A remark on oscillatory integrals associated with fewnomials
Shaoming Guo












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    A uniform estimate]A remark on oscillatory integrals associated with fewnomials831 E. Third St., Bloomington, 47405, IN, USAshaoguo@iu.edu[2010]42B20, 42B25
    
    We prove that the L^2 bound of an oscillatory integral associated with a polynomial phase depends only on the number of monomials that this polynomial consists of. 
    [
        Shaoming Guo
        December 30, 2023
    =====================
    
    
    
    § INTRODUCTION
    
    
    Let d∈. Consider the operator 
    
    H_Q f(x):=∫_ f(x-t)e^i Q(t)dt/t,
    
    with 
    
    Q(t)=a_1 t^α_1+… +a_d t^α_d.
    
    Here a_i∈ and α_i is a positive integer for each 1≤ i≤ d. 
    
    Given d∈, we have 
    H_Q f_2 ≤ C_d f_2.
    
    Here C_d is a constant that depends only on d, but not on any a_i or α_i. 
    
    On ^2, define the Hilbert transform along the polynomial curve (t, Q(t))_t∈ by 
    H_Q f(x, y)=∫_ f(x-t, y-Q(t))dt/t.
    
    As a corollary of Theorem <ref>, we have 
    
    Given d∈, we have 
    H_Q f_2 ≤ C_d f_2.
    
    Here C_d is a constant that depends only on d, but not on any a_i or α_i. 
    
    Corollary <ref> follows from Theorem <ref> via applying Plancherel's theorem to the second variable of H_Q f. We leave out the details. 
    
    
    Denote by n  the degree of the polynomial Q given by (<ref>). Then it is well-known (see Stein and Wainger <cit.>) that the estimate (<ref>) holds true if we replace C_d by C_n, a constant that is allowed to depend on the degree n. Moreover, Parissis <cit.> proved that 
    sup_P∈P_n|p.v.∫_ e^iP(t)dt/t| ≃log n,
    
    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 (<ref>) can be made to (log d)^c for some c>0.
     Acknowledgements. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author 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 suggestions on how to improve the exposition of the paper. 
    
    
    
    
    § REDUCTION TO MONOMIALS
    
    We start the proof. In this section, we will split  into 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 <cit.>. Here we follow the formulation of Li and Xiao <cit.>.
    
    Notice that we can always let the function f absorb the linear term of Q. Hence we assume that 
    1<α_1< … < α_d.
    Denote by n the degree of the polynomial Q, that is n=α_d. Let λ=2^1/n.
    Define b_j∈ such that 
    λ^b_j≤ |a_j|< λ^b_j+1.
    
    We define a few bad scales. For 1≤ j_1< j_2≤ d, define 
    J^(0)_bad(Γ_0, j_1, j_2):={l∈: 2^-Γ_0 |a_j_2λ^α_j_2l|≤ |a_j_1λ^α_j_1l|≤ 2^Γ_0 |a_j_2λ^α_j_2l|}.
    
    Here Γ_0:=2^10d!.
    
    
    Notice that l satisfies 
    
    -2-nΓ_0 + b_j_2-b_j_1≤ (α_j_1-α_j_2) l≤ nΓ_0 + b_j_2-b_j_1+2.
    
    Hence J^(0)_bad(Γ_0, j_1, j_2) is a connected set whose cardinality is smaller than 4nΓ_0.
    Define 
    J^(0)_good:=( ⋃_j_1≠ j_2J^(0)_bad(Γ_0, j_1, j_2) )^c
    
    Notice that J^(0)_good has at most d^2 connected components. Moreover, on each component, there is exactly one monomial which is “dominating”. 
    
    
    Similarly, we define 
    J^(1)_bad(Γ_0, j_1, j_2):=   {l∈: 2^-Γ_0 |α_j_2(α_j_2-1) a_j_2λ^α_j_2l|
       ≤ |α_j_1(α_j_1-1) a_j_1λ^α_j_1l|≤ 2^Γ_0 |α_j_2(α_j_2-1) a_j_2λ^α_j_2l|}.
    
    Moreover,  
    J^(1)_bad:= ⋃_j_1≠ j_2J^(1)_bad(Γ_0, j_1, j_2)  and J_good:=J^(0)_good∖J^(1)_bad.
    
    Analogously, J_good has at most d^4 connected components. 
    
    
    
    § BAD SCALES
    
    
    Due to the control on the cardinalities of various bad sets, the contributions from those l∉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 scales J^(0)_bad(Γ_0, j_1, j_2) for some j_1 and j_2. Define 
    
    H_l f(x)=∫_ f(x-t) e^i Q(t)ψ_l(t)dt/t.
    
    Here ψ_0  is a non-negative smooth bump function supported on  [-λ^2, -λ^-1]∪[λ^-1, λ^2] such that 
    ∑_l∈ψ_l(t)=1  for every  t≠ 0,  with ψ_l(t):=ψ_0(t/λ^l).
    
    By the triangle inequality, we have 
    |∑_l∈J^(0)_bad(Γ_0, j_1, j_2) H_l f(x)| ≤∑_l∈J^(0)_bad(Γ_0, j_1, j_2)∫_ |f(x-t)| ψ_l(t)dt/|t|.
    
    Recall that the cardinality of J^(0)_bad(Γ_0, j_1, j_2) is at most 4nΓ_0. Now we partition the set J^(0)_bad(Γ_0, j_1, j_2) into subsets of consecutive elements, and such that each subset  contains exactly n elements, with possibly one exception which can be handled in the same way. The scale that these n elements can see is about λ^n=2, in the sense that for every l_0∈, supp(∑_l=l_0^l_0+nψ_l)  has Lebesgue measure about λ^n. Hence the contribution from each of these subsets can be controlled by 2 Mf(x). Here M denotes the Hardy-Littlewood maximal operator. Hence the right hand side of (<ref>) can be controlled by 8Γ_0 · Mf(x). This takes care of the contribution from bad scales. 
    
    
    § GOOD SCALES
    
    
    
    
    
    Suppose we are working on one connected component of J_good, and for each integer l in such a component, we assume that 
    a_j_1 t^α_j_1 dominates  Q(t) in the sense of (<ref>), that is,
    
    |a_j_1λ^α_j_1l|≥ 2^Γ_0 |a_j'_1λ^α_j'_1l|  for every  j'_1≠ j_1,
    
    and a_j_2α_j_2(α_j_2-1) t^α_j_2-2 dominates  Q”(t) in the sense of (<ref>), that is,
    
    |a_j_2α_j_2(α_j_2-1)λ^α_j_2l|≥ 2^Γ_0 |a_j'_2α_j'_2(α_j'_2-1)λ^α_j'_2l|  for every  j'_2≠ j_2.
     
    Let us call such a set J_good(j_1, j_2). Under this assumption, we have the estimates 
    
    |Q(t)|≤ 2 |a_j_1 t^α_j_1|  and  |Q”(t)|≥ |a_j_1t^α_j_1-2|,
    
    for every t∈ [λ^l-2, λ^l+1]  with  l∈J_good(j_1, j_2).
    Recall that λ=2^1/n is the smallest scale that we will work with. This scale is only visible when a_n t^n 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 (<ref>) below). Define 
    λ_j_1:
    
    =2^1/α_j_1.
    We choose this scale because the monomial a_j_1 t^α_j_1 dominates. Let 
    Φ_j_1, j_2(t)= ∑_l∈J_good(j_1, j_2)ψ_l(t).
    
    Notice that here we join all the small scales from J_good(j_1, j_2) to form a larger scale. Next we will apply a new partition of unity to the function Φ_j_1, j_2. Define 
    
    H^(j_1)_l' f(x)=∫_ f(x-t) e^i Q(t)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t.
    
    Here ψ^(j_1)_0 is a non-negative smooth bump function supported on [-λ_j_1^2, -λ_j_1^-1]∪[λ_j_1^-1, λ_j_1^2] such that 
    ∑_l'∈ψ^(j_1)_l'(t)=1  for every  t≠ 0,  with ψ^(j_1)_l'(t):=ψ^(j_1)_0(t/λ_j_1^l').
    
    
    We define B_j_1∈ such that  
    λ_j_1^-B_j_1≤ |a_j_1|<λ_j_1^-B_j_1+1,
    
    denote γ_j_1=B_j_1/α_j_1 and split the sum in l' into two cases. 
    ∑_l'∈H^(j_1)_l f=∑_l'≤γ_j_1H^(j_1)_l' f+∑_l'> γ_j_1H^(j_1)_l' f.
    
    The  first summand in (<ref>) can be controlled by the maximal function and the maximal Hilbert transform. To be precise, we have a bound 
       ∑_l'≤γ_j_1|∫_ f(x-t)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t|
        +∑_l'≤γ_j_1|∫_ f(x-t) (e^i Q(t)-1)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t|
        H^* f(x)+ Mf(x)+ ∑_l'≤γ_j_1|∫_ f(x-t) (e^i Q(t)-1)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t|.
    
    Here H^* stands for the maximally truncated Hilbert transform. The last summand in (<ref>) can be further controlled by 
       ∑_l'≤γ_j_1∫_ |f(x-t)| |a_j_1t^α_j_1|ψ^(j_1)_l'(t)dt/|t|≤∑_l∈∫_λ^γ_j_1-l-2_j_1^λ^γ_j_1-l+1_j_1 |f(x-t)| |a_j_1| |t|^α_j_1-1dt
       ≤∑_l∈λ^(γ_j_1-l+1)(α_j_1-1)_j_1∫_λ^γ_j_1-l-2_j_1^λ^γ_j_1-l+1_j_1 |f(x-t)| |a_j_1|dt
     ≤ 8Mf(x). 
    
    Hence it remains to handle the latter term from (<ref>). 
    We will prove that there exists δ>0 such that 
    H^(j_1)_γ_j_1+l f_2 ≤ C_d 2^-δ lf_2,  for every  l≥ 0,
    
    with a constant C_d depending only on d. This amounts to proving a decay for the multiplier 
    ∫_ e^iQ(t)+itξψ^(j_1)_γ_j_1+l(t)dt/t=∫_ e^iQ(λ^γ_j_1+l_j_1t)+iλ^γ_j_1+l_j_1tξψ^(j_1)_0(t)dt/t.
    
    We calculate the second order derivative of the phase function:
    λ^2γ_j_1+2l_j_1 |Q”(λ^γ_j_1+l_j_1t)|≥1/2 |a_j_1|λ^B_j_1+α_j_1 l_j_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 <cit.>
    20[FW12]FW M. Folch-Gabayet and J. Wright. Weak-type (1,1) bounds for oscillatory singular integrals with rational phases.  Studia Math. 210 (2012), no. 1, 57–76. 
    [LX16]LiXiao X. Li and L. Xiao. Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials. Amer. J. Math. 138 (2016), no. 4, 907–962. 
    [Par08]Par I. Parissis. A sharp bound for the Stein-Wainger oscillatory integral.  Proc. Amer. Math. Soc. 136 (2008), no. 3, 963-972.
    [Stein93]Stein E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals.  With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5 
    [SW70]SW E. Stein and S. Wainger. The estimation of an integral arising in multiplier transformations. Studia Math. 35 1970 101–104.
