Uniform boundedness of oscillatory singular integrals on Hardy spaces

Publicacions Matem`atiques, Vol. 44 (2000), 605–611 UNIFORM BOUNDEDNESS OF OSCILLATORY SINGULAR INTEGRALS ON HARDY SPACES Leslie C. Cheng and Yibiao Pan Abstract We prove the uniform H1boundedness of oscillatory singular integrals with degenerate phase functions. 1. Introduction In a recent paper [1], Carbery, Christ, and Wright established a series of important estimates for oscillatory integrals and sublevel sets. As one of the applications they also obtained the uniform boundedness on Lpspaces of oscillatory singular integrals with possibly flat phase functions. The purpose of this paper is to apply their results to the study of the uniform boundedness of oscillatory singular integrals on the Hardy space H1(Rn). Oscillatory singular integral operators are singular integral operators which carry oscillatory factors in their kernels. They have arisen in many problems in harmonic analysis and related areas and have been studied extensively ([10], [11]). For previous work on the boundedness on Lp and Hardy spaces of such operators, see Phong-Stein [7], Ricci-Stein [8], Stein [10], Pan [5], [6], and Carbery-Christ-Wright [1]. Let n∈N,ϕ∈C∞ 0(Rn), and Φ ∈C∞(Rn) be a real-valued function satisfying ∇Φ(0) = 0. Let K(·) be a Calder´on-Zygmund kernel on Rn, i.e. |K(x)|+|x||∇K(x)|≤A|x|−nand a<|x|<b K(x)dx = 0 for b>a>0. For each λ∈Rwe define the (localized) oscillatory singular integral operator Tλon Rnby (Tλf)(x) = p.v. Rn eiλΦ(x−y)K(x−y)ϕ(x−y)f(y)dy.(1) The following was proved in [6]: 606 L. C. Cheng, Y. Pan Theorem A. Let K,ϕ,Φ, and Tλbe given as above. If DαΦ(0) =0for some multi-index αwith |α|≥2, then there exists a positive constant C such that TλfH1(Rn)≤CfH1(Rn) (2) for f∈H1(Rn)and λ∈R. Since it follows easily from the standard theory of singular integrals that Tλis a bounded operator on H1(Rn) for each λ∈R, the importance of Theorem A lies in the uniformity of bounds on the operator norms TλH1→H1. It was also shown in [6] that (2) may not hold if DαΦ(0) vanishes for every α. Theorem B. Let x, y ∈R. Define (Tλf)(x)=p.v. |x−y|≤1 eiλΦ(x−y)f(y) x−ydy.(3) There exists a nonconstant function Φ∈C∞(R)with Φ(k)(0)=0for all k=0,1,2,..., such that sup λ∈R TλH1→H1=∞.(4) A phase function Φ is said to be flat at 0 if DαΦ(0) = 0 for all α. Can the uniform boundedness of Tλ’s on H1(Rn) still hold when the phase function Φ is flat at 0? Inspired by the work of Carbery, Christ, and Wright we address this issue by establishing the following: Theorem C. Let K,ϕ,Φ, and Tλbe given as in Theorem A. Let αbe a multi-index with |α|≥3. Suppose that at least one coordinate of αis strictly greater than 1and there exist δ, A > 0such that max |β|=|α|sup |x|≤s |DβΦ(x)|≤Ainf s≤|x|≤δ|DαΦ(x)|(5) holds for all s∈(0,δ). Then there exists a positive constant Csuch that TλfH1(Rn)≤CfH1(Rn) (6) for f∈H1(Rn)and λ∈R. Condition (5) is simply Condition (8.1) of [1] in the convolutional setting. As pointed out in [1], (5) is always satisfied if DαΦ(0) =0 and it may also be satisfied even if Φ is flat at 0. For example, when n=1,Φ(x)=η(x) exp(ω(x)/x2) satisfies (5) if ηand ωare smooth and η(0) =0,ω(0) <0. Throughout the rest of the paper the letter Cwill stand for a constant but not necessarily the same one in each occurrence. Oscillatory Singular Integrals 607 2. Some Lemmas Lemma 2.1. Let K,ϕ,Φ, and Tλbe given as in Theorem A. Let αbe a multi-index with |α|≥3. Suppose that at least one coordinate of αis strictly greater than 1and that (5) holds for some δ, A > 0, and every s∈(0,δ). Then for every p∈(1,∞)there exists a positive constant Cp such that TλfLp(Rn)≤CpfLp(Rn) (7) for f∈Lp(Rn)and λ∈R. Proof: Let ψ∈C∞ 0(Rn) such that m∈Znψm≡1 where ψm(x)= ψ(x−m). Let Tλ,m(f)=Tλ(ψmf). It follows from Theorem 8.2 in [1] that the operators {Tλ,m :λ∈R,m∈Zn}are uniformly bounded on Lp(Rn) for 1 <p<∞. One then obtains (7) by observing that the sets {supp(Tλ,m(f)) : m∈Zn}have the finite overlapping property. Let Qn=[−1/2,1/2]n. The following result is due to Carbery, Christ, and Wright ([1]). Lemma 2.2. For n=n+n, each 1≤p, q ≤∞(p=1,q=∞) and each β∈(N∪{0})nwith at least one nonzero entry in each of {1,... ,n } and {n+1,... ,n }and with at least one entry greater than 1, there exist ε=ε(n, β, p, q)>0and C=C(ε, n, β, p, q)such that for every real-valued, integrable function usatisfying Dβu≥1on Qn, for every λ∈R, the operator (Sλf)(x)=Qn eiλu(x,x )f(x)dx (8) satisfies SλfLq(Qn)≤Cλ−εfLp(Qn ).(9) 3. Proof of Theorem C Let K,ϕ, Φ, and Tλbe given as in Theorem C. By the atomic decomposition of H1(Rn) and the characterization of H1(Rn) using Riesz transforms (see [2]-[4], [11]), it suffices to show that Tλfis in L1(Rn) uniformly in λfor all atoms f. 608 L. C. Cheng, Y. Pan Let αbe the multi-index such that (5) holds and set J(s)=s|α|−1inf s≤|x|≤δ|DαΦ(x)|. Let ρ>0 be small (say, ρ<δ/2), fbe an atom which is supported in ρQnand satisfies f∞≤ρ−nand Rnf(x)dx = 0. Let λ>0. We shall first prove that |x|≥2ρ |Tλf(x)|dx ≤C(10) holds for some constant Cindependent of λ,ρ, and f. If λρ ≤(J(δ))−1, then (10) follows from |Tλf(x)|≤ A |x|nρQn |eiλΦ(x−y)−eiλΦ(x)||ϕ(x−y)f(y)|dy +|T0f(x)| ≤A(λρ) |x|n−1ρQn |ϕ(x−y)f(y)|dy +|T0f(x)| for |x|≥2ρ. Thus we may assume that λρ>(J(δ))−1. Let t= max{J−1(1/(λρ)),2ρ}.Forj≥1 define the operator Qjby (Qjf)(x)=χ[1,2](|x|)Qn eiλΦ(2jtx−ρy)ϕ(2jtx −ρy)g(y)dy.(11) By our assumption there exist multi-indices α(1),α(2) such that α= α(1) +α(2),|α(2)|= 1, and at least one entry of α(1) is greater than 1. Thus |Dα(1) xDα(2) y[Φ(2jtx −ρy)]|≥[2j(|α|−1)ρ]J(t) holds for 1 ≤|x|≤2 and y∈Qn. Since the factor ϕ(2jtx −ρy)is harmless, it follows from Lemma 2.2 that 1≤|x|≤2 |Qjg(x)|2dx1/2 ≤C[2j(|α|−1)λρJ(t)]−εgL2(Qn) (12) Oscillatory Singular Integrals 609 for some positive εindependent of α,ρ,jand g. By letting fρ(x)= ρnf(ρx) and rescaling, we have |x|≥2t |Tλf(x)|dx ≤C|x|≥2tRn eiλΦ(x−y)ϕ(x−y)f(y)dy dx |x|n +|x|≥2ρρQn |K(x−y)−K(x)||f(y)|dydx ≤C j≥1 Qj(fρ)2+C ≤C 1+(λρJ(t))−ε j≥1 2−ε(|α|−1)j ≤C. (13) Let P(x)=  |β|≤|α|−1 (β!)−1DβΦ(0)xβ and Ψ(x)=Φ(x)−P(x). By Theorem A, RnRn eiλP (x−y)K(x−y)ϕ(x−y)f(y)dy dx ≤C.(14) Thus, by (14) 2ρ≤|x|<2t |Tλf(x)|dx ≤2ρ≤|x|≤t/2ρQn |eiλ(Ψ(x−y)−Ψ(x)) −1||K(x−y)ϕ(x−y)f(y)|dydx+C ≤Cλ( max |β|=|α|sup |x|≤t |DβΦ(x)|) ×2ρ≤|x|≤t/2 |x||α|−1−ndxρQn |y||f(y)|dy+C ≤CλρJ(t)+C≤C, (15) where we took t=J−1(1/(λρ)) (otherwise t=2ρand the integral of |Tλf(x)|on {2ρ≤|x|<2t}is trivially bounded). By combining (13) and (15) we see that (10) holds. 610 L. C. Cheng, Y. Pan By H¨older’s inequality and Lemma 2.1, |x|<2ρ |Tλf(x)|dx ≤Cρn/2f2≤C.(16) It follows from (10) and (16) that Tλf1≤C(17) holds for those atoms fdescribed at the beginning of our proof. By using translation and (16) (for large ρ’s) we see that (17) holds for an arbitrary atom f. Theorem C is proved. References [1] A. Carbery, M. Christ and J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12(4) (1999), 981–1015. [2] R. R. Coifman, A real variable characterization of Hp,Studia Math. 51 (1974), 269–274. [3] C. Fefferman and E. M. Stein,Hpspaces of several variables, Acta Math. 129(3–4) (1972), 137–193. [4] R. H. Latter, A characterization of Hp(Rn) in terms of atoms, Studia Math. 62(1) (1978), 93–101. [5] Y. B. Pan, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100(1) (1991), 207–220. [6] Y. B. Pan, Boundedness of oscillatory singular integrals on Hardy spaces. II, Indiana Univ. Math. J. 41(1) (1992), 279–293. [7] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157(1-2) (1986), 99–157. [8] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal. 73(1) (1987), 179–194. [9] E. M. Stein,“Singular integrals and differentiability properties of functions”, Princeton Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970. [10] E. M. Stein, Oscillatory integrals in Fourier analysis, in “Beijing lectures in harmonic analysis” (Beijing, 1984), Ann. of Math. Stud. 112, Princeton Univ. Press, Princeton, N.J., 1986, pp. 307–355. Oscillatory Singular Integrals 611 [11] E. M. Stein,“Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, Princeton Mathematical Series 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. Leslie C. Cheng: Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010 U.S.A. E-mail address:[email protected] Yibiao Pan: Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260 U.S.A. E-mail address:[email protected] Primera versi´o rebuda el 21 de desembre de 1999, darrera versi´o rebuda el 31 de gener de 2000.