SINGULAR INTEGRALS ASSOCIATED TO HYPERSURFACES: L^{2} THEORY

STEPHEN WAINGER, JAMES WRIGHT, AND SARAH ZIESLER

Abstract. We consider singular integrals associated to a classical Calderón-Zygmund kernel K and a hypersurface given by the graph of φ(ψ(t)) where φ is an arbitrary C^{1} function and ψ is a smooth convex function of finite type. We give a characterization of those Calderón-Zygmund kernels K and convex functions ψ so that the associated singular integral operator is bounded on L^{2} for all C^{1} functions φ.

1. Introduction

The main purpose of this paper is to investigate the L^{2} boundedness of singular integral operators associated to hypersurfaces in ℝ^{n}, n ≥ 3. Let Γ(t) be a C^{1} mapping from a neighborhood of the origin in ℝ^{n−1} into ℝ^{n} with Γ(0) = 0. For x in ℝ^{n} and f a C^{1} function with compact support in ℝ^{n}, we set

Hf(x) = lim_{ϵ→0} ∫_{ϵ≤|t|≤1} f(x − Γ(t))K(t) dt

where K(t) is a Calderón-Zygmund kernel in ℝ^{n−1}. That is, K is smooth (C^{∞}) away from the origin,

∫_{a≤|t|≤b} K(t) dt = 0

for every 0 < a < b, and

K(λt) = λ^{−n+1}K(t)

for every λ > 0.

It is known that if Γ(t) is smooth and the vectors \{\frac{∂^{α}Γ}{∂t^{α}}(0)\}, given by the derivatives of Γ at the origin, span ℝ^{n}, then

∥Hf∥_{L^{p}} ≤ A_{p} ∥f∥_{L^{p}}, 1 < p < ∞.

See [St] for this result. Our main interest is studying what happens when the vectors \frac{∂^{α}Γ}{∂t^{α}}(0) do not span ℝ^{n}. We shall consider surfaces of the form

Γ(t) = (t, φ(ψ(t)))

Wainger was supported in part by an NSF grant. Wright was supported in part by an ARC grant.

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

where t = (t_{1}, …, t_{n−1}) and ψ(t) is a smooth convex function of finite type with ψ(0) = ∇ψ(0) = 0. (We say that ψ(t) is of finite type if the graph t_{n} = ψ(t) has no lines tangent to infinite order.) If ψ(t) = |t|^{2} = t_{1}^{2} + ⋯ + t_{n−1}^{2}, then

∥Hf∥_{L^{2}} ≤ A ∥f∥_{L^{2}}

for any φ. The details of this easy calculation can be found in [KWWZ].

The main purpose of this paper is to decide for what convex functions of finite type ψ and Calderón-Zygmund kernels K do we have

∥Hf∥_{L^{2}} ≤ A∥f∥_{L^{2}}

for all C^{1} functions φ with φ(0) = 0. To give the answer to this problem we introduce certain sets which were considered by Schulz, [Sch]. Let

E_{ℓ} = {v ∈ ℝ^{n−1} | ψ(sv) = 𝒪(s^{ℓ+1}) for small s > 0}.

From the convexity of ψ, each E_{ℓ} is a linear subspace of ℝ^{n−1}. Clearly E_{1} = ℝ^{n−1}, E_{ℓ+1} ⊆ E_{ℓ} and ⋂ E_{ℓ} = {0} (from the finite type condition). We let ℓ_{0} be the smallest value of ℓ such that E_{ℓ} is not all of ℝ^{n−1}. We then have the following theorem.

Theorem 1. If the codimension of E_{ℓ_{0}} in ℝ^{n−1} is at least 2, then

∥Hf∥_{L^{2}} ≤ A ∥f∥_{L^{2}}

for all C^{1} functions φ with φ(0) = 0.

If the codimension of E_{ℓ_{0}} is 1, then H is bounded on L^{2} for all C^{1} functions φ if and only if K satisfies an additional cancellation condition.

Theorem 2. Suppose the codimension of E_{ℓ_{0}} is 1, and let v be a non-zero vector in E_{ℓ_{0}}^{⊥}. Then

∥Hf∥_{L^{2}} ≤ A ∥f∥_{L^{2}}

for all C^{1} functions φ with φ(0) = 0 if and only if K(t) satisfies the additional cancellation condition

∫_{{v·t≥0 \atop a≤|t|≤b}} K(t) dt = 0 (1.1)

for all 0 < a < b.

Remarks.

(1) The positive assertions in Theorems 1 and 2 hold for the more general operators

Hf(x) = ∫ b(ψ(t))K(t)f(x − Γ(t)) dt

for any bounded function b, with no change in the proof.

(2) Theorem 1 is vacuous and Theorem 2 is trivially true when n = 2, and so nothing new is being proved for singular integrals along curves in the plane.

Examples.

(1) ψ(x, y, z) = x^{2} + y^{2} + z^{4} is a convex function of finite type where ℓ_{0} = 2 and

E_{ℓ_{0}} = {(0, 0, z) | z ∈ ℝ}

has codimension 2 and so Theorem 1 applies.

(2) ψ(x, y, z) = x^{2} + y^{4} + z^{4} is also a convex function of finite type where ℓ_{0} = 2 but

E_{ℓ_{0}} = {(0, y, z) | y, z ∈ ℝ}

has codimension 1 and so Theorem 2 applies with v = (1, 0, 0).

Next we turn to examine what happens when the cancellation condition (1.1) is not satisfied.

Theorem 3. Let φ̄(s) = φ(s^{ℓ_{0}}). Assume the codimension of E_{ℓ_{0}} is 1, and the cancellation condition (1.1) fails. Then if φ̄(s) is convex,

∥Hf∥_{L^{2}} ≤ A ∥f∥_{L^{2}}

if and only if

φ̄′(Cs) ≥ 2φ̄′(s)

for some C ≥ 1 and all 0 < s ≤ 1.

Remarks.

(1) The significance of the power ℓ_{0} is that \frac{1}{ℓ_{0}} is the smallest power α such that [ψ(t)]^{α} is a convex function.

(2) When ϕ(s) = |s|^{2} and so n = 1, Theorem 3 was proved in [NVWW].

If E_{ℓ_{0}} = {0}, which means that ψ is approximately homogeneous of degree ℓ_{0}, we obtain L^{p} results for H and the corresponding maximal function

Mf(x) = sup_{0<h≤1} \frac{1}{h^{n−1}} ∫_{|t|≤h} |f(x − Γ(t))| dt.

We again set φ̄(s) = φ(s^{ℓ_{0}}).

Theorem 4. Suppose n ≥ 3, E_{ℓ_{0}} = {0} and φ̄(s) is convex. Then

∥Hf∥_{L^{p}} ≤ A_{p} ∥f∥_{L^{p}}, 1 < p < ∞,

and

∥Mf∥_{L^{p}} ≤ A_{p} ∥f∥_{L^{p}}, 1 < p ≤ ∞.

Remark. It is known that the assertion of Theorem 4 fails in general if the hypothesis that φ̄ is convex is dropped, even if ψ(t) = |t|^{2}. See [SWWZ].

Finally we make one observation in ℝ^{3} in the case that ψ(t) is not of finite type. Let t_{0} be a point on the curve ψ(t) = 1 and ℓ(t_{0}) denote the line tangent to ψ(t) = 1 at t_{0}. Set

E(t_{0}, ϵ) = {s ∈ ℝ^{2} | ψ(s) = 1 and dist(s, ℓ(t_{0})) ≤ ϵ}.

Theorem 5. Assume ψ(t) is convex and homogeneous of degree 1. Then if

sup_{{t_{0} \atop ψ(t_{0})=1}} ∫_{0}^{1} |E(t_{0}, ϵ)| \frac{dϵ}{ϵ} < ∞,

∥Hf∥_{L^{2}} ≤ A ∥f∥_{L^{2}} for every C^{1} function φ.

Example. Consider a smooth convex function ψ(x, y), homogeneous of degree 1, such that for |x| << |y|,

ψ(x, y) = \sqrt{x^{2} + y^{2}} exp (− (\frac{\sqrt{x^{2} + y^{2}}}{|x|} )^{α}) .

Clearly ψ is not of finite type and the integrability condition in Theorem 5 is satisfied exactly when α < 1.

In section 2 we will prove Theorems 1 and 2 in the special cases where ψ(x, y, z) = x^{2} + y^{2} + z^{4} (for Theorem 1) and ψ(x, y, z) = x^{2} + y^{4} + z^{4} (for Theorem 2), where the main direction of the proof is not clouded by intricate estimates. The proof for Theorem 1 in the general case will be given in section 3. Theorems 2 and 3 will be proved in section 4 and sections 5 and 6 contain the proofs of Theorems 4 and 5 respectively.

Our work is heavily dependent on ideas of Schulz, [Sch]. We would like to thank A. Iosevich for bringing the paper [Sch] to our attention. We would also like to thank Professor A. Carbery for evaluating a determinant for us.

2. Special Cases

In this section we will prove Theorems 1 and 2 in the special cases ψ(x, y, z) = x^{2} + y^{2} + z^{4} and ψ(x, y, z) = x^{2} + y^{4} + z^{4}. We begin with ψ(x, y, z) = x^{2} + y^{2} + z^{4}. Here no further cancellation condition is required for the Calderón-Zygmund kernel K. We need to show

| ∫_{ϵ≤x^{2}+y^{2}+z^{2}≤1} e^{iγφ(x^{2}+y^{2}+z^{4})}e^{iηz}e^{i(ξ_{1}x+ξ_{2}y)}K(x, y, z) dx dy dz| ≤ B (2.1)

uniformly in ξ = (ξ_{1}, ξ_{2}), η, γ and ϵ > 0. Introducing polar coordinates in the (x, y) integral, the integral in (2.1) becomes

(2.2) ∫_{ϵ≤r^{2}+z^{2}≤1} e^{iγφ(r^{2}+z^{4})}e^{iηz}r ∫_{0}^{2π}e^{ir(ξ_{1} cos θ+ξ_{2} sin θ)}K(r cos θ, r sin θ, z) dθ dr dz.

We split the integral in (2.2) as a sum of two integrals I_{1}+I_{2} where the r integration in I_{1} is restricted to r|ξ| ≥ 1 and where the integration in I_{2} is over the complementary range. Using the fact that the θ integral in (2.2) is the Fourier transform of a smooth density on the unit circle, we see that

|I_{1}| ≤ C ∫_{r|ξ|≥1} \frac{r^{2}}{(r|ξ|)^{1/2}} ∫_{−∞}^{∞} \frac{1}{(r^{2} + z^{2})^{4/2}} + \frac{1}{r(r^{3} + |z|^{3})} dz dr = C ∫_{r|ξ|≥1} \frac{1}{(r|ξ|)^{1/2}} \frac{dr}{r} ≤ C.

In I_{2} we replace e^{ir(ξ_{1} cos θ+ξ_{2} sin θ)} with 1, creating an error at most a multiple of

∫_{r|ξ|≤1} r^{2}|ξ| ∫_{−∞}^{∞} \frac{1}{(r^{2} + z^{2})^{3/2}} dz dr = C ∫_{r|ξ|≤1} |ξ| dr = C.

Therefore the integral in (2.2) is

∫_{0}^{2π}∫ ∫_{{ϵ≤r^{2}+z^{2}≤1 \atop r|ξ|≤1}} e^{iγφ(r^{2}+z^{4})}e^{iηz}K(r cos θ, r sin θ, z)r dr dz dθ + 𝒪(1).

Furthermore the (r, z) integration may be further restricted to the region where |z| ≤ δr^{1/2} since integrating K over the complementary region is at most

∫ ∫_{δr^{1/2}≤|z|≤1} \frac{1}{(r^{2} + z^{2})^{3/2}} r dr dz ≤ ∫_{|z|≤1} \frac{1}{|z|^{3}} ∫_{r≤( \frac{1}{δ} |z|)^{2}} r dr ≤ C.

With the restriction |z| ≤ δr^{1/2} for small δ > 0, we may make the change of variables λ = \sqrt{r^{2} + z^{4}} (so that λ ∼ r) in the r integral to reduce matters to showing that the integral

I = ∫_{0}^{2π} ∫_{λ|ξ|≤1} e^{iγφ(λ^{2})}λ ∫_{{|z|≤δλ^{1/2} \atop ϵ≤λ^{2}+z^{2}≤1}} e^{iηz}K(\sqrt{λ^{2} − z^{4}} cos θ, \sqrt{λ^{2} − z^{4}} sin θ, z) dz dλ dθ

is uniformly bounded in γ, η, ξ and ϵ > 0. Replacing \sqrt{λ^{2} − z^{4}} by λ in I creates an error at most

C ∫_{0}^{1} ∫_{|z|≤λ^{1/2}} \frac{z^{4}}{(z^{2} + λ^{2})^{4/2}} dz dλ ≤ C ∫_{0}^{1} λ^{1/2} dλ ≤ C

and so

I = ∫_{0}^{2π} ∫_{λ|ξ|≤1} e^{iγφ(λ^{2})}λ ∫_{{|z|≤δλ^{1/2} \atop ϵ≤λ^{2}+z^{2}≤1}} e^{iηz}K(λ cos θ, λ sin θ, z) dz dλ dθ + 𝒪(1).

Next we will see that we can replace the oscillatory factor e^{iηz} with 1 in the above integral if we further restrict the λ integration to λ ≤ \frac{1}{|η|}. In fact we can integrate by parts in the z integral to see that the part of the integral where λ|η| ≥ 1 is at most

C \frac{1}{|η|} ∫_{λ|η|≥1} λ ∫_{−∞}^{∞} \frac{1}{(z^{2} + λ^{2})^{4/2}} dz ≤ C \frac{1}{|η|} ∫_{λ|η|≥1} \frac{1}{λ^{2}} ≤ C.

For λ|η| ≤ 1, replacing e^{iηz} by 1 creates an error at most

C |η| ∫_{λ|η|≤1} λ ∫_{−∞}^{∞} \frac{|z|}{(λ^{2} + z^{2})^{3/2}} dz dλ ≤ C |η| ∫_{λ|η|≤1} dλ ≤ C.

Therefore

I = ∫_{0}^{2π} ∫_{{λ|ξ|≤1 \atop λ|η|≤1}} e^{iγφ(λ^{2})}λ ∫_{{|z|≤δλ^{1/2} \atop ϵ≤λ^{2}+z^{2}≤1}} K(λ cos θ, λ sin θ, z) dz dλ dθ + 𝒪(1)

= ∫_{0}^{2π} ∫_{{λ|ξ|≤1 \atop λ|η|≤1}} e^{iγφ(λ^{2})} \frac{1}{λ} ∫_{{|s|≤δλ^{−1/2} \atop ϵ≤λ^{2}(1+s^{2})≤1}} K(cos θ, sin θ, s) ds dλ dθ + 𝒪(1)

= ∫_{0}^{2π} ∫_{{λ|ξ|≤1 \atop λ|η|≤1}} e^{iγφ(λ^{2})} \frac{1}{λ} ∫_{ϵ≤λ^{2}(1+s^{2})≤1} K(cos θ, sin θ, s) ds dλ dθ + 𝒪(1)

= − ∫_{{λ|ξ|≤1 \atop λ|η|≤1}} e^{iγφ(λ^{2})} \frac{1}{λ} ∫_{0}^{2π} ∫_{{0 \atop λ≤sin ψ≤ \frac{λ}{\sqrt{ϵ}}}}^{π} K(sin ψ cos θ, sin ψ sin θ, cos ψ) sin ψ dψ dθ dλ + 𝒪(1).

Here we made the change of variables z = sλ followed by s = cot ψ in the z integral. Using the fact that

0 = ∫_{0}^{2π}∫_{0}^{π} K(sin ψ cos θ, sin ψ sin θ, cos ψ) sin ψ dψ dθ

we easily see (by splitting the λ integration at λ = \sqrt{ϵ}) that I is uniformly bounded in γ, ξ, η and ϵ > 0. This finishes the proof of Theorem 1 in the case where ψ(x, y, z) = x^{2} + y^{2} + z^{4}.

For the example ψ(x, y, z) = x^{2} + y^{4} + z^{4} we will show that the integral

∫_{ϵ≤x^{2}+y^{2}+z^{2}≤1} e^{iγφ(x^{2}+y^{4}+z^{4})}e^{iηx}e^{i(ξ_{1}y+ξ_{2}z)}K(x, y, z) dx dy dz (2.3)

is uniformly bounded in γ, η, ξ = (ξ_{1}, ξ_{2}) and ϵ > 0 under the additional hypothesis that for all 0 < a < b

∫_{{a≤x^{2}+y^{2}+z^{2}≤b \atop x≥0}} K(x, y, z) dx dy dz = 0. (2.4)

We would like to make the change of variables λ^{2} = x^{2} + y^{4} + z^{4} in the x integral. In order to do this first observe that the integral in (2.3) over the region δ|x|^{1/2} ≤ \sqrt{y^{2} + z^{2}} is uniformly bounded. In fact

∫ ∫ ∫_{δ|x|^{1/2}<\sqrt{y^{2}+z^{2}}≤1} |K(x, y, z)| dx dy dz

≤ C ∫ ∫_{\sqrt{y^{2}+z^{2}}≤1} dy dz ∫_{δ|x|^{1/2}≤\sqrt{y^{2}+z^{2}}} \frac{1}{(y^{2} + z^{2})^{3/2}} dx

≤ C ∫ ∫_{\sqrt{y^{2}+z^{2}}≤1} \frac{1}{\sqrt{y^{2} + z^{2}}} dy dz ≤ C.

Hence it suffices to show the uniform boundedness of

II = ∫_{{ϵ≤x^{2}+|ȳ|≤1 \atop |ȳ|≤δ|x|^{1/2}}} e^{iγφ(x^{2}+y^{4}+z^{4})}e^{iηx}e^{iξ·ȳ}K(x, ȳ) dx dȳ

where ȳ = (y, z). We write II = II_{+} + II_{−} where the integration in II_{+} is over positive values of x. We first concentrate on II_{+}, making the change of variables

λ = \sqrt{x^{2} + y^{4} + z^{4}}, x = x(λ, ȳ) = \sqrt{λ^{2} − y^{4} − z^{4}}

in the x integral so that x ∼ λ. Then

II_{+} = ∫_{0}^{1} e^{iγφ(λ^{2})} ∫_{{ϵ≤λ^{2}+|ȳ|^{2}≤1 \atop |ȳ|≤δλ^{1/2}}} e^{iηx(λ,ȳ)}e^{iξ·ȳ}K(x(λ, ȳ), ȳ) \frac{∂x}{∂λ} dȳdλ + 𝒪(1).

In order to analyze this integral we make the following simple observations regarding x(λ, ȳ) in the region |ȳ| ≤ δλ^{1/2}: (a)

x(λ, 0) = λ, \frac{∂x}{∂λ}(λ, 0) = 1,

(b)

|\frac{∂x}{∂ȳ}| ∼ \frac{|ȳ|^{3}}{λ} ,  |\frac{∂x}{∂λ∂ȳ}| ∼ \frac{|ȳ|^{3}}{λ^{2}} ,

and (c)

\frac{∂^{4}x}{∂y^{4}} ∼ \frac{1}{λ}, \frac{∂^{4}x}{∂z^{4}} ∼ \frac{1}{λ}.

SINGULAR INTEGRALS ASSOCIATED TO HYPERSURFACES: L^{2} THEORY 9

Using (a) and (b) we may replace \frac{∂x}{∂λ} by 1 in II_{+} with an error at most

\begin{aligned}C ∫_{0}^{1} \frac{1}{λ^{2}} ∫_{|ȳ|≤λ^{1/2}} \frac{|ȳ|^{4}}{λ^{3} + |ȳ|^{3}} dȳdλ & ≤ C ∫_{0}^{1} λ ∫_{|s̄|≤λ^{−1/2}} \frac{|s̄|^{4}}{1 + |s̄|^{3}} ds̄dλ \\ & ≤ C ∫_{0}^{1} \frac{1}{\sqrt{λ}} dλ ≤ C. \end{aligned}

Also replacing x(λ, ȳ) with λ in the kernel K creates an error at most

\begin{aligned}C ∫_{0}^{1} \frac{1}{λ} ∫_{|ȳ|≤λ^{\frac{1}{2}}} \frac{|ȳ|^{4}}{λ^{4} + |ȳ|^{4}} dȳdλ & ≤ C ∫_{0}^{1} λ ∫_{|s̄|≤λ^{−1/2}} \frac{|s̄|^{4}}{1 + |s̄|^{4}} ds̄dλ \\ & ≤ C ∫_{0}^{1} dλ = C. \end{aligned}

Therefore

II_{+} = ∫_{0}^{1} e^{iγφ(λ^{2})} ∫_{{ϵ≤λ^{2}+|ȳ|^{2}≤1 \atop |ȳ|≤δλ^{1/2}}} e^{iηx(λ,ȳ)}e^{iξ·ȳ}K(λ, ȳ) dȳdλ + 𝒪(1).

Next we will show that we can replace the oscillation e^{iηx(λ,ȳ)} with e^{iηλ} provided that the λ integration is restricted to where λ|η|^{1/3} ≤ 1. In fact using the fact that \frac{∂^{4}x}{∂y^{4}} ∼ \frac{1}{λ} and Van der Corput’s lemma (see e.g., [St]) in the y integral we see that the part of the integral where λ|η|^{1/3} ≥ 1 is at most

C \frac{1}{|η|^{1/4}} ∫_{λ|η|^{1/3}≥1} λ^{1/4} [∫ \frac{1}{λ^{4} + |ȳ|^{4}} dȳ + ∫ \frac{1}{λ^{3} + z^{3}} dz] dλ ≤ C \frac{1}{|η|^{1/4}} ∫_{λ|η|^{1/3}≥1} \frac{λ^{1/4}}{λ^{2}} dλ ≤ C.

For the part where λ|η|^{1/3} ≤ 1 we expect only to replace e^{iηx(λ,ȳ)} with e^{iηλ} in the region where |ȳ| ≤ (\frac{λ}{|η|})^{1/4} since using (b)

|e^{iηx(λ,ȳ)} − e^{iηλ}| ≤ |η| |x(λ, ȳ) − x(λ, 0)| ≤ |η| \frac{|ȳ|^{4}}{λ} .

10 STEPHEN WAINGER, JAMES WRIGHT, AND SARAH ZIESLER

In the complementary region, λ|η|^{1/3} ≤ 1 and |ȳ| ≥ (\frac{λ}{|η|})^{1/4} we see that K is uniformly integrable. In fact

∫_{λ|η|^{1/3}≤1} ∫_{|ȳ|≥(\frac{λ}{|η|})^{1/4}} |K(λ, ȳ)| dȳdλ ≤ ∫_{λ|η|^{1/3}≤1} ∫_{|ȳ|≥(\frac{λ}{|η|})^{1/4}} \frac{1}{λ^{3} + |ȳ|^{3}} dȳdλ

≤ C ∫_{λ|η|^{1/3}≤1} \frac{1}{λ} ∫_{|s̄|≥(\frac{λ}{|η|})^{1/4} \frac{1}{λ}} \frac{1}{1 + |s̄|^{3}} ds̄dλ

≤ C|η|^{1/4} ∫_{λ|η|^{1/3}≤1} \frac{1}{λ^{1/4}} dλ ≤ C.

Replacing e^{iηx(λ,ȳ)} with e^{iηλ} in the region λ|η|^{1/3} ≤ 1 and |ȳ| ≤ (\frac{λ}{|η|})^{1/4} creates an error at most

C|η| ∫_{λ|η|^{1/3}≤1} \frac{1}{λ} ∫_{|ȳ|≤(\frac{λ}{|η|})^{1/4}} \frac{|ȳ|^{4}}{λ^{3} + |ȳ|^{3}} dȳdλ

≤ C|η| ∫_{λ|η|^{1/3}≤1} λ^{2} ∫_{|s̄|=(\frac{λ}{|η|})^{1/4} \frac{1}{λ}} \frac{|s̄|^{4}}{1 + |s̄|^{3}} ds̄dλ

≤ C \frac{|η|}{|η|^{3/4}} ∫_{λ|η|^{1/3}≤1} \frac{1}{λ^{1/4}} dλ ≤ C.

Therefore

II_{+} = ∫_{λ|η|^{1/3}≤1} e^{iγφ(λ^{2})}e^{iηλ} ∫_{{ϵ≤λ^{2}+|ȳ|^{2}≤1 \atop |ȳ|≤δλ^{1/2}}} e^{iξ·ȳ}K(λ, ȳ) dȳdλ + 𝒪(1).

A similar but easier argument allows us to replace e^{iξ·ȳ} with 1 if we further restrict the λ integration where λ|ξ| ≤ 1. Hence making the change of variables ȳ = λs̄,

\begin{aligned}II_{+} & = ∫_{{0≤λ≤1 \atop λ≤min(|ξ|^{−1},|η|^{−1/3})}} e^{iγφ(λ^{2})}e^{iηλ} ∫_{{ϵ≤λ^{2}+|ȳ|^{2}≤1 \atop |ȳ|≤δλ^{1/2}}} K(λ, ȳ) dȳdλ + 𝒪(1) \\ & = ∫_{{0≤λ≤1 \atop λ≤min(|ξ|^{−1},|η|^{−1/3})}} e^{iγφ(λ^{2})}e^{iηλ} \frac{1}{λ} ∫_{ϵ≤λ^{2}(1+|s̄|^{2})≤1} K(1, s̄) ds̄dλ + 𝒪(1). \end{aligned}

Here we used the fact that

∫_{|s̄|≥λ^{−1/2}} |K(1, s̄)| ds̄ = 𝒪(λ^{1/2}).

Making a polar change of coordinates s̄ = (r cos θ, r sin θ) followed by r = tan ψ, 0 ≤ ψ < π/2, in the s̄ integral allows us to write

II_{+} = ∫_{{0≤λ≤1 \atop λ≤min(|ξ|^{−1},|η|^{−1/3})}} e^{iγφ(λ^{2})}e^{iηλ} \frac{1}{λ} ∫_{0}^{2π}  ∫_{{0≤ψ≤π/2 \atop λ≤cos ψ≤λ/\sqrt{ϵ}}} K(cos ψ, sin ψ cos θ, sin ψ sin θ)

sin ψ dψ dθ dλ + 𝒪(1)

= ∫_{{\sqrt{ϵ}≤λ≤1 \atop λ≤min(|ξ|^{−1},|η|^{−1/3})}} e^{iγφ(λ^{2})}e^{iηλ} \frac{1}{λ} ∫_{0}^{2π}∫_{0}^{π/2}K(cos ψ, sin ψ cos θ, sin ψ sin θ)

sin ψ dψ dθ dλ + 𝒪(1).

A similar analysis for II_{−} shows

II_{−} = ∫_{{\sqrt{ϵ}≤λ≤1 \atop λ≤min(|ξ|^{−1}|η|^{−1/3})}} e^{iγφ(λ^{2})}e^{−iηλ} \frac{1}{λ} ∫_{0}^{2π} ∫_{π/2}^{π} K(cos ψ, sin ψ cos θ, sin ψ sin θ)

sin ψ dψ dθ dλ + 𝒪(1).

Therefore

II = II_{+} + II_{−}

= ∫_{{\sqrt{ϵ}≤λ≤1 \atop λ≤min(|ξ|^{−1}|η|^{−1/3})}} e^{iγφ(λ^{2})} sin(ηλ) \frac{1}{λ} ∫_{0}^{2π}∫_{0}^{π} K(cos ψ, sin ψ cos θ, sin ψ sin θ)

sin ψ dx dθ dλ + 𝒪(1)

= 0 + 𝒪(1)

by (2.4). Note that when the additional cancellation condition for K is not satisfied, we are left with a truncated Hilbert transform along the curve (λ, φ(λ^{2})) and so we might expect to be able to use the analysis in [NVWW] when φ(λ^{2}) is convex.

3. Proof of Theorem 1

According to Schulz [Sch], after a rotation of coordinates, we may write

ψ(t) = P (t) + R(t) (3.1) where

P (t) = ∑_{j=1}^{r} a_{j}t_{j}^{ℓ_{0}} + ∑_{j=r+1}^{n−1} a_{j}t_{j}^{m_{j}} + P_{1}(t).

P (t) is a convex polynomial, P (t) > 0 for t ≠ 0, a_{j} > 0 for 1 ≤ j ≤ n − 1, ℓ_{0} < m_{j} for r + 1 ≤ j ≤ n − 1, P_{1}(t) has no pure powers of t, and if At_{1}^{α_{1}} … t_{n−1}^{α_{n−1}} is a monomial of P_{1}(t),

\frac{1}{ℓ_{0}} ∑_{j=1}^{r} α_{j} + ∑_{j=r+1}^{n−1} \frac{α_{j}}{m_{j}} = 1.

R(t) is smooth and if At_{1}^{a_{1}} … t_{n−1}^{a_{n−1}} is a term in the Taylor expansion of R(t)

\frac{1}{ℓ_{0}} ∑_{j=1}^{r} a_{j} + ∑_{j=r+1}^{n−1} \frac{a_{j}}{m_{j}} > 1.

To prove our theorems, we may assume ψ(t) has the form (3.1). The hypothesis of Theorem 1 asserts that r ≥ 2. Let H(t) be the part of P (t) which is homogeneous of degree ℓ_{0}. Then H(t) is a function of only t_{1}, …, t_{r}. In fact if

A t_{1}^{α_{1}} … t_{r}^{α_{r}}t_{r+1}^{α_{r+1}} … t_{n−2}^{α_{n−2}}t_{n−1}^{ℓ_{0}−(α_{1}+···+α_{n−2})}

were a monomial of H, then

\frac{1}{ℓ_{0}} ∑_{j=1}^{r} α_{j} + ∑_{j=r+1}^{n−2} \frac{α_{j}}{m_{j}} + \frac{ℓ_{0} − (α_{1} + ⋯ + α_{n−2})}{m_{n−1}} = 1.

This identity would clearly hold if m_{r+1} = … m_{n−1} = ℓ_{0}, so it could not hold if one of the m’s were bigger than ℓ_{0}. Similarly every monomial of P (t) which depends only on t_{1}, …, t_{r} belongs to H. So

H(t) = H(t_{1}, …, t_{r}) = P (t_{1}, …, t_{r}, 0, …, 0)

is convex and positive if some t_{j} is nonzero.

We write y = (t_{1}, …, t_{r}) in ℝ^{r} and x = (t_{r+1}, …, t_{n−1}) in ℝ^{n−1−r}. We shall suppose n − r − 1 ≥ 1, otherwise the proof is similar but simpler. We then write

P (x, y) = H(y) + P_{2}(x, y).

To prove Theorem 1, we must show for ξ in ℝ^{r} and η in ℝ^{n−1−r},

| ∫_{ϵ≤|x|^{2}+|y|^{2}≤1} e^{iγφ(ψ(x,y))}e^{iη·x}e^{iξ·y}K(y, x) dy dx| ≤ B (3.2)

uniformly in ξ, η, γ and ϵ > 0.

We begin by introducing polar coordinates in the y variables. That is, we write y = sω where s goes from 0 to 1 and ω runs over the surface H(w) = 1. The integral in (3.2) becomes

∫_{H(ω)=1} ∫_{ϵ≤|x|^{2}+s^{2}|ω|^{2}≤1} e^{iγφ(ψ(x,sω))}e^{i(sξ·ω+η·x)}K(sω, x)s^{r−1}h(ω) ds dx dω (3.3)

where h is a smooth function. We let m be the smallest of the values among the m_{j}’s, r + 1 ≤ j ≤ n − 1, and choose σ > 0 so that \frac{ℓ_{0}}{m} + σ < 1. We may restrict the integration in (3.3) to |x| ≤ s^{ℓ_{0}/m+σ} since integrating K over the complementary region is at most

\begin{aligned}C ∫ ∫_{s^{\frac{ℓ_{0}}{m} +σ}≤|x|≤1} \frac{1}{s^{n−1} + |x|^{n−1}} s^{r−1} ds dx & ≤ C ∫_{|x|≤1} \frac{1}{|x|^{n−1}} ∫_{s≤|x|^{λ}} s^{r−1} ds dx \\ & ≤ C ∫_{|x|≤1} \frac{1}{|x|^{n−rλ−1}} dx ≤ C \end{aligned}

since

λ = \frac{1}{\frac{ℓ_{0}}{m} + σ} > 1 and x ∈ ℝ^{n−r−1}.

Furthermore in the region |x| ≤ s^{\frac{ℓ_{0}}{m} +σ},

ψ(x, sω) = s^{ℓ_{0}} + 𝒪(s^{ℓ_{0}+σ}) (3.4)

and

\frac{∂ψ}{∂s} (x, sω) = ℓ_{0}s^{ℓ_{0}−1} + 𝒪(s^{ℓ_{0}−1+σ}). (3.5)

In fact

ψ(x, sω) = s^{ℓ_{0}} + P_{2}(x, sω) + R(x, sω)

and every monomial in P_{2} or any monomial in R of the form x^{α}(sω)^{β} with |α| > 0 has the bound

|x^{α}(sω)^{β}| ≤ s^{(\frac{ℓ_{0}}{m} +σ)|α|+|β|} ≤ s^{ℓ_{0}+σ}.

Also any monomial in R of the form (sω)^{β} is 𝒪(s^{ℓ_{0}+1}). Therefore we may make the change of variables

λ^{ℓ_{0}} = ψ(x, sω) (3.6) in s for fixed x and ω and write (3.3) as

∫_{H(ω)=1} h(ω) ∫_{0}^{1} e^{iγφ(λ^{ℓ_{0}})} ∫_{{ϵ≤|x|^{2}+s^{2}|ω|^{2}≤1 \atop |x|≤s^{\frac{ℓ_{0}}{m} +σ}}} e^{iη·x}e^{isξ·ω}K(sω, x)s^{r−1} \frac{∂s}{∂λ} dx dλ dω + 𝒪(1) (3.7)

where s = s(λ, ω, x). From (3.4) and (3.5) we have, for some ϵ > 0,

s = λ + 𝒪(λ^{1+ϵ}) and \frac{∂s}{∂λ} = 1 + 𝒪(λ^{ϵ}).

Also we have

|\frac{∂s}{∂x}| ≤ Cλ^{1− \frac{ℓ_{0}}{m}} (3.8)

which follows by differentiating (3.6) with respect to x, giving

0 = \frac{∂ψ}{∂x} + \frac{∂ψ}{∂s} \frac{∂s}{∂x}.

(3.5) implies \frac{∂ψ}{∂s} ∼ λ^{ℓ_{0}−1} and a similar argument which established (3.4) and (3.5) shows \frac{∂ψ}{∂x} = 𝒪(λ^{ℓ_{0}− \frac{ℓ_{0}}{m}}) and thus (3.8).

Arguing as in section 2, using the above estimates on the derivatives of s(λ, ω, x) and s itself, shows that we may replace s(λ, ω, x) by λ (except in the oscillation e^{isξ·ω}) and \frac{∂s}{∂λ} by 1 with a uniformly bounded error. Thus the integral in (3.7) is

∫_{0}^{1} e^{iγφ(λ^{ℓ_{0}})}λ^{r−1} ∫_{|x|≤λ^{\frac{ℓ_{0}}{m} +σ}} e^{iη·x}  ∫_{{H(ω)=1 \atop ϵ≤|x|^{2}+λ^{2}|ω|^{2}≤1}} e^{is(λ,ω,x)ξ·ω}K(λω, x)h(ω) dω dx dλ + 𝒪(1). (3.9)

Consider first the contribution to (3.9) from those values of λ where λ|ξ| ≥ 1. Since H(ω) = 1 is of finite type we may for each ω_{0} on H(ω) = 1 parametrize H(ω) = 1 in a neighborhood of ω_{0} as

ω_{0} + (τ_{1}, …, τ_{r−1}, g(τ_{1}, …, τ_{r−1}))

where g(0) = 0, ∇g(0) = 0, and for some j_{0} ≥ 2,

\frac{∂^{j}g}{∂τ_{1}^{j}} (0) = 0

for 1 ≤ j ≤ j_{0} − 1 and

\frac{∂^{j_{0}}g}{∂τ_{1}^{j_{0}}} (0) ≠ 0.

It follows that we may assume

\frac{∂^{j_{0}}g}{∂τ_{1}^{j_{0}}} ≠ 0

for all τ in a neighborhood of 0. Therefore since s(λ, ω, x) ∼ λ,

| ∫_{{|ω−ω_{0}|≤δ \atop H(ω)=1}} e^{is(λ,ω,x)ξ·ω}h(ω) dw| ≤ C \frac{1}{(λ|ξ|)^{δ}}

for some positive δ by Van der Corput’s lemma . Integrating by parts now shows that the contribution to the integral in (3.9) from those λ where λ|ξ| ≥ 1 is at most

C ∫_{λ|ξ|≥1} λ^{r−1} \frac{λ}{(λ|ξ|)^{δ}} ∫_{ℝ^{n−r−1}} \frac{1}{λ^{n} + |x|^{n}} dx dλ ≤ C ∫_{λ|ξ|≥1} \frac{1}{(λ|ξ|)^{δ}} \frac{dλ}{λ} ≤ C.

Thus the proof of Theorem 1 reduces to showing that the integral

I = ∫_{λ|ξ|≤1} e^{iγφ(λ^{ℓ_{0}})}λ^{r−1} ∫_{H(ω)=1} ∫_{{ϵ≤|x|^{2}+λ^{2}|ω|^{2}≤1 \atop |x|≤λ^{\frac{ℓ_{0}}{m} +σ}}} e^{iη·x}K(λω, x)h(ω) dx dω dλ

is uniformly bounded in γ, ξ, η and ϵ > 0. Putting x = λz makes

I = ∫_{λ|ξ|≤1} e^{iγφ(λ^{ℓ_{0}})} \frac{1}{λ} ∫_{H(ω)=1} ∫_{{ϵ≤λ^{2}(|z|^{2}+|ω|^{2})≤1 \atop |z|≤λ^{\frac{ℓ_{0}}{m} +σ−1}}} e^{iλη·z}K(ω, z)h(ω) dz dω dλ

and using the fact

∫_{(\frac{C}{λ})^{δ}≤|z|} |K(ω, z)| dz = 𝒪(\frac{λ}{C})^{rδ}

three times, first with δ = 1 and C = \sqrt{ϵ}, then with δ = 1 and C = 1, and finally with δ = 1 − ((\frac{ℓ_{0}}{m}) + σ) and C = 1, we see that

I = ∫_{A\sqrt{ϵ}≤λ≤ \frac{1}{|ξ|}} e^{iγφ(λ_{0}^{ℓ})} \frac{1}{λ} ∫_{H(ω)=1} ∫ e^{iλη·z}K(ω, z)h(ω) dz dω dλ + 𝒪(1)

if A is chosen large enough. An integration by parts in the z integral shows that the part of the integral where λ|η| ≥ 1 is at most (up to boundary terms)

C \frac{1}{|η|} ∫_{λ|η|≥1} \frac{1}{λ^{2}} ∫_{ℝ^{n−r−1}} sup_{ω} |∇K(ω, z)| dz dλ ≤ C \frac{1}{|η|} ∫_{λ|η|≥1} \frac{1}{λ^{2}} ∫_{|z|≥1} \frac{1}{|z|^{n}} dz dλ ≤ C,

and so

I = ∫_{{A\sqrt{ϵ}≤λ≤1 \atop λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|})}} e^{iγφ(λ^{ℓ_{0}})} \frac{1}{λ} ∫ H(ω) = 1∫ e^{iλη·z}K(ω, z)h(ω) dzdω dλ + 𝒪(1).

The boundary terms are handled similarly. Replacing e^{iλη·z} by 1 creates an error at most

C|η| ∫_{λ|η|≤1} ∫_{H(ω)=1} ∫ \frac{|z|}{|ω|^{n−1} + |z|^{n−1}} dz dω dλ ≤ C

since r ≥ 2. Therefore

I = ∫_{{A\sqrt{ϵ}≤λ≤1 \atop λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|})}} e^{iγφ(λ^{ℓ_{0}})} \frac{1}{λ} ∫_{H(ω)=1} ∫ K(ω, z)h(ω) dzdω dλ + 𝒪(1),

and so it suffices to show

∫_{H(ω)=1} ∫_{ℝ^{n−r−1}} K(ω, z)h(ω) dz dω = 0. (3.10)

Now for any δ > 0

0 =  ∫_{1−δ≤|y|^{2}+|x|^{2}≤1} K(y, x) dy dx

= ∫_{H(ω)=1} ∫_{1−δ≤λ^{2}|ω|^{2}+|x|^{2}≤1} λ^{r−1}K(λω, x)h(ω) dλ dω dx

= ∫_{H(ω)=1} ∫_{1−δ≤λ^{2}(|ω|^{2}+|z|^{2})≤1} K(ω, z)h(ω) \frac{dλ}{λ} dω dz

= ∫_{H(ω)=1} h(ω) ∫_{ℝ^{n−r−1}} K(ω, z) ∫_{\frac{1−δ}{|ω|^{2}+|z|^{2}} ≤λ^{2}≤ \frac{1}{|ω|^{2}+|z|^{2}}} \frac{dλ}{λ} dz dω.

Dividing by δ and letting δ → 0 gives (3.10) and this finishes the proof of Theorem

4. The proof of Theorems 2 and 3.

We may again assume ψ(t) is of the general form (3.1), where now r = 1. The cancellation condition (1.1) now becomes

∫_{Σ^{+}} K(t) dσ(t) = 0

where Σ^{+} = {t ∈ ℝ^{n−1} | |t| = 1, t_{1} > 0} is the “upper” hemisphere of S^{n−2}. It will be convenient to let t_{1} be denoted by y and (t_{2}, …, t_{n−1}) = x ∈ ℝ^{n−2}. We are then concerned with the uniform boundedness of

\begin{aligned}∫ ∫ e^{iγφ(ψ(y,x))}e^{iξ·x}e^{iηy}K(y, x) dy dx & = ∫ ∫_{y>0} + ∫ ∫_{y<0} \\ & = I + II. \end{aligned}

Theorems 2 and 3 will then follow if we can prove for some b, 0 < b < 1,

I = ∫_{{0≤λ≤1 \atop λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|^{b}})}} e^{iγφ(λ^{ℓ_{0}})}e^{iηq(λ)} \frac{dλ}{λ} ∫_{Σ^{+}} K(ω)dσ(ω) + 𝒪(1), (4.1)

II = − ∫_{{o≤λ≤1 \atop λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|^{b}})}} e^{iγφ(λ^{ℓ_{0}})}e^{−iηq(λ)} \frac{dλ}{λ} \begin{matrix}∫ \\ ∑ & ^{+} \end{matrix} K(ω)dσ(ω) + 𝒪(1), (4.2)

and for φ̄(λ) = φ(λ^{ℓ_{0}}) convex,

∫_{{0≤λ≤1 \atop λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|^{b}})}} e^{iγφ̄(λ)} sin(ηq(λ)) \frac{dλ}{λ} (4.3)

is uniformly bounded in γ, η and ξ if and only if

φ̄′(Cλ) ≥ 2φ̄′(λ)

for some C ≥ 1 and 0 < λ ≤ 1. Here q(λ) = λ + 𝒪(λ^{1+ϵ}) and q′(λ) = 1 + 𝒪(λ^{ϵ}). We begin with the proof of (4.1). It will convenient to write (3.1) in the form

ψ(y, x) = Ay^{ℓ_{0}} + ∑_{j=1}^{n−2} a_{j}x_{j}^{m_{j}} + ∑_{j=1}^{n−2} b_{j}x_{j}^{α_{j}}y^{β_{j}} + P_{2}(y, x) + R(y, x) (4.4)

where ℓ_{0} < m_{j}, 1 ≤ j ≤ n − 2, A > 0, a_{j} > 0, b_{j} ≠ 0, 1 ≤ j ≤ n − 2 and each monomial of P_{2}(y, x) has the form x_{j}^{α}y^{β} with α > α_{j}, or contains powers of at least two different x_{j}’s. Let m = min_{1≤j≤n−2}(m_{j}) and choose σ > 0 such that \frac{ℓ_{0}}{m} + σ < 1. Again

∫_{|x|≥y^{\frac{ℓ_{0}}{m} +σ}} |K(x, y)| dx dy = 𝒪(1)

and so it suffices to study I in the region |x| ≤ y^{\frac{ℓ_{0}}{m} +σ}. In this region we wish to make a change of variables

λ^{ℓ_{0}} = ψ(y, x) (4.5)

in the y integral. As in section 3, |x| ≤ y^{\frac{ℓ_{0}}{m} +σ} implies that y = y(x, λ) defined implicitly by (4.5) satisfies

y(x, λ) = A^{− \frac{1}{ℓ_{0}}} λ + 𝒪(λ^{1+σ}), (4.6)

\frac{∂y}{∂λ} = A^{− \frac{1}{ℓ_{0}}} + 𝒪(λ^{σ}), (4.7)

and

|\frac{∂y}{∂x}| ≤ Cλ^{1− \frac{ℓ_{0}}{m}} . (4.8)

Therefore, as before, making the change of variables (4.5), shows

I = ∫_{0≤λ≤1} e^{iγφ(λ^{ℓ_{0}})} ∫_{|x|≤y^{\frac{ℓ_{0}}{m} +σ}} e^{ix·ξ}e^{iηy(x,λ)}K(x, λ) dx dλ + 𝒪(1). (4.9)

To study (4.9), it is necessary to have information on the derivatives of y with respect to the x variables.

Lemma 6. Suppose |x| ≤ λ^{\frac{ℓ_{0}}{m} +σ}.

(1) For δ > 0 small,

|\frac{∂^{k}y}{∂x_{j}^{k}} (x, λ)| ≤ δλ^{1−k \frac{ℓ_{0}}{m_{j}}} , 1 ≤ k ≤ α_{j} − 1, 1 ≤ j ≤ n − 2.

(2)

\frac{∂^{α_{j}}y}{∂x_{j}^{α_{j}}} (x, λ) ∼ λ^{1−α_{j} \frac{ℓ_{0}}{m_{j}}} , 1 ≤ j ≤ n − 2.

(3) For every β = (β_{1}, …, β_{n−2}) with 0 ≤ β_{j} ≤ α_{j}, 1 ≤ j ≤ n − 2,

|\frac{∂^{β}y}{∂x^{β}} (x, λ)| ≤ Cλ^{1−ℓ_{0} ∑_{j=1}^{n−2} \frac{β_{j}}{m_{j}}} .

(4) For every β = (β_{1}, …, β_{n−2}) with 0 ≤ β_{j} ≤ α_{j} − 1, 1 ≤ j ≤ n − 2, either

\frac{∂^{β}y}{∂x^{β}} (0, λ) ∼ λ^{p_{β}}

for some p_{β} > −|β|, or

\frac{∂^{β}y}{∂x^{β}} (0, λ) = 𝒪(λ^{N})

for every N.

Proof of lemma. For M large, write

ψ(y, x) = Ay^{ℓ_{0}} + ∑_{j=1}^{n−2} b_{j}x_{j}^{α_{j}}y^{β_{j}} + ∑_{u,β} c_{u,β}x^{u}y^{β} + 𝒪(|x|^{M}) + 𝒪(y^{M})

where A > 0, b_{1}, …, b_{n−2} ≠ 0, \frac{α_{j}}{m_{j}} + \frac{β_{j}}{ℓ_{0}} = 1 for 1 ≤ j ≤ n − 2, each x^{u}y^{β} satisfies ∑_{j=1}^{n−2} \frac{u_{j}}{m_{j}} + \frac{β}{ℓ_{0}} ≥ 1 and if u = (0, …, u_{j}, …, 0), then u_{j} > α_{j}. To prove (1), we will

show inductively that in the larger region, |x_{j}| ≤ ϵλ^{\frac{ℓ_{0}}{m_{j}}}, 1 ≤ j ≤ n − 2,

|\frac{∂^{k}y}{∂x_{j}^{k}} | ≤ δλ^{1−k \frac{ℓ_{0}}{m_{j}}} , 1 ≤ k ≤ α_{j} − 1, (4.10)

provided ϵ = ϵ(δ) > 0 is small enough. We first prove (4.10) for k = 1 (so α_{j} ≥ 2 or there is nothing to prove). If we differentiate (4.5) with respect to x_{j}, noting y ∼ λ from (4.6), we obtain

0 = C_{1} \frac{∂y}{∂x_{j}} + C_{2}

where

C_{1} = Aℓ_{0}y^{ℓ_{0}−1} + 𝒪(λ^{ℓ_{0}+1}) + E

and E is a finite sum of terms of the form x^{u}y^{β−1} where u = (u_{1}, …, u_{n−2}) ≠ 0 and ∑_{j=1}^{n−2} \frac{u_{j}}{m_{j}} + \frac{β}{ℓ_{0}} ≥ 1. Hence for ϵ > 0 small enough, |x^{u}y^{β−1}| ≤ δλ^{ℓ_{0}−1} and therefore

C_{1} ∼ λ^{ℓ_{0}−1}.

C_{2} is 𝒪(λ^{ℓ_{0}}) plus a finite sum of terms of the form x_{j}^{−1}x^{u}y^{β} with u_{j} ≥ 2 and \frac{|u|}{m}+\frac{β}{ℓ_{0}} ≥ 1. Thus for ϵ > 0 small enough,

\begin{aligned}|x_{j}^{−1}x^{u}y^{β}| & ≤ δλ^{|u| \frac{ℓ_{0}}{m} +β}λ^{− \frac{ℓ_{0}}{m_{j}}} \\ & ≤ δλ^{ℓ_{0}− \frac{ℓ_{0}}{m_{j}}} \end{aligned}

and so C_{2} = 𝒪(δλ^{ℓ_{0}− \frac{ℓ_{0}}{m_{j}}}) which proves (4.10) with k = 1.

Next we assume (4.10) for k ≤ k_{0} − 1 where k_{0} ≤ α_{j} − 1, and prove (4.10) for k = k_{0}. Differentiating (4.5) k_{0} times with respect to x_{j}, we again obtain

0 = D_{1} \frac{∂^{k_{0}}y}{∂x_{j}^{k_{0}}} + D_{2}

where as before D_{1} ∼ λ^{ℓ_{0}−1}. D_{2} consists of a finite sum of products of terms involving either a positive power of x or a derivative of order at most k_{0} − 1 of y with respect to x_{j}. In the first case we pick up an ϵ from the powers of x and in the second case we pick up a δ from the induction hypothesis. So we only need to determine the magnitude of each term in D_{2}. Since each term in the expression for ψ(y, x) is 𝒪(λ^{ℓ_{0}}), we only need to understand how the powers of λ decrease when we differentiate a product involving x^{u} and

(\frac{∂^{k}y}{∂x_{j}^{k}} )^{p} , 1 ≤ k ≤ k_{0} − 2,

with respect to x_{j}. Differentiating x^{u} gives x_{j}^{−1}x^{u}, losing λ^{− \frac{ℓ_{0}}{m_{j}}} and differentiating

(\frac{∂^{k}y}{∂x_{j}^{k}} )^{p} gives (\frac{∂^{k}y}{∂x_{j}^{k}} )^{p−1} \frac{∂^{k+1}y}{∂x_{j}^{k+1}} , losing λ^{−(1−k \frac{ℓ_{0}}{m_{j}} )}λ^{1−(k+1) \frac{ℓ_{0}}{m_{j}}} = λ^{− \frac{ℓ_{0}}{m_{j}}}

by induction. Therefore each term in D_{2} is 𝒪(δλ^{ℓ_{0}−k_{0} \frac{ℓ_{0}}{m_{j}}}) and this finishes the proof of (4.10) and thus (1) of the lemma.

The proof of (2) follows in the same way as the proof of (1). The only difference is that differentiating the term b_{j}x_{j}^{α_{j}}y^{β}, b_{j} ≠ 0, contributes a term b_{j}α_{j}! y^{β_{j}} ∼ λ^{β_{j}} and so

D_{2} ∼ λ^{β_{j}} + 𝒪(δλ^{ℓ_{0}−α_{j} \frac{ℓ_{0}}{m_{j}}}) ∼ λ^{ℓ_{0}−α_{j} \frac{ℓ_{0}}{m_{j}}}

since \frac{α_{j}}{m_{j}} + \frac{β_{j}}{ℓ_{0}} = 1. This shows (2).

The proof of (3) follows similarly. We use induction on the partial ordering u = (u_{1}, …, u_{n−2}) ≤ β = (β_{1}, …, β_{n−2}) if and only if u_{j} ≤ β_{j}, 1 ≤ j ≤ n − 2. (1) and (2) show (3) is true for all pure derivatives, β = (0, …, β_{j}, …, 0). The arguments used in proving (1) and (2) show that if (3) is true for all u ≨ β, then differentiating (4.5) shows

0 = D_{1} \frac{∂^{β}y}{∂x^{β}} + D_{2}

where D_{1} ∼ λ^{ℓ_{0}−1} and

D_{2} = 𝒪 (λ^{ℓ_{0}−ℓ_{0} ∑_{j=1}^{n−2} \frac{β_{j}}{m_{j}}} ) ,

proving (3).

Finally to prove (4), we first note that (3) implies that it is enough to show (4) for any power p_{β}. Again we use induction on the partial ordering ≤, supposing (4) is true for all u ≨ β. Rewriting (4.4) expresses (4.5) as

λ^{ℓ_{0}} = Ay^{ℓ_{0}} + ∑_{{u \atop u≤β}} a_{u}(y)x^{u} + 𝒪(|x|^{|β|+1}) (4.11)

where a_{0}(y) = 𝒪(y^{ℓ_{0}+1}), a′_{0}(y) = 𝒪(y^{ℓ_{0}}) and the a_{u}’s are smooth. Taking the β-th derivative of (4.11) gives

0 = [Aℓ_{0}y^{ℓ_{0}−1}(0, λ) + a′_{0}(y(0, λ))] \frac{∂^{β}y}{∂x^{β}} (0, λ) + C(λ)

where C(λ) is a finite sum of terms of the form

a^{(s)}(y(0, λ)) ∏_{{u \atop u≨β}} (\frac{∂^{u}y}{∂x^{u}} (0, λ))^{q_{u}}

for some non-negative integers q_{u}. Here a(y) is either a power of y or one of the a_{u}’s. Using the fact that y(0, λ) ∼ λ and the inductive hypothesis, we see that C(λ) ∼ λ^{p} for some p or C(λ) = 𝒪(λ^{N}) for every N. Since

\frac{∂^{β}y}{∂x^{β}} (0, λ) = −\frac{C(λ)}{[Aℓ_{0}y^{ℓ_{0}−1}(0, λ) + a′_{0}(y(0, λ))]}

and a′_{0}(y) = 𝒪(λ^{ℓ_{0}}), we have shown (4) and this finishes the proof of the lemma.

We now turn back to the proof of (4.1) where we are examining the integral in (4.9). Let us write

y(x, λ) = M_{1}(x, λ) + M_{2}(x, λ)

where M_{1}(x, λ) is a polynomial in x_{1} of degree α_{1} − 1 and M_{2} is that part of the Taylor expansion of y(x, λ) in the variable x_{1} that is 𝒪(|x_{1}|^{α_{1}}). We wish to replace the integral in (4.9) by a similar integral where y(x, λ) is replaced by M_{1}(x, λ) and the λ integral is restricted to

λ ≤ (\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}

where κ(α_{1}) = 1 − α_{1} \frac{ℓ_{0}}{m_{1}} . Note that α_{1} + κ(α_{1}) > 1. Since

\frac{∂^{α_{1}}y}{∂x_{1}^{α_{1}}} ∼ λ^{1−α_{1} \frac{ℓ_{0}}{m_{1}}}

by part (2) of Lemma 1 and since α_{1} ≥ 2, an application of Van der Corput’s lemma together with integration by parts shows

| ∫_{λ≥(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} e^{iγφ(λ^{ℓ_{0}})} ∫_{|x|≤y^{\frac{ℓ_{0}}{m} +σ}} e^{iξ·x}e^{iηy(x,λ)}K(x_{1}λ) dx dλ|

≤ C (\frac{1}{|η|})^{1/α_{1}} ∫_{λ≥(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} (\frac{1}{λ})^{^{\frac{1}{α_{1}}} [1−α_{1} \frac{ℓ_{0}}{m_{1}} ]} ∫_{ℝ^{n−2}} \frac{1}{|x|^{n} + λ^{n}} dx dλ

≤ C (\frac{1}{|η|})^{\frac{1}{α_{1}}} ∫_{λ≥(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} \frac{λ^{\frac{ℓ_{0}}{m_{1}}}}{λ^{2+ \frac{1}{α_{1}}}} dλ ≤ C (\frac{1}{|η|})^{\frac{1}{α_{1}}} |η|^{^{\frac{1}{α_{1}+κ(α_{1})}} [1+ \frac{1}{α_{1}} − \frac{ℓ_{0}}{m_{1}} ]}

= C (\frac{1}{|η|})^{\frac{1}{α_{1}}} |η|^{\frac{1}{α_{1}} ^{[ \frac{α_{1}+κ(α_{1})}{α_{1}+κ(α_{1})} ]}} = C.

In the region

λ ≤ (\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}

we would like to replace e^{iηy(x,λ)} by e^{iηM_{1}(x,λ)}. We expect to be able to replace e^{iηy(x,λ)} by e^{iηM_{1}(x,λ)} with a bounded error when

|x| ≤ (\frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}}

since

|e^{iηy(x,λ)} − e^{iηM_{1}(x,λ)}| ≤ C|η|λ^{1−α_{1} \frac{ℓ_{0}}{m_{1}}} |x|^{α_{1}} ≤ C

when |x| ≤ (\frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}} . However in the complementary region, when

λ ≤ (\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{2})}} and |x| ≥ (\frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}} ,

SINGULAR INTEGRALS ASSOCIATED TO HYPERSURFACES: L^{2} THEORY 23

K is uniformly integrable. In fact

∫_{λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} ∫_{|x|≥( \frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}}} |K(x, λ)| dx dλ

≤ C  ∫_{λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} ∫_{|x|≥( \frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}}} \frac{1}{|x|^{n−1}} dx dλ

≤ ∫_{λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} (|η|λ^{κ(α_{1})})^{\frac{1}{α_{1}}} dλ

= C ∫_{|η|^{\frac{1}{α_{1}+κ(α_{1})}} λ≤1} (|η|^{\frac{1}{α_{1}+κ(α_{1})}} λ)^{\frac{κ(α_{1})}{α_{1}} +1} \frac{dλ}{λ} ≤ C

since

\frac{κ(α_{1})}{α_{1}} = \frac{1 − α_{1} \frac{ℓ_{0}}{m_{1}}}{α_{1}} > −1.

Replacing e^{iηy(x,λ)} by e^{iηM_{1}(x,λ)} when

λ ≤ (\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}} and |x| ≤ (\frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}}

creates an error at most

C|η|  ∫_{λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}+κ(α_{1})}}} λ^{κ(α_{1})}  ∫_{|x|≤( \frac{1}{|η|λ^{κ(α_{1})}} )^{\frac{1}{α_{1}}}} \frac{|x|^{α_{1}}}{|x|^{n−1}} dx dλ

≤ C|η| ∫_{λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}} +κ(α_{1})}} \frac{λ^{κ(α_{1})}}{(|η|λ^{κ(α_{1})})^{\frac{1}{α_{1}} (α_{1}−1)}} dλ

≤ C ∫_{|η|^{\frac{1}{α_{1}+κ(α_{1})}} λ≤1} (|η|^{\frac{1}{α_{1}+κ(α_{1})}} λ)^{\frac{α_{1}+κ(α_{1})}{α_{1}}} \frac{dλ}{λ} ≤ C

since

\frac{α_{1} + κ(α_{1})}{α_{1}} = 1 − \frac{ℓ_{0}}{m_{1}} + \frac{1}{α_{1}} > 0.

24 STEPHEN WAINGER, JAMES WRIGHT, AND SARAH ZIESLER

Therefore

I = ∫_{{0≤λ≤1 \atop λ≤(\frac{1}{|η|})^{\frac{1}{α_{1}+K(α_{1})}}}} e^{iγφ(λ^{ℓ_{0}})} ∫_{|x|≤y^{\frac{ℓ_{0}}{m} +σ}} e^{iξ·x}e^{iηM_{1}(x,λ)} K(x, λ) dx dλ + 𝒪(1).

Since

M_{1}(x, λ) = ∑_{k=0}^{α_{1}−1} \frac{1}{k!} \frac{∂^{k}y}{∂x_{1}^{k}} (0, x_{2}, …, x_{n−2})x_{1}^{k},

we see that for 2 ≤ j ≤ n − 2,

\frac{∂^{α_{j}}M_{1}}{∂x_{j}^{α_{j}}} (x, λ) = \frac{∂^{α_{j}}y}{∂x_{j}^{α_{j}}} (0, x_{2}, …, x_{n−2}) + ∑_{k=1}^{α_{1}−1} \frac{1}{k!} \frac{∂^{k+α_{j}}y}{∂x_{1}^{k}∂x_{j}^{α_{j}}} (0, x_{2}, …, x_{n−2})x_{1}^{k}.

Also since |x_{1}| ≤ ϵλ^{\frac{ℓ_{0}}{m_{1}}} , we have by part (3) of Lemma 1,

| ∑_{k=1}^{α_{1}−1} \frac{1}{k!} \frac{∂^{k+α_{j}}y}{∂x_{1}^{k}∂x_{j}^{α_{j}}} (0, x_{2}, …, x_{n−2})x_{1}^{k}| ≤ ϵλ^{k \frac{ℓ_{0}}{m_{1}}} λ^{1−ℓ_{0}( \frac{k}{m_{1}} + \frac{α_{j}}{m_{j}} )} ≤ ϵλ^{1−ℓ_{0} \frac{α_{j}}{m_{j}}} ,

and since

\frac{∂^{α_{j}}y}{∂x_{j}^{α_{j}}} ∼ λ^{1−ℓ_{0} \frac{α_{j}}{m_{j}}}

by part (2) of Lemma 1, we conclude that for 2 ≤ j ≤ n − 2,

\frac{∂^{α_{j}}M_{1}}{∂x_{j}^{α_{j}}} (x, λ) ∼ λ^{1−ℓ_{0} \frac{α_{j}}{m_{j}}} .

Therefore we may proceed in the same manner to find that up to a bounded error

I = ∫_{λ≤ \frac{1}{|η|^{δ}}} e^{iγφ(λ^{ℓ_{0}})} ∫_{|x|≤λ^{\frac{ℓ_{0}}{m} +σ}} e^{iξ·x}e^{iηQ(x,λ)} K(x, λ) dx dλ

for some 0 < δ < 1 where

Q(x, λ) = ∑_{{β \atop β_{j}≤α_{j}−1}} \frac{1}{β!} \frac{∂^{β}y}{∂x^{β}} (0, λ) x^{β}.

By part (4) of Lemma 1, we have for each β with β_{j} ≤ α_{j} − 1, 1 ≤ j ≤ n − 2, either

\frac{∂^{β}y}{∂x^{β}} (0, λ) ∼ λ^{p_{β}}

for some p_{β} > −|β| or

\frac{∂^{β}y}{∂x^{β}} (0, λ) = 𝒪 (λ^{N})

for every N. If the latter occurs, then up to a bounded error, we may clearly replace

e^{iη \frac{∂^{β} y}{∂x^{β}} (0,λ)x^{β}} by 1. When the former occurs, that is, when \frac{∂^{β}y}{∂x^{β}} (0, λ) behaves like a power of λ, we can repeat the above argument to see that for some (other) δ, 0 < δ < 1,

I = ∫_{λ≤ \frac{1}{|η|^{δ}}} e^{iγφ(λ^{ℓ_{0}})}e^{iηy(0,λ)} ∫_{|x|≤λ^{\frac{ℓ_{0}}{m} +σ}} e^{i ∑_{j=1}^{n−2}(ξ_{j}+λ^{p_{j}} η_{j})x_{j}} K(x, λ) dx dλ + 𝒪(1).

For each x_{j} integral, by splitting the λ integral where λ is smaller or larger than

|ξ_{j}/η_{j}|^{\frac{1}{p_{j}}}, we can once again repeat the same argument to conclude that

\begin{aligned}I & =  ∫_{λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|^{b}})} e^{iγφ(λ^{ℓ_{0}})}e^{iηy(0,λ)} ∫_{|x|≤λ^{\frac{ℓ_{0}}{m} +σ}} K(x, λ) dx dλ + 𝒪(1) \\ & =  ∫_{λ≤min( \frac{1}{|ξ|} , \frac{1}{|η|^{b}})} e^{iγφ(λ^{ℓ_{0}})}e^{iηy(0,λ)} \frac{1}{λ} ∫_{ℝ^{n−2}} K(z, 1) dz dλ + 𝒪(1) \end{aligned}

for some 0 < b < 1. Thus the proof of (4.1) will be finished once we establish the identity

∫_{ℝ^{n−2}} K(x, 1) dx = ∫_{Σ^{+}} K(ω) dσ(ω). (4.12)

This is done by making the change of variables

x_{j} = \frac{s_{j}}{1 − |s|^{2}} , 1 ≤ j ≤ n − 2.

In evaluating the Jacobian of this change of variables, we need to observe that if an r × r matrix (α_{j,k}) is defined by α_{j,k} = s_{j}s_{k} for j ≠ k and α_{j,j} = 1 − |s|^{2} − s_{j}^{2}, then

det(α_{j,k}) = (1 − |s|^{2})^{r−1}.

This calculation was shown to us by A. Carbery and is carried out in the appendix. This establishes (4.12) and finishes the proof of (4.1). The proof of (4.2) is similar. It remains to prove (4.3).

Suppose first that there is no constant C_{0} so that φ̄′(C_{0}λ) ≥ 2φ̄′(λ) for 0 < λ ≤ 1. Then there exists a sequence of points λ_{j} ↘ 0 such that

\frac{λ_{j}φ̄′(λ_{j})}{λ_{j}φ̄′(λ_{j}) − φ̄(λ_{j})} → ∞.

See, e.g., [NVWW]. Let

γ_{j} = \frac{π}{4} \frac{1}{λ_{j}φ̄^{1}(λ_{j}) − φ̄(λ_{j})}, η_{j} = γ_{j}φ̄′(λ_{j}),

and choose ξ_{j} so that

\frac{1}{ξ_{j}} = min (λ_{j}, \frac{1}{η_{j}^{b}} , \frac{1}{η_{j}^{1+ϵ}} )

where ϵ > 0 is chosen so that q(λ) = λ + 𝒪(λ^{1+ϵ}). Then

|∫_{0}^{\frac{1}{ξ}} e^{iγ_{j} φ̄(λ)} sin(η_{j}q(λ)) \frac{dλ}{λ}| = |∫_{\frac{1}{η_{j}}}^{\frac{1}{ξ}} e^{i(γ_{j} φ̄(λ)−η_{j}λ)} \frac{dλ}{λ}| + 𝒪(1)

≥ A log (\frac{η_{j}}{ξ_{j}} ) ≥ A log(λ_{j}η_{j}) → +∞

for some A > 0 since λ_{j}η_{j} → ∞ and

0 ≤ η_{j}λ − γ_{j}φ̄(λ) ≤ \frac{π}{4}

for all 0 ≤ λ ≤ λ_{j}.

Finally let us turn to the proof of the sufficiency of (4.3) and assume φ̄(0) = φ̄′(0) = 0, φ̄′(C_{0}t) ≥ 2φ̄′(t) for some C_{0} ≥ 1. It suffices to show that the integral

II = ∫_{\frac{1}{η} ≤t≤1} e^{iγφ̄(t)}e^{−iηq(t)} \frac{dt}{t} (4.13)

is uniformly bounded in γ, η > 0. First assume 10γ > η. Choosing t_{0} such that φ̄′(t_{0}) = \frac{η}{γ} we write

II = ∫_{\frac{1}{η} ≤t≤ \frac{t_{0}}{C_{0}}} + ∫_{\frac{t_{0}}{C_{0}} ≤t≤C_{0}t_{0}} + ∫_{C_{0}t_{0}≤t≤1} = A + B + D.

For \frac{1}{η} ≤ t ≤ \frac{t_{0}}{C_{0}} , \frac{d}{dt}(ηt − γφ̄(t)) = η − γφ̄′(t) ≥ η − γφ̄′(\frac{t_{0}}{C_{0}} ) ≥ \frac{η}{2}, and so integrating by parts shows

\begin{aligned}|A| & ≤ \frac{1}{η} ∫_{\frac{1}{η} ≤t≤1} η|q′(t) − 1| \frac{dt}{t} + \frac{1}{η} ∫_{\frac{1}{η} ≤t} \frac{1}{t^{2}} dt + C \\ & ≤ C ∫_{0}^{1} t^{ϵ} \frac{dt}{t} + C ≤ C. \end{aligned}

Also

|B| ≤ ∫_{\frac{t_{0}}{C_{0}} ≤t≤C_{0}t_{0}} \frac{1}{t} dt ≤ 2 log(C_{0}).

For C_{0}t_{0} ≤ t, \frac{d}{dt}(γφ̄(t) − ηt) = γφ̄′(t) − η ≥ \frac{γ}{2}φ̄′(t), and so integrating by parts show

|D| ≤ \frac{1}{γ} ∫_{\frac{1}{η} ≤t_{0}≤t} (\frac{φ̄′′(t)}{t[φ̄′(t)]^{2}} + \frac{1}{φ̄′(t)t^{2}} + \frac{η|q′(t) − 1|}{φ̄′(t)t}) dt + \frac{η}{γ} \frac{1}{φ̄′(t_{0})}

≤ \frac{η}{γ} ∫_{t_{0}≤t} \frac{φ̄′′(t)}{[φ̄′(t)]^{2}} dt + \frac{1}{φ̄′(t_{0})} [\frac{η}{γ} + C \frac{η}{γ} ∫_{0}^{1} \frac{t^{ϵ}}{t} dt] + \frac{η}{γ} \frac{1}{φ̄′(t_{0})}

≤ C \frac{η}{γ} \frac{1}{φ̄′(t_{o})} ≤ C

since \frac{η}{γ} = φ̄′(t_{0}). Next suppose 10γ ≤ η. Then in a neighborhood of the origin, \frac{d}{dt}[ηt − γφ̄(t)] ≥ \frac{η}{2}, and so integrating by parts shows

\begin{aligned}|II| & ≤ \frac{1}{η} [ ∫_{\frac{1}{η} ≤t} η|q′(t) − 1| \frac{dt}{t} + ∫_{\frac{1}{η} ≤t} \frac{1}{t^{2}} ] + C \\ & ≤ C ∫_{0}^{1} t^{ϵ} \frac{dt}{t} + C ≤ C. \end{aligned}

This completes the proof of Theorems 2 and 3.

5. Proof of Theorem 4

We will prove the L^{p} boundedness of the maximal function

Mf(x′, x_{n}) = sup_{0<h≤1} \frac{1}{h^{n−1}} | ∫_{|t|≤h} f(x′ − t, x_{n} − φ(ψ(t)) dt| .

The proof for the singular integral is similar.

When E_{ℓ_{0}} = {0}, the main term P (t), in the decomposition (3.1) for ψ(t), ψ(t) = P (t) + R(t), is a positive homogeneous polynomial of degree ℓ_{0}. R(t) consists of all the terms in the Taylor expansion of ψ with degree greater than ℓ_{0}. The proof of L^{p} boundedness for M in the case P (t) = |t|^{2} and R(t) ≡ 0 is carried out in [KWWZ]. We will see that slight modifications of the arguments given in [KWWZ] work for the general case.

It will be convenient for us to use polar coordinates with respect to the surface P (ω) = 1. That is, every t ≠ 0 ∈ ℝ^{n−1} can be written uniquely as t = rω where r > 0 and P (ω) = 1. We also introduce a norm ∥ · ∥ so that ∥t∥ = ∥rw∥ = r. Since the Euclidean norm of ω, |ω|, is bounded above and below as ω runs over the surface P (ω) = 1, it is clear that the maximal function Mf(x) is pointwise comparable to the maximal function defined in terms of averages with respect to the norm ∥ · ∥. Therefore it suffices to consider

ℳf(x) = sup_{k>0} 2^{k(n−1)} |∫ χ(2^{k}∥t∥)f(x − Γ(t)) dt| =^{def} sup_{k>0} |f ∗ dμ_{k}(x)|

where χ is a smooth cut-off function supported in [1, 2] and chosen so that

2^{k(n−1)} ∫_{ℝ^{n−1}} χ(2^{k}∥t∥) dt ≡ 1.

To prove L^{p} bounds for ℳ we introduce dilations {δ(t)}_{t>0}, defined by δ(t)(ξ, γ) = (tξ, φ̄(t)γ). Although the “balls” generated with respect to these dilations do not in general form a space of homogeneous type with respect to Lebesgue measure, an appropriate singular integral and Littlewood–Paley theory for the dilations {δ(t)}_{t>0} has been worked out in [CCVWW]. Using this theory and well-known techniques, following the arguments detailed in [KWWZ], we reduce ourselves to proving two basic estimates for the Fourier transform of the measures {dμ_{k}} defined above:

|d̂μ_{k}(ξ, γ) − 1| ≤ C|δ(2^{−k+3})(ξ, γ)|, (5.1)

and

|d̂μ_{k}(ξ, γ)| ≤ C|δ(2^{−k−1})(ξ, γ)|^{−ϵ} (5.2)

for some ϵ > 0. Using polar coordinates t = rω,

d̂μ_{k}(ξ, γ) = 2^{k(n−1)} ∫_{ℝ} ∫_{P (ω)=1} χ(2^{k}r)e^{iξr·ω}e^{iγφ(ψ(rω))}r^{n−2}h(ω) dω dr (5.3)

where h(ω) is some smooth function. Since ψ(rω) = r^{ℓ_{0}} +𝒪(r^{ℓ_{0}+1}), we have for k > 0 large, φ(ψ(rω)) ≤ φ̄(2^{−k+3}) when 2^{−k} ≤ r ≤ 2^{−k+1}. Therefore

|d̂μ_{k}(ξ, γ) − 1| ≤ C[2^{−k}|ξ| + φ̄(2^{−k+3})|γ|] ≤ C|δ(2^{−k+3})(ξ, γ)|,

establishing (5.1). To prove (5.2) we make the change of variables

λ^{ℓ_{0}} = ψ(rω) = r^{ℓ_{0}} + R(rω) (5.4)

in the r integral in (5.3) for fixed ω. For k > 0 large this is a good change of variables and so

d̂μ_{k}(ξ, γ) = 2^{k(n−1)} ∫_{ℝ} e^{iγφ̄(λ)} ∫_{P (ω)=1} e^{ir(λ,ω)ξ·ω}χ(2^{k}r(λ, ω)) \frac{∂r}{∂λ} r^{n−2}(λ, ω) h(ω) dω dλ. (5.5)

SINGULAR INTEGRALS ASSOCIATED TO HYPERSURFACES: L^{2} THEORY 29

From (5.4) one easily deduces the following estimates on the derivatives of r(λ, ω):

r(λ, ω) = λ + 𝒪(λ^{2}), \frac{∂r}{∂λ} = 1 + 𝒪(λ), (5.6)

∇_{ω}r = 𝒪(λ), \frac{∂r}{∂λ∂ω} = 𝒪(1), (5.7)

\frac{∂^{2}r}{∂λ^{2}} = 𝒪 (\frac{1}{λ}) . (5.8)

Since P (ω) = 1 is of finite type, we can argue as in section 3 to find an ϵ > 0 such that

| ∫_{P (ω)=1} e^{ir(λ,ω)ξ·ω}h(ω) dω| ≤ C \frac{1}{|λξ|^{2ϵ}} .

Now integrating by parts, using (5.6) and (5.7), shows

|d̂μ_{k}(ξ, γ)| ≤ C (\frac{1}{2^{−k−1}|ξ|})^{2ϵ} ,

establishing (5.2) if

\sqrt{|γ|φ̄(2^{−k−1})} ≤ C2^{−k−1}|ξ|.

On the other hand, if

C2^{−k−1}|ξ| ≤ \sqrt{|γ|φ̄(2^{−k−1})},

we perform the λ integration first, writing (5.5) as

d̂μ_{k}(ξ, γ) = 2^{k(n−1)} ∫_{P (ω)=1} h(ω) ∫_{ℝ} e^{i[γφ̄(λ)+λξ·ω]}e^{i[r(λ,w)−λ]ξ·ω}χ(2^{k}r) \frac{∂r}{∂λ} r^{n−2} dλ dω.

For 2^{−k} ≤ r(λ, ω) ≤ 2^{−k+1}, we have

|\frac{∂}{∂λ}[γφ̄(λ) + λξ · ω]| ≥ \frac{|γ|}{2}φ̄′(2^{−k−1}) ≥ |γ| \frac{φ̄(2^{−k−1})}{2^{−k}}

since 2^{−k}|ξ| ≪ |γ|φ̄(2^{−k−1}). Thus integrating by parts, using (5.6) and (5.8), shows

|d̂μ_{k}(ξ, γ)| ≤ C [\frac{1}{|γ|φ̄(2^{−k−1})} + \frac{|ξ|2^{−k}}{|γ|φ̄(2^{−k−1})}]

≤ C \frac{1}{\sqrt{|γ|φ̄(2^{−k−1})}}

≤ C \frac{1}{\sqrt{|δ(2^{−k−1})(ξ, γ)|}}

since C2^{−k}|ξ| ≤ \sqrt{|γ|φ̄(2^{−k−1})}.

This completes the proof of (5.1) and (5.2) from which the L^{p} boundedness of the maximal function follows as in [KWWZ].

6. Proof of Theorem 5

We need to show that the multiplier for H,

m(ξ, γ) = ∫ ∫_{{|t|≤1 \atop t∈ℝ^{2}}} e^{iγφ(ψ(t))}e^{iξ·t}K(t) dt (6.1)

is uniformly bounded for ξ ∈ ℝ^{2} and γ ∈ ℝ. Introducing polar coordinates with respect to the convex curve ψ(t) = 1, we may write (6.1) as

∫_{0}^{1} e^{iγφ(r)} \frac{1}{r} ∫_{ψ(ω)=1} e^{irξ·ω}K(ω) h(ω) dω dr (6.2)

for some smooth function h(ω). The argument used in the proof of Theorem 1 to establish (3.10) shows

∫_{ψ(ω)=1} K(ω) h(ω) dω = 0

and so the part of the integral in (6.2) where r ≤ \frac{1}{|ξ|} is at most

C ∫_{r≤ \frac{1}{|ξ|}} \frac{1}{r} | ∫_{ψ(ω)=1} (e^{irξ·ω} − 1)K(ω)h(ω) dω| dr ≤ C|ξ| ∫_{r≤ \frac{1}{|ξ|}} dr ≤ C.

For the region where r ≥ \frac{1}{|ξ|}, we observe that the inner integral in (6.2) is the Fourier transform of a smooth density on the convex curve ψ(ω) = 1 evaluated at rξ. This Fourier transform can be estimated in terms of the “balls,” E(t, ϵ), introduced in section 1. In fact

| ∫_{ψ(ω)=1} e^{irξ·ω} K(ω)h(ω) dω| ≤ C [|E(t_{1}(ξ), \frac{1}{r|ξ|})| + |E(t_{2}(ξ), \frac{1}{r|ξ|})|]

where t_{1}(ξ) and t_{2}(ξ) are the two points on the curve ψ(t) = 1 whose tangent lines are normal to ξ. See [BNW]. Therefore the part of the integral in (6.2) where r ≥ \frac{1}{|xi|} can be estimated by

C sup_{{t \atop ψ(t)=1}} ∫_{\frac{1}{|ξ|} ≤r} |E(t, \frac{1}{r|ξ|})| \frac{dr}{r} ≤ C sup_{{t \atop ψ(t)=1}} ∫_{0}^{1} |E(t, δ)| \frac{dδ}{δ} .

Hence the multiplier m(ξ, γ) is uniformly bounded in ξ and γ if the quantity

sup_{{t \atop ψ(t)=1}} ∫_{0}^{1} |E(t, δ)| \frac{dδ}{δ}

is finite. This completes the proof of Theorem 5.

7. Appendix

In this appendix we will compute the determinant of an r × r matrix A = {α_{j,k}} of the form A = cI + B where B = {b_{j,k}} and b_{j,k} = b s_{j}t_{k}. We will show that

det(A) = c^{r} + c^{r−1} b ∑_{j=1}^{r} s_{j}t_{j}. (7.1)

For the example we need in this paper, α_{j,k} = s_{j}s_{k} for j ≠ k and α_{j,j} = 1 − |s|^{2} + s_{j}^{2}. Therefore taking t_{j} = s_{j}, c = 1 − |s|^{2} and b = 1 in the above formula (7.1) gives us the desired result det(A) = (1 − |s|^{2})^{r−1} in this case. To prove (7.1) first note that as a function of s = (s_{1}, …, s_{r}), det(A) is an affine function in each of the variables s_{j} separately. Also computing any pure mixed derivative, e.g., \frac{∂^{3}}{∂s_{1}∂s_{2}∂s_{3}} , of det(A) gives rise to two or more rows being identical and therefore zero. Hence expanding det(A) in its Taylor series in s about the origin, we see that (7.1) follows from the fact that for each 1 ≤ j ≤ r, the partial derivative of det(A) with respect to s_{j} at the origin is c^{r−1} b t_{j}. This is a straightforward computation.

References

[BNW] J. Bruna, A. Nagel, and S. Wainger. Convex hypersurfaces and Fourier transforms. Ann. of Math., 127:333–365, 1988.

[CCVWW] Anthony Carbery, Michael Christ, James Vance, Stephen Wainger, and David K. Watson. Operators associated to flat plane curves: L^{p} estimates via dilation methods. Duke Math. J., 59(3):675–700, 1989.

[KWWZ] Weon-Ju Kim, Stephen Wainger, James Wright, and Sarah Ziesler. Singular integrals and maximal functions associated to surfaces of revolution. Bull. London Math. Soc., 28(3):291–296, 1996.

[NVWW] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. Hilbert transforms for convex curves. Duke Math. J., 50:735–744, 1983.

[Sch] Helmut Schulz. Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms. Indiana Univ. Math. J., 40(4):1267–1275, 1991.

[SWWZ] A. Seeger, S. Wainger, J. Wright, and S. Ziesler. Classes of singular integrals along curves and surfaces. Trans. Amer. Math. Soc. To appear.

[St] E. Stein. Harmonic Analysis Princeton University Press, Princeton New Jersey, 1993.

Stephen Wainger, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

  E-mail address: wainger@math.wisc.edu

James Wright, School of Mathematics, University of New South Wales, Sydney 2052, Australia

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

Sarah Ziesler, Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland

  E-mail address: zies@ollamh.ucd.ie