x_{1}, for each fixed x_{2}. Then one may use parabolic polar co-ordinates to obtain a representation of the parabolic Calderón commutator in terms of T_{A}^{γ}. Indeed,

\begin{aligned}C_{A}f(x) & ≡ p.v. ∫_{ℝ^{2}}[A(x) − A(y)]K(x − y)f(y)dy \\ & ≡ p.v. ∫_{S^{1}} K(σ)T_{A}^{γ_{σ}}f(x)(1 + σ_{2}^{2})dσ, \end{aligned} (1.5)

where T_{A}^{γ_{σ}} is defined as in (1.3), with γ_{σ}(t) ≡ (tσ_{1}, t^{2}σ_{2}), (σ_{1}, σ_{2}) ∈ S^{1}. We have used here the parabolic polar coordinates

x_{1} = ρσ_{1}, x_{2} = ρ^{2}σ_{2}
dx = ρ^{2}dρ(1 + σ_{2}^{2})dσ.

When σ = (±1, 0), L^{2} boundedness of T_{A}^{γ_{σ}} reduces to that of Calderón’s original commutator on the line [1]; when σ = (0, ±1), matters reduce to a result of Murray [17]. Otherwise, L^{2} boundedness of T_{A}^{γ_{σ}} is new and will be treated in this paper, although to simplify the notation we shall take σ_{1} = 1 = σ_{2}. The proof in the general case is identical and one obtains bounds independent of σ ∈ S^{1}.

In [12], it was shown that for the special case

K(x_{1}, x_{2}) ≡ x_{2}^{−2} exp \{\frac{−|x_{1}|^{2}}{4x_{2}} \} χ_{\{x_{2}>0\}}, (1.6)

a necessary and sufficient condition for the L^{2} boundedness of C_{A} is that, for some B < ∞,

\begin{matrix}(i) & |A(x_{1} + h, x_{2}) − A(x_{1}, x_{2})| ≤ B|h| \\ (ii) & ∥𝔻_{2}A∥_{BMO} ≤ B, \end{matrix} (1.7)

where

𝔻_{2}A ≡ (\frac{ξ_{2}}{\sqrt{|ξ_{1}|^{2} − iξ_{2}}} Â(ξ))^{∨} ,

and f̂ and f^{∨} denote the Fourier and inverse Fourier transforms of f, respectively. The BMO norm is defined as usual by

∥b∥_{BMO} = sup ∫_{I} |b − m_{I}b|,

where m_{I}b denotes the mean value of b over I, and where the sup runs over all parabolic “cubes” of the form

I ≡ I_{r}(a, b) ≡ [a − \frac{r}{2}, a + \frac{r}{2}] × [b − \frac{r^{2}}{2} , b + \frac{r^{2}}{2} ] . (1.8)

The higher dimensional case was also treated in [12]. The kernel (1.6) arises in the multilinear expansion of the parabolic double layer potential on a time varying graph x_{3} = A(x_{1}, x_{2}). A condition similar to (1.7) (later shown to be equivalent in [15]) had previously been shown to be sufficient for L^{2} boundedness in [16]. It was also shown in [12] that (1.7) implies the Lip_{1,1/2} condition (1.2), with bound B_{0} ≡ CB.

In this paper we prove the following:

Theorem 1. Suppose A satisfies (1.7). Then T_{A}^{γ} is bounded on L^{2}.

Remarks.

(1) By our previous comments, one may then obtain a “method of rotations” which permits us to deduce L^{2} boundedness of C_{A} (defined in (1.5)) for K as in (1.4) satisfying only that K ∈ L^{1}(S^{1}), and that K(x_{1}, x_{2}) is even in x_{1}, for each fixed x_{2}. Previously, it had been required that K have some smoothness on S^{1}.

(2) One can also show that T_{A}^{γ} is bounded on L^{p}, 1 < p < ∞, by using the key estimate (2.4) below and Littlewood-Paley arguments adapted to rough singular integral operators. See e.g., [14].

In the next section we begin the proof of Theorem 1, and complete it in Section 3.

2. Proof of Theorem 1

We may assume, and do, that the constant B in (1.7) is one. By the parabolic version of [14] (whose proof is virtually identical that given in [14], and is therefore omitted here), it is enough to check the following “rough operator” T 1 criterion, which consists of three parts. We need to prove that

T_{A}^{γ}1, (T_{A}^{γ})^{∗} 1 ∈ BMO, (2.1)

where (T_{A}^{γ})^{∗} is defined as a mapping from test functions to distributions by

⟨(T_{A}^{γ})^{∗} f, g⟩ ≡ ⟨f, T_{A}^{γ}g⟩,

for all f, g ∈ C_{0}^{∞}(ℝ^{2}). Second, for all x ∈ ℝ^{2}, r > 0, let Φ(x, r) denote the class of all φ ∈ C_{0}^{∞}, supported in I_{r}(x) (see (1.7)), and satisfying

(2.2)  (i)  ∥φ∥_{∞} ≤ 1
(ii)  |φ(y + h) − φ(y)| ≤ ρ/r,

whenever |h_{1}| ≤ ρ, |h_{2}| ≤ ρ^{2}, h ≡ (h_{1}, h_{2}), and

(iii)  sup_{k+m≤2} ∥ (\frac{∂}{∂x_{1}} )^{k} (\frac{∂}{∂x_{2}} )^{m} φ∥_{∞} ≤ 1.

We shall need to establish the Weak Boundedness Property (WBP):

|⟨ψ, T_{A}^{γ}ϕ⟩| ≤ Cr^{3}, (2.3)

for all ψ, ϕ ∈ Φ(x, r), for any r > 0, x ∈ ℝ^{2}. Since the homogeneous dimension of parabolic ℝ^{2} is d = 3, (2.1) and (2.3) are the usual David-Journé conditions [7] in this context. However, in lieu of the standard Calderón-Zygmund kernel conditons considered in [7],we shall establish instead the weak smoothness condition of [14], which had also appeared in a similar connection in [3]. Let Q_{s} denote the operator defined by convolution with a smooth function ψ_{s} which has mean value zero, is supported in I_{s}(0), and is normalized so that ∥ψ_{s}∥_{1} = 1. We shall prove that, whenever s ≤ 2^{j},

∥Q_{s}T_{j}∥_{L^{2}→L^{2}} ≤ C (\frac{s}{2^{j}} )^{ϵ} (2.4)

for some ϵ > 0, where

T_{j}f(x) ≡ ∫_{ℝ}[A(x) − A(x − γ(t))]f(x − γ(t))η (\frac{t}{2^{j}} ) \frac{dt}{t^{2}},

and where η ∈ C_{0}^{∞} [(\frac{1}{2}, 2) ∪ (−2, −\frac{1}{2})], 0 ≤ η ≤ 1, and ∑_{j=−∞}^{∞} η (\frac{·}{2^{j}} ) ≡ 1 away from 0.

By [14] (or rather its parabolic analogue), Theorem 1 follows immediately from (2.1), (2.3) and (2.4). In this section we shall prove (2.1) and (2.3). In the next section we prove (2.4).

To establish WBP (2.3), by dilation invariance we may assume r = 1, so it is clearly enough to prove that, for any x_{0} ∈ ℝ^{2},

T_{A}^{γ}φ ∈ L^{2}(I) + L^{∞}(I), ∀φ ∈ Φ(x_{0}, 4), (2.5)

where I = I_{1}(x_{0}). We claim that (2.5) also implies that T_{A}^{γ}1 ∈ BMO (we omit the proof of the fact that (T_{A}^{γ})^{∗}1 ∈ BMO, since it is identical). To prove the claim, we need to consider

\frac{1}{|I|} ∫_{I} |T_{A}^{γ}1 − C_{I}|, (2.6)

where I = I_{r}(x_{0}), and by dilation invariance we may take r = 1. We write

1 = ϕ + (1 − ϕ),

where ϕ ≡ 1 in I_{3}(x_{0}), and ϕ ∈ C_{0}^{∞}(I_{4}(x_{0})). Then up to a normalizing constant, ϕ ∈ Φ(x_{2}, 4), so by (2.5)

∫_{I} |T_{A}^{γ}ϕ| ≤ C.

We now set C_{I} ≡ T_{A}^{γ}(1 − φ)(x_{0}), so that, for all x ∈ I,

|T_{A}^{γ}(1 − ϕ)(x) − T_{A}^{γ}(1 − ϕ)(x_{0})|

≤ ∫ |A(x) − A(x − γ(t)) − [A(x_{0}) − A(x_{0}) − γ(t)]||1 − ϕ(x − γ(t))|\frac{dt}{t^{2}}

+ ∫ |A(x_{0}) − A(x_{0} − γ(t))||ϕ(x − γ(t) − ϕ(x_{0} − γ(t))|\frac{dt}{t^{2}}

≤ c (∫_{|t|>1} \frac{dt}{t^{2}} + ∫_{1<|t|<5} \frac{dt}{t}) = C,

by (1.2) (with B_{0} ≡ C), and the definition of ϕ. Consequently, (2.6) is no larger than C, which proves the claim. Thus, we have reduced the proofs of (2.1) and (2.3) to that of (2.5).

We establish (2.5) under the a priori assumption that A ∈ C^{∞}, but our quantitative estimates will depend only on the bounds in (1.7) (which we have normalized to be 1). Let ϕ ∈ Φ(x_{0}, 4) and then integrate by parts; to do this rigorously requires truncation of the principal value integral in (1.3), but it is routine to verify that the boundary terms (which arise when integrating the truncated integrals by parts) are harmless. We shall therefore argue formally, and ignore all such truncations and boundary terms, in order not to tire the reader with minutiae. Formally then,

\begin{aligned}T_{A}^{γ}ϕ(x) & = ∫_{ℝ} \frac{∂A}{∂x}(x − γ(t))ϕ(x − γ(t))\frac{dt}{t} + 2 ∫_{ℝ} \frac{∂A}{∂t}(x − γ(t))ϕ(x − γ(t))dt \\ & + ∫ [A(x) − A(x − γ(t)] \{\frac{∂ϕ}{∂x}(x − γ(t)) + 2\frac{∂ϕ}{∂t}(x − γ(t))t\} \frac{dt}{t} ≡ I + II + III. \end{aligned}

But I is precisely the Hilbert transform along the curve (t, t^{2}), acting on the L^{2} function \frac{∂A}{∂x}ϕ. Thus, I ∈ L^{2}. Also III ∈ L^{∞}(I), by (1.2) and the fact that ϕ ∈ Φ(x_{0}, 4) (see 2.2). The only delicate term is II. We recall that \frac{∂A}{∂x} ∈ L^{∞}, 𝔻_{2}A ∈ BMO, with norm B = 1 (given our normalization). Define a ≡ 𝔻A ≡ (\sqrt{|ξ_{1}|^{2} − iξ_{2}}Â(ξ))^{∨}, which is therefore in BMO, with norm CB = C by parabolic Calderón-Zygmund Theory [9, 10]. Then

\frac{∂A}{∂t} ≡ 𝔻_{2}(𝔻A) ≡ 𝔻_{2}a,

so that

II = ∫ 𝔻_{2}a(x − γ(t))ϕ(x − γ(t))dt. (2.7)

By [9, Lemma 1 and Lemma 2, pp. 111-113],

𝔻_{2}a(x) ≡ p.v. ∫ k(x − y)a(y)dy,

where k(x) is odd, belongs to C^{∞}(ℝ^{2}\{0}) and satisfies the homogeneity property

k(λx, λ^{2}t) ≡ λ^{−d−1}k(x, t) (2.8)

(we recall that d = 3 is the homogeneous dimension of parabolic ℝ^{2}). By the oddness of k, we may assume that a has mean value zero on I_{1}(x_{0}) ≡ I. Let ∥x∥ ≡ |x_{1}|+|x_{2}|^{1/2} be the parabolic “norm” of x. Let η ∈ C_{0}^{∞}[−10, 10], η ≡ 1 on [−9, 9], and set a_{1}(x) = a(x)η(∥x − x_{0}∥), a_{2} = a − a_{1}. By (2.8) and the parabolic version of a standard estimate of [11],

\begin{aligned}|𝔻_{2}a_{2}(x)| & ≤ ∫ \frac{c}{1 + ∥x − y∥^{d+1}} |a(y)|dy \\ & ≤ C∥a∥_{BMO} ≤ C, \end{aligned}

whenever x ∈ supp ϕ ⊆ I_{4}(x_{0}). Consequently the contribution of a_{2} to (2.7) yields a bounded term, as desired. The contribution of a_{1} is

∫ ϕ(x − γ(t))η(t)𝔻_{2}a_{1}(x − γ(t))dt

(since x ∈ I_{1}(x_{0}), x − γ(t) ∈ I_{4}(x_{0}))

= ∫ η(t) ∫ k(x − γ(t) − y)[ϕ(x − γ(t)) − ϕ(y)]a_{1}(y)dydt

+ ∫ η(t)𝔻_{2}(ϕa_{1})(x − γ(t))dt

≡ II_{1} + II_{2}.

By [12] (or even [9], since ϕ is smooth), the commutator [𝔻_{2}, ϕ] defines a bounded operator on L^{2}. Hence, by Minkowksi’s inequality,

\begin{aligned}∥II_{1}∥_{2} & ≤ ∫ η(t)∥[𝔻_{2}, ϕ]a_{1}(· − γ(t)∥_{2}dt \\ & ≤ c∥a_{1}∥_{2} ≤ c∥a∥_{BMO} ≤ C. \end{aligned}

Finally, since ϕa_{1} ∈ L^{2}, and 𝔻_{2} is given by the multiplier ξ_{2}(|ξ_{1}|^{2} − iξ_{2})^{−1/2}, it is enough to show that the operator

f → ∫ f(x − γ(t))η(t)dt

is smoothing on L^{2} of parabolic order 1; i.e., that the multiplier

m(ξ) ≡ ∫ e^{i(ξ_{1}t+ξ_{2}t^{2})}η(t)dt

satisfies

|m(ξ)| ≤ C(|ξ_{1}| + |ξ_{2}|^{1/2})^{−1}

This estimate is well known, and is an easy consequence of standard integration by parts arguments as may be found in [18, Chapter VIII]. For the sake of completeness, we sketch the argument here.

Case 1. |ξ_{1}| > 40|ξ_{2}|.

In this case,

|n(ξ)| = |∫ e^{i(ξ_{1}t+ξ_{2}t^{2})} \frac{d}{dt} (\frac{1}{ξ_{1} + 2 + ξ_{2}} η^{0}(t)) dt| ≤ C|ξ_{1}|^{−1},

Since supp η ≤ [−10, 10].

Case 2. |ξ_{1}| < 40|ξ_{2}|.

In this case we seek the estimate |m(ξ)| ≤ C|ξ_{2}|^{−1/2}. We write

m(ξ) = ∫ e^{iλφ_{ξ}(t)}η(t)dt,

where λ ≡ |ξ_{2}|, and ϕ_{ξ}(t) ≡ (\frac{ξ_{1}}{|ξ_{2}|}t + sgnξ_{2}t^{2}). Note that |ϕ′′_{ξ}(t)| = 2. The desired bound now follows immediately from Vander Corput’s Lemma, or to be more precise, its corollary given in [18, p. 334, inequality (6)]. This concludes the proof of (2.5), and therefore also the proofs of (2.1) and (2.3). We finish the proof of our Theorem in the next section, in which we prove (2.4).

3. Proof of Theorem 1 (continued): estimate (2.4)

In this section, we give the proof of estimate (2.4), which will complete the proof of Theorem 1. This is the most technical part of the proof, and follows ideas from [6]; see also [4] and [5] .

By dilation invariance, we may take j = 0. We recall that T_{0}f is a sum of two terms

T̃_{0}f(x) ≡ ∫_{ℝ}[A(x) − A(x − γ(t))]f(x − γ(t))η(t)\frac{dt}{t^{2}}

where η is an even smooth cut-off function supported in the “half annulus” \frac{1}{2} < t < 2, plus another term with −2 < t < −\frac{1}{2}. By symmetry, we treat only the former. Let K_{0} denote the kernel of T̃_{0}^{∗}T̃_{0}^{∗}, where T̃_{0}^{∗} denotes the adjoint of T̃_{∗}. Following [6], we see that the desired bound (2.4) then follows easily (we omit the routine details) from

Claim.

∫ |K_{0}(x + h, y) − K(x, y)|dy ≤ Cλ^{1/3}, (3.1)

whenever |h_{1}| ≤ λ, |h_{2}| ≤ λ^{2}, λ ≤ 1/1000. Now, it is a routine matter to see that

T̃_{0}^{∗}g(x) = − ∫ [A(x) − A(x + γ(s))]g(x + γ(s))η(s)\frac{ds}{s^{2}}

Thus,

\begin{aligned}T̃_{0}T̃_{0}^{∗}f(w) & = ∫ [A(w) − A(w − γ(t))]T_{0}^{∗}f(ω − γ(t))ϕ(t)dt \\ & = − ∫∫ [A(ω) − A(ω − γ(t))] \\ & [A(ω − γ(t)) − A(ω + γ(s) − γ(t)]f(ω + γ(s) − γ(t))ϕ(t)ϕ(s)dtds, \end{aligned} (3.2)

where for convenience of notation we have defined

ϕ(t) ≡ η(t)t^{−2}.

Now, the claim (3.1) amounts to saying that the operator

f → ∫ [K_{0}(· + h, y) − K_{0}(·, y)]f(y)dy

maps L^{∞} → L^{∞} with norm Cλ^{1/3}. Hence, we may with impunity excise (in (3.2)) any set in (s, t) space of measure no larger than Cλ^{1/3}. Indeed, we shall restrict the domain of integration in (3.2) to the set

\{(s, t) ∈ [\frac{1}{2}, 2]^{2} : |s − t| ≥ 15λ^{\frac{1}{3}} \} .

By symmetry it suffices to consider only the case s > t. Note that in the set where s − t ≥ λ^{\frac{1}{3}} , the map

\begin{matrix}Φ_{ω} : (s, t) & ↦ ω + γ(s) − γ(t) = y \\ \frac{1}{4} ≤ s < t ≤ 4 & ↦ ℝ^{2} \end{matrix} (3.3)

is injective, and that the Jacobian matrix is

[ \begin{matrix}1 & −1 \\ 2s & −2t \end{matrix} ]

with determinant 2(s − t) ≥ 2λ^{1/3}. Let S_{0} denote the part of the operator T̃_{0}T̃_{0}^{∗} in (3.2), with (s − t) > 15λ^{1/3}. We then have that

|S_{0}f(x + h) − S_{0}f(x)| = | ∫∫ _{Eλ} ϕ(s)ϕ(t) {[A(x + h) − A(x + h − γ(t)][A(x + h − γ(t)]

[A(x + h − γ(t)) − A(x + h + γ(s) − γ(t))]f(x + h + γ(s) − γ(t))

− [A(x) − A(x − γ(t))]

[A(x − γ(t)) − A(x + γ(s) − γ(t)]f(x + γ(s) − γ(t))} dsdt|

≡ | ∫∫ _{E_{λ}} ϕ(s)ϕ(t) {B(x + h, s, t)f(x + h + γ(s) − γ(t))
− B(x, s, t)f(x + γ(s) − γ(t)} dsdt|

(where E_{λ} ≡ \{(s, t) : s − t ≥ 15λ^{1/3}, \frac{1}{2} ≤ t < s ≤ 2\}, and B(x, s, t) ≡ [A(x) − A(x − γ(t))][A(x − γ(t)) − A(x + γ(s) − γ(t))])

= | ∫∫ _{E_{λ}} ϕ(s)ϕ(t) {[B(x + h, s, t) − B(x, s, t)]f(x + h + γ(s) − γ(t))
+ B(x, s, t)[f(x + h + γ(s) − γ(t)) − f(x + γ(s) − γ(t))]dsdt|
≡ | ∫∫ (I + II)dsdt|.

Now suppose that ∥f∥_{L^{∞}=1}. Taking the supremum over all such f, we see that the contribution of ∫∫ Idsdt to (3.1) satisfies the claim, by virtue of the Lip(1, \frac{1}{2}) character of A.

Next,

\begin{aligned}| ∫∫ IIdsdt| & = | ∫_{Φ_{x+h}(E_{λ})} B(x, Φ_{x+h}^{−1}(y))f(y)JΦ_{x+h}^{−1}(y)dy \\ & − ∫_{Φ_{x}(ξ_{λ})} B(x, Φ_{x}^{−1}(y))f(y)JΦ_{x}^{−1}(y)dy \end{aligned}

(where Φ_{x} is the mapping defined in (3.3))

= | ∫_{Φ_{x+h}(E_{λ})} \frac{[B(x, Φ_{x+h}^{−1}(y)) − B(x, Φ_{x}^{−1}(y)]}{JΦ_{x+h}(Φ_{x+h}^{−1}(y)]}f(y)dy

+ ∫_{Φ_{x+h}(E_{λ})} [\frac{1}{JΦ_{x}(Φ_{x+h}^{−1}(y))} − \frac{1}{JΦ_{x}(Φ_{x}^{−1}(y))}] B(x, Φ_{x}^{−1}(y))f(y)dy

+ ∫_{Φ_{x+h}(E_{λ})\Φ_{x}(E_{λ})} \frac{B(x, Φ_{x}^{−1}(y))}{JΦ_{x}(Φ_{x}^{−1}(y))}f(y)dy − ∫_{Φ_{x}(E_{λ})\Φ_{x+h}(E_{λ})} \frac{B(x, Φ_{λ}^{−1}(y))}{JΦ_{x}(Φ_{x}^{−1}(y))}f(y)dy|

≡ | ∫ (III + IV + V + V I)dy|,

where we have used that JΦ_{x}(s, t) is independent of x. Then

(3.4) | ∫ IIIdy| ≤ c∥f∥_{∞} ∫_{Φ_{x+h}(E_{λ})} |B(x, Φ_{x+h}^{−1}(y)) − B(x, Φ_{x}^{−1}(y))|JΦ_{x+h}^{−1}(y)dy
= C∥f∥_{∞} ∫_{E_{λ}} |B(x, s, t) − B(x, s_{h}, t_{h})|dsdt,

where

(3.5)  (s_{h}, t_{h}) ≡ Φ_{x}^{−1}(Φ_{x+h}(s, t)),

i.e.

x + γ(s_{h}) − γ(t_{h}) = x + h + γ(s) − γ(t)
⇔  γ(s_{h}) − γ(t_{h}) = h + γ(s) − γ(t)
⇔  (s_{h} − t_{h}, s_{h}^{2} − t_{h}^{2}) = (h_{1} + s − t, h_{2} + s^{2} − t^{2}).

(by definition of γ) i.e.,

s_{h} = \frac{1}{2} \{h_{1} + s − t + \frac{h_{2} + s^{2} − t^{2}}{h_{1} + s − t} \}
t_{h} = \frac{1}{2} \{t − s − h_{1} + \frac{h_{2} + s^{2} − t^{2}}{h_{1} + s − t} \}

Now,

\begin{aligned}|B(x, s, t) − B(x, s_{h}, t_{h})| & ≡ |[A(x) − A(x − γ(t))][A(x − γ(t)) − A(x + γ(s) − γ(t)] \\ & − [A(x) − A(x − γ(t_{h}))][A(x − γ(t_{h}) − A(x + γ(s_{h}) − γ(t_{h})]| \\ & = |[A(x) − A(x − γ(t)) − \{A(x) − A(x − γ(t_{h}))\}][A(x − γ(t)) \\ & − A(x + γ(s) − γ(t)] \\ & + [A(x) − A(x − γ(t_{h})][A(x − γ(t)) − A(x + γ(s) − γ(t)) \\ & − \{A(x − γ(t_{h}) − A(x + γ(s_{h}) − γ(t_{h})\}]| \\ & ≤ C(∥γ(t_{h}) − γ(t)∥ + ∥γ(s_{h}) − γ(s)∥), \end{aligned} (3.6)

where ∥ · ∥ denotes the parabolic metric ∥(u, v)∥ ≅ |u| + |v|^{1/2}. Notice that

|t_{h}^{2} − t^{2}| ≅ |t_{h} − t|,

and similarly for s_{h}^{2} − s^{2}, so that

∥γ(t_{h}) − γ(t)∥ ≤ C|t_{h} − t|^{1/2}
∥γ(s_{h}) − γ(s)∥ ≤ C|s_{h} − s|^{1/2}.

Furthermore

\begin{aligned}|t^{·} − t_{h}| & = |\frac{1}{2} \{t + s + h_{1} − (\frac{h_{2} + s^{2} − t^{2}}{h_{1} + s − t} )\}| \\ & = |\frac{h_{1}}{2} + \frac{1}{2} \{\frac{(h_{1} + (s − t))(t + s) − (h_{2} + s^{2} − t^{2})}{h_{1} + s − t}\}| \\ & = |\frac{h_{1}}{2} + \frac{1}{2} \{\frac{h_{1}(t + s) − h_{2}}{h_{1} + s − t} \}| \\ & ≤ λ^{2/3}, \end{aligned} (3.7)

since |h_{1}| ≤ λ ≤ 1, |h_{2}| ≤ λ^{2}, and s − t > 15λ^{1/3}, \frac{1}{2} ≤ s, t ≤ 2, on E_{λ}. A similar estimate holds for |s − s_{h}|. Hence ∫ (III)dy ≤ Cλ^{1/3} as desired, by virtue of (3.4) and (3.6). Next, we observe that

\begin{aligned}| ∫ IV dy| & ≤ C∥f∥_{∞} ∫_{Φ_{x+h}(E_{λ})} |\frac{1}{JΦ_{x}(Φ_{x+h}^{−1}(y))} − \frac{1}{JΦ_{x}(Φ_{x}^{−1}(y))}| dy \\ & = C∥f∥_{∞} ∫∫ _{E_{λ}} |\frac{1}{s − t} − \frac{1}{s_{h} − t_{h}} | (s − t)dsdt \end{aligned}

where (s_{h}, t_{h}) is defined as above (see (3.5)), and where we have used that JΦ_{x}(s, t) = 2(s − t). As we have observed (see (3.7)), |t − t_{h}| + |s − s_{h}| ≤ 2λ^{2/3} << 15λ^{\frac{1}{3}} ≤ s − t, so that

\begin{aligned}| ∫ IV dy| & ≤ C∥f∥_{∞} ∫ ∫_{E_{λ}} \frac{λ^{2/3}}{(s − t)^{2}} (s − t)dsdt \\ & ≤ C∥f∥_{∞}λ^{1/3} \end{aligned}

as desired.

Turning last to the term | ∫ V dy| (the term | ∫ V Idy| can be handled by similar arguments, which we omit) we see that, since JΦ_{x}(s, t) = 2(s − t),

\begin{aligned}| ∫ V dy| & ≤ C∥f∥_{∞} ∫_{Φ_{x+h}(E_{λ})\Φ_{x}(E_{λ})} \frac{1}{JΦ_{x}(Φ_{x}^{−1}(y))}dy \\ & ≤ C∥f∥_{∞} ∫∫ _{F_{λ}} \frac{1}{s_{h} − t_{h}} (s − t)dsdt, \end{aligned}

where F_{λ} = Φ_{x+h}^{−1}(Φ_{x+h}(E_{λ}) − Φ_{x}(E_{λ})), and where (s_{h}, t_{h}) is defined as above (see (3.5)). Since s_{h} − t_{h} ≥ cλ^{1/3}, it is enough to show that |F_{λ}| ≤ Cλ^{2/3}.

To this end, suppose that (s, t) ∈ F_{λ}. Then, in particular, (s, t) ∈ E_{λ}, and y ≡ Φ_{x+h}(s, t) ∈ Φ_{x+h}(E_{λ})\Φ_{x}(E_{λ}), i.e., y ≠ Φ_{λ}(s′, t′) for any (s′, t′) ∈ E_{λ}. On the other hand, we have observed previously that y = Φ_{x}(s_{h}, t_{h}) for some (s_{h}, t_{h}) satisfying

|s_{h} − s| + |t_{h} − t| ≤ 12λ^{2/3}.

Since (s_{h}, t_{h}) ∉ E_{λ}, (s, t) ∈ E_{λ}, we must have dist((s, t), ∂E_{λ}) ≤ Cλ^{2/3}. Thus |F_{λ}| ≤ Cλ^{2/3} as desired, and the proof is done.

References

[1] A.P. Calderón, Commutators of Singular Integral Operators, Proc. Nat. Acad. Sci. USA 53 (1965), 1092-1099.

[2] A.P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math 78 (1956), 249-271.

[3] A. Carbery and A. Seeger, Conditionally convergent series of linear operators on L^{p} spaces and L^{p} estimates for pseudodifferential operators, Proc. Lond. Math. Soc. 57 (1988), 481-510.

[4] A. Carbery, J. Vance, S. Wainger, D. Watson and J. Wright, L^{p} estimates for operators associated to flat curves without the Fourier transform, Pacific J. Math. 167 (1995), 243-262.

[5] A. Carbery, S. Wainger and J. Wright, The Hilbert transform and maximal function along flat curves in the Heisenberg group, J. Amer. Math. Soc. 8 (1995), 141-179.

[6] M. Christ, Hilbert transforms along curves I: nilpotent groups, Ann. of Math. 122 (1985), 575-596.

[7] G. David and J.L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397.

[8] E.B. Fabes, Singular integrals and partial differential equaitons of parabolic type, Studia Math. 28 (1966), 81-131.

[9] E.B. Fabes and N. Riviere, Symbolic calculus of kernels with mixed homogeneity, in Singular Integrals, A.P. Calderon, Ed., Proc. Symp. Pure Math. 10, Amer. Math. Soc., Providence 1967, pp. 106-127.

[10] E.B. Fabes and N. Riviere, Singular integrals with mixed homogeneity, Studia. Math. 27 (1966), 19-38.

[11] C. Fefferman and E.M. Stein, H^{p} spaces of several variables, Acta Math. 129 (1972), 137-193.

[12] S. Hofmann, A characterization of commutators of parabolic singular integrals, in Fourier Analysis and Partial Differential Equaitons, J. Garcia-Cuerva, E. Hernandez, F. Soria, and J.-L. Torrea, editors, CRC Press, Boca Raton, 1995.

[13] S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, to appear in Duke Math. J.

[14] S. Hofmann, On certain nonstandard Calderón-Zygmund operators. Studia Math 109 (1994), 105-131.

[15] S. Hofmann and John Lewis, L^{2} solvability and representation by caloric layer potentials in time varying domains, Ann. of Math 144 (1996), 349-420.

[16] J. Lewis and M. Murray, The method of layer potentials for the heat equation in time varying domains, Memoirs of the Amer. Math. Soc. 545 (1995).

[17] M. Murray, Commutators with fractional differentiation and BMO Sobolev spaces, Indiana University Math. J. 34 (1985), 205-215.

[18] E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, N.J. 1993.

[19] S. Wainger, Averages and singular integrals over lower dimensional sets, in Beijing Lectures in Harmonic Analysis, Ann. of Math Studies 112, E.M. Stein, Ed., Princeton Univ. Press, 1986, pp. 357-421.

Anthony Carbery, Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, Scotland EH9 3JZ

E-mail address: carbery@maths.ed.ac.uk

Steve Hofmann, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address: hofmann@math.missouri.edu

James Wright, Department of Mathematics, University of New South Wales, Sydney, N.S.W. 2052, Australia

E-mail address: jimw@maths.unsw.edu.au