Shaoming Guo

New York Journal of Mathematics

New York J. Math. 23(2017) 1733–1738.

A remark on oscillatory integrals associated with fewnomials

Shaoming Guo

Abstract. We prove that theL²bound of an oscillatory integral asso- ciated with a polynomial phase depends only on the number of mono- mials that this polynomial consists of.

Contents

1. Introduction 1733

2. Reduction to monomials 1734

3. Bad scales 1735

4. Good scales 1736

References 1738

1. Introduction

Letd∈N. Consider the operator

(1) H_Qf(x) :=

Z

R

f(x−t)e^iQ(t)dt t , with

(2) Q(t) =a1t^α1+· · ·+a_dt^αd.

Here a_i ∈R and α_i is a positive integer for each 1≤i≤d.

Theorem 1.1. Givend∈N, we have

(3) kH_Qfk₂ ≤C_dkfk₂.

Here Cd is a constant that depends only ond, but not on any ai or αi.

Received July 13, 2017.

2010Mathematics Subject Classification. 42B20, 42B25.

Key words and phrases. Uniform estimate, oscillatory integral, Stein–Wainger, fewnomial.

Partially supported by the National Science Foundation under Grant No. DMS- 1440140.

ISSN 1076-9803/2017

1733

1734 SHAOMING GUO

OnR², define the Hilbert transform along the polynomial curve (t, Q(t))t∈R

by

(4) H_Qf(x, y) =

Z

R

f(x−t, y−Q(t))dt t . As a corollary of Theorem1.1, we have:

Corollary 1.2. Given d∈N, we have

(5) kH_Qfk₂≤Cdkfk₂.

Here C_d is a constant that depends only ond, but not on any ai or αi. Corollary1.2follows from Theorem1.1via applying Plancherel’s theorem to the second variable ofH_Qf. We leave out the details.

Denote by n the degree of the polynomial Q given by (2). Then it is well-known (see Stein and Wainger [SW70]) that the estimate (3) holds true if we replaceC_d by Cn, a constant that is allowed to depend on the degree n. Moreover, Parissis [Par08] proved that

(6) sup

P∈Pn

p.v.

Z

R

e^iP(t)dt t

'logn,

where P_n is the collection of all real polynomials of degree at most n. It would also be interesting to know whether the constant C_d in (3) can be made to (logd)^cfor some c >0.

Acknowledgements. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the au- thor was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2017. The author thanks the anonymous reviewers for their careful reading of the manuscript and sug- gestions on how to improve the exposition of the paper.

2. Reduction to monomials

We start the proof. In this section, we will splitRinto different intervals, and show that for all but finitely many of these intervals, there always exists a monomial which “dominates” our polynomial Q. In dimension one, this idea has been used extensively in the literature, for instance Folch-Gabayet and Wright [FW12]. Here we follow the formulation of Li and Xiao [LX16].

Notice that we can always let the functionf absorb the linear term of Q.

Hence we assume that 1 < α1 < · · · < αd. Denote by n the degree of the polynomial Q, that is n=αd. Letλ= 2ⁿ¹.Definebj ∈Zsuch that

(7) λ^bj ≤ |a_j|< λ^bj+1.

We define a few bad scales. For 1≤j1< j2 ≤d, define

(8) J_bad⁽⁰⁾(Γ0, j1, j2) :={l∈Z: 2^−Γ0|a_j2λ^αj2l| ≤ |a_j1λ^αj1l| ≤2^Γ0|a_j2λ^αj2l|}.

Here Γ₀ := 2^10d!. Notice that lsatisfies

(9) −2−nΓ₀+b_j2 −b_j1 ≤(α_j1 −α_j2)l≤nΓ₀+b_j2 −b_j1+ 2.

Hence J_bad⁽⁰⁾(Γ0, j1, j2) is a connected set whose cardinality is smaller than 4nΓ₀.Define

(10) J_good⁽⁰⁾ :=



 [

j16=j₂

J_bad⁽⁰⁾(Γ0, j1, j2)





c

.

Notice thatJ_good⁽⁰⁾ has at mostd² connected components. Moreover, on each component, there is exactly one monomial which is “dominating”.

Similarly, we define J_bad⁽¹⁾(Γ₀, j₁, j₂) :=

(11)

l∈Z: 2^−Γ0|α_j2(α_j2 −1)a_j2λ^αj2l| ≤ |α_j1(α_j1 −1)a_j1λ^αj1l|

≤2^Γ0|α_j2(α_j2 −1)a_j2λ^αj2l| . Moreover,

(12) J_bad⁽¹⁾ := [

j16=j2

J_bad⁽¹⁾(Γ₀, j₁, j₂) and J_good:=J_good⁽⁰⁾ \ J_bad⁽¹⁾. Analogously,J_good has at mostd⁴ connected components.

3. Bad scales

Due to the control on the cardinalities of various bad sets, the contribu- tions from those l 6∈ J_good can be controlled by a multiple of the Hardy–

Littlewood maximal function.

Let us be more precise. Suppose that we are working on the collection of bad scalesJ_bad⁽⁰⁾(Γ0, j1, j2) for some j1 and j2. Define

(13) Hlf(x) =

Z

R

f(x−t)e^iQ(t)ψl(t)dt t .

Here ψ₀ is a nonnegative smooth bump function supported on [−λ²,−λ⁻¹]∪[λ⁻¹, λ²]

such that

(14) X

l∈Z

ψ_l(t) = 1 for everyt6= 0, withψ_l(t) :=ψ₀ t

λ^l

.

By the triangle inequality, we have (15)

X

l∈J_bad⁽⁰⁾(Γ0,j1,j2)

H_lf(x)

≤ X

l∈J_bad⁽⁰⁾(Γ0,j1,j2)

Z

R

|f(x−t)|ψ_l(t)dt

|t|.

1736 SHAOMING GUO

Recall that the cardinality of J_bad⁽⁰⁾(Γ0, j1, j2) is at most 4nΓ0. Now we partition the set J_bad⁽⁰⁾(Γ₀, j₁, j₂) into subsets of consecutive elements, and such that each subset contains exactly n elements, with possibly one ex- ception which can be handled in the same way. The scale that these n elements can see is about λⁿ = 2, in the sense that for every l₀ ∈ Z, supp

Pl0+n l=l0 ψ_l

has Lebesgue measure about λⁿ. Hence the contribution from each of these subsets can be controlled by 2M f(x). Here M denotes the Hardy–Littlewood maximal operator. Hence the right hand side of (15) can be controlled by 8Γ₀·M f(x). This takes care of the contribution from bad scales.

4. Good scales

Suppose we are working on one connected component of J_good, and for each integerl in such a component, we assume that a_j1t^αj1 dominatesQ(t) in the sense of (8), that is,

(16) |a_j1λ^αj1l| ≥2^Γ0|a_j⁰

1λ^αj01l| for every j₁⁰ 6=j₁,

and a_j2α_j2(α_j2 −1)t^αj2−2 dominatesQ⁰⁰(t) in the sense of (11), that is, (17) |a_j2αj2(αj2 −1)λ^αj2l| ≥2^Γ0|a_j⁰

2α_j⁰

2(α_j⁰

2 −1)λ^αj02l|for every j₂⁰ 6=j2. Let us call such a set J_good(j₁, j₂). Under this assumption, we have the estimates

(18) |Q(t)| ≤2|a_j1t^αj1|and |Q⁰⁰(t)| ≥ |a_j1t^αj1−2|,

for every t ∈ [λ^l−2, λ^l+1] with l ∈ J_good(j₁, j₂). Recall that λ = 2¹ⁿ is the smallest scale that we will work with. This scale is only visible when antⁿ dominates. When some other monomial dominates, at such a small scale, our polynomial will not have enough room to see the oscillation. This will be reflected when we come to the stage of applying van der Corput’s lemma (see (27) below). Define

λ_j1 := 2

1 αj1.

We choose this scale because the monomialaj1t^αj1 dominates. Let

(19) Φj1,j2(t) = X

l∈J_good(j1,j2)

ψl(t).

Notice that here we join all the small scales from J_good(j₁, j₂) to form a larger scale. Next we will apply a new partition of unity to the function Φj1,j2. Define

(20) H_l^(j0¹⁾f(x) = Z

R

f(x−t)e^iQ(t)ψ_l^(j0¹⁾(t)Φ_j1,j2(t)dt t .

Here ψ₀^(j1) is a nonnegative smooth bump function supported on [−λ²_j

1,−λ⁻¹_j

1 ]∪[λ⁻¹_j

1 , λ²_j1] such that

(21) X

l⁰∈Z

ψ_l^(j0¹⁾(t) = 1 for everyt6= 0, withψ^(j_l0¹⁾(t) :=ψ^(j₀¹⁾ t λ^l_j⁰₁

! .

We defineB_j1 ∈Zsuch that

(22) λ^−B_j ^j1

1 ≤ |a_j1|< λ^−B_j ^j1+1

1 ,

denote γ_j1 =B_j1/α_j1 and split the sum inl⁰ into two cases.

(23) X

l⁰∈Z

H_l^(j1)f = X

l⁰≤γ_j

1

H_l^(j0¹⁾f+ X

l⁰>γj1

H_l^(j0¹⁾f.

The first summand in (23) can be controlled by the maximal function and the maximal Hilbert transform. To be precise, we have a bound

X

l⁰≤γ_j1

Z

R

f(x−t)ψ_l^(j0¹⁾(t)Φj1,j2(t)dt t

(24)

+ X

l⁰≤γj1

Z

R

f(x−t)(e^iQ(t)−1)ψ^(j_l0¹⁾(t)Φ_j1,j2(t)dt t

.H^∗f(x) +M f(x)

+ X

l⁰≤γ_j1

Z

R

f(x−t)(e^iQ(t)−1)ψ^(j_l0¹⁾(t)Φj1,j2(t)dt t

.

Here H^∗ stands for the maximally truncated Hilbert transform. The last summand in (24) can be further controlled by

X

l⁰≤γ_j1

Z

R

|f(x−t)||a_j1t^αj1|ψ_l^(j0¹⁾(t)dt (25) |t|

≤X

l∈N

Z λ^γj_j^1−l+1

1

λ^γj_j ^1−l−2

1

|f(x−t)||a_j1||t|^αj1−1dt

≤X

l∈N

λ^(γ_j1^{j1−l+1)(αj1−1)}

Z λ^γj_j^1−l+1

1

λ^γj_j ^1−l−2

1

|f(x−t)||a_j1|dt≤8M f(x).

Hence it remains to handle the latter term from (23). We will prove that there existsδ >0 such that

(26) kH_γ^(j1)

j1+lfk₂ ≤C_d2^−δlkfk₂, for everyl≥0,

1738 SHAOMING GUO

with a constantC_d depending only on d. This amounts to proving a decay for the multiplier

(27) Z

R

e^iQ(t)+itξψ_γ^(j1)

j1+l(t)dt t =

Z

R

e^iQ(λ

γj1+l

j1 t)+iλ^γj_j ^1+l

1 tξ

ψ₀^(j1)(t)dt t . We calculate the second order derivative of the phase function:

(28) λ^2γ_j ^j1+2l

1 |Q⁰⁰(λ^γ_j^j1+l

1 t)| ≥ 1

2|a_j1|λ^B_j^j1+αj1l

1 ≥2^l−2.

Hence the desired estimate follows from van der Corput’s lemma, for which we refer to Proposition 2 in Page 332 [Stein93].

References

[FW12] Folch-Gabayet, Magali; Wright, James. Weak-type (1,1) bounds for os- cillatory singular integrals with rational phases.Studia Math.210(2012), no. 1, 57–76.MR2949870,Zbl 1258.42011, doi:10.4064/sm210-1-4.

[LX16] Li, Xiaochun; Xiao, Lechao. Uniform estimates for bilinear Hilbert trans- forms and bilinear maximal functions associated to polynomials.Amer. J. Math.

138 (2016), no. 4, 907–962. MR3538147, Zbl 1355.42011, arXiv:1308.3518, https://muse.jhu.edu/article/628316.

[Par08] Parissis, Ioannis R.A sharp bound for the Stein–Wainger oscillatory integral.

Proc. Amer. Math. Soc.136(2008), no. 3, 963–972.MR2361870,Zbl 1215.42010, arXiv:0709.1466, doi:10.1090/S0002-9939-07-09013-2.

[Stein93] Stein, Elias M.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Math- ematical Series, 43. Monographs in Harmonic Analysis, III.Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5.MR1232192,Zbl 0821.42001.

[SW70] Stein, Elias M.; Wainger, Stephen.The estimation of an integral arising in multiplier transformations.Studia Math.35 (1970). 101–104.MR0265995,Zbl 0202.12401, doi:10.4064/sm-35-1-101-104.

(Shaoming Guo)831 E. Third St., Bloomington, 47405, IN, USA [email protected]

This paper is available via http://nyjm.albany.edu/j/2017/23-77.html.