Jiman Zhao

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 345 (2007) 377–380http://france.elsevier.com/direct/CRASS1/Mathematical Analysis/Harmonic AnalysisLittlewood–Paley and Lusin functions on stratified groupsJiman ZhaoSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR ChinaReceived 10 May 2007; accepted 12 September 2007Presented by Jean-Pierre KahaneAbstractIn this Note, we define the Littlewood–Paley and Lusin functions associated with the sub-Laplacian operator on stratified groups.The Lp (1 < p < ∞) boundedness of Littlewood–Paley and Lusin functions are proved. To cite this article: J. Zhao, C. R. Acad.Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméLes fonctions de Littlewood–Paley et Lusin sur les groupes stratifiés. Dans cette Note, nous définissons les fonctions deLittlewood–Paley et de Lusin sur les groupes stratifiés. Nous prouvons que pour 1 < p < ∞, elles sont bornées sur Lp . Pour citercet article : J. Zhao, C. R. Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. PreliminariesIn classical harmonic analysis, the Littlewood–Paley functions play an important role in the study of non-tangentialconvergence of Fatou type and the boundedness of Riesz transforms and multipliers [9–11].In [9], Stein extended the Lp boundedness of the vertical Littlewood–Paley G-function to the context of compactLie groups, and the Lp boundedness of the horizontal Littlewood–Paley g-function to a general setting of symmetricMarkov semigroups, for 1 < p < ∞. For the latter see [8] and the references therein. These facts have been generalisedfurther. One direction is the Littlewood–Paley theory on Coifman–Weiss’s spaces of homogeneous type, see [5].Another direction is the study of the Littlewood–Paley functions on non-compact complete Riemannian manifolds,in connection with the study Riesz transforms: some results have been obtained by N. Lohoué for Cartan–Hadamardmanifolds and non-amenable Lie group, see [6,7], and by J.C. Chen for Riemannian manifolds with non-negative Riccicurvature, see [1]. T. Coulhon, X. Duong, X.D. Li studied Littlewood–Paley-Stein functions on complete Riemannianmanifolds for 1  p  2, see [2]. The literature is so vast that we do not give exhaustive references.The aim of this Note is to study the Lp boundedness of Littlewood–Paley functions defined on the stratifiedgroup, where 1 < p < ∞. To prove this, we also need to define the Littlewood–Paley g∗λ function and prove itsLp boundedness. The difficult point is to prove the Lp boundedness where 2 < p < ∞.E-mail addresses: jzhao@bnu.edu.cn, jzhao@pku.org.cn.1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.09.007378 J. Zhao / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 377–380First let us to recall some properties on the stratified group which we will use in the sequel, for more details,see [3,4].A stratified group is a simply connected nilpotent Lie group G endowed with a graded Lie algebra g, which isdecomposed into a direct sum of subspaces Vj :g = V1 ⊕ · · · ⊕ Vm such that Vj+1 = [V1,Vj ] for every j < m, and[V1,Vm] = {0}.The elements of g will be considered as left-invariant vector fields on G, and we fix a basis X1, . . . ,Xn for V1 ⊂g. The operator  = −∑nj=1 X2j is called the sub-Laplacian of G and the associated gradient is defined by ∇ =(X1, . . . ,Xn).We define a one-parameter family {γr : r > 0} of automorphisms of g, called dilation, by the formulaγr(m∑1Yj)=m∑1rjYj (Yj ∈ Vj ).The dilations {γr} on g induce automorphisms of G, still called dilations and defined by φr(x) = rx =exp(γr(exp−1 x)), r > 0, x ∈ G.The number Q = ∑m1 j (dimVj ) is called the homogeneous dimension of G, since d(rx) = rQdx for r > 0, wheredx is bi-invariant Haar measure on G.Let Y → ‖Y‖ be a Euclidean norm on g. If x ∈ G, we set ‖x‖ = ‖ exp−1 x‖. Let x → |x| on G be a homogeneousnorm defined by∣∣∣∣∣expm∑1Yj∣∣∣∣∣ =(m∑1‖Yj‖2m!j) 12m!(Yj ∈ Vj ).The homogeneous norm is continuous on G, C∞ on G − {0}, homogeneous of degree 1, and satisfies (a) |x| > 0 ifx = 0, (b) |x| = |x−1|, where m is the number of steps in the stratification of g.Consider the group G × R, whose Lie algebra has a natural stratification ⊕m1 Wj , where W1 is the span of V1 and∂t and Wj = Vj for j > 1.The corresponding dilations are given by r(x, t) = (rx, rt), the second factor being the ordinary multiplication,and the homogeneous dimension of G × R is Q + 1.The sub-Laplacian of G × R is defined by H = − ∂2∂t2 −∑nj=1 X2j , and the associated gradient is defined by∇H = ( ∂∂t ,X1, . . . ,Xn).Before defining the Poisson kernel, we give the following facts due to [3,4].There is a unique C∞ function K on G × R − {(0,0)} which satisfies (a) K(rx, rt) = r1−QK(x, t), (b) H Kis the Dirac distribution at (0,0)(Q > 1). (This result holds only if Q > 1. If Q = 1, then G = R and H is minusthe classical Laplacian on R2, and we take K to be the usual logarithmic potential.) K is real and satisfies K(x, t) =K(x−1,−t), K(x, t) = K(x,−t),K(x, t) = K(x−1, t).Now we define the Poisson kernel p(x, t) = pt(x) by p(x, t) = A−1q(x, t), (t > 0, x ∈ G), where q(x, t) =∂tK(x, t), and A =∫Gq(x, t)dx = ∫Gq(x,1)dx = 0, then the corresponding operator Pt is defined by Ptf (x) =pt ∗ f (x).We have the following estimate of the Poisson kernel. For further properties of the Poisson kernel, see [4].Proposition 1.1. pt(x) = p(x, t)  Ct(t+|x|)Q+1 , t > 0, x ∈ G.Now we define the Littlewood–Paley and Lusin functions as follows:g(f )(x) =( +∞∫0∣∣∇H (Ptf )(x)∣∣2t dt) 12,g∗λ(f )(x) =( +∞∫ ∫ (tt + |y|)λQ∣∣∇H (Ptf )(xy)∣∣2t−Q+1 dy dt) 12,0 GJ. Zhao / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 377–380 379S(f )(x) =( +∞∫0∫|x−1y|t∣∣∇H (Ptf )(y)∣∣2t−Q+1 dy dt) 12.We will study the boundedness of Littlewood–Paley g-function in the following section.2. The Littlewood–Paley g-functionThe basic results for g are the following, and the proof of the second theorem is more complicated.Theorem 2.1. g is Lp(G) bounded, where 1 < p  2.Theorem 2.2. g is Lp(G) bounded, where 2 < p < ∞.Following Stein’s argument, first we prove four lemmas.Lemma 2.3.H(up) = p(p − 1)up−2|∇H u|2,where u(x, t) = (Ptf )(x), x ∈ G, t > 0,1 < p  2.Using the fact that H u = 0, we can prove it easily.Lemma 2.4.+∞∫0∫GtH (Ptf )(x)dx dt =∫Gf (x)dx.Lemma 2.5. supt>0 |(Ptf )(x)|  CAM(f )(x), where f ∈ Lp(G), p  1, A =∫Gψ(x)dx, ψ(x) = sup|y||x| |Q(y)|,Q(x) = 1(1+|x|)Q+1 .Lemma 2.6. Let f ∈ Lp(G), p  μ,μ  1, then∣∣(Ptf )(xy)∣∣  C(1 + |y|t)QM(f )(x),more generally, we have∣∣(Ptf )(xy)∣∣  Cμ(1 + |y|t)QμMμ(f )(x),where Mμ(f )(x) = (supr>0 1|B(x,r)|∫B(x,r)|f (y)|μ dy) 1μ .By Lemmas 2.4 and 2.5, we can prove Theorem 2.1. To prove Theorem 2.2, we need to prove the followingvector-valued singular integral theorem first.Theorem 2.7. Let B1, B2 be two Hilbert spaces, suppose thatTf (x) =∫GK(x−1y)f (y)dy,is a bounded operator from L2(G,B1) into L2(G,B2). Assume that K satisfies∣∣∇K(x)∣∣B1→B2  C|x|Q+1 ,then there exists a constant Ap such that ‖Tf ‖p  Ap‖f ‖p,1 < p < +∞.380 J. Zhao / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 377–3803. Lusin functionIn this section, we will prove the Lp-boundedness of the Lusin function. By the definition, it is easy to see thatS(f )  Cg∗λ(f ). The main theorem of this part is the following:Theorem 3.1. S(f ) ∈ Lp(G),f ∈ Lp(G),1 < p < ∞.The proof of this theorem depends on the following theorem.Theorem 3.2. g∗λ(f ) ∈ Lp(G), where f ∈ Lp(G),1 < p < ∞.By Lemmas 2.3, 2.4, 2.6, we can prove this theorem.AcknowledgementsThe author would like to express her thanks to Department of Mathematics, Paris-Sud University for its hospitality.References[1] J.C. Chen, Heat kernels on positively curved manifolds and applications, Ph.D. thesis, Hangzhou Univ., 1987.[2] T. Coulhon, X.T. Duong, X.D. Li, Littlewood−Paley−Stein functions on complete Riemannian manifolds for 1  p  2, Studia Math. 154 (1)(2003) 37–57.[3] G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Math. 13 (2) (1975) 161–207.[4] G.B. Folland, Lipschitz classes and Poisson integrals on stratified groups, Studia Math. 66 (1) (1979) 37–55.[5] Y.S. Han, E.T. Sawyer, Littlewood–Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math.Soc. 110 (530) (1994).[6] N. Lohoué, Estimation des fonctions de Littlewood–Paley–Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. École Norm.Sup. (4) 20 (1987) 505–544 (in French).[7] N. Lohoué, Transformées de Riesz et fonctions de Littlewood–Paley sur les groupes non moyennables, C. R. Acad. Sci. Paris Sér. IMath. 306 (7) (1988) 327–330 (in French).[8] S. Meda, On the Littlewood–Paley–Stein g-function, Trans. Amer. Math. Soc. 347 (6) (1995) 2201–2212.[9] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood–Paley Theory, Annals of Mathematics Studies, vol. 63, Princeton Univ.Press, Univ. of Tokyo Press, 1970.[10] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton Univ. Press,1970.[11] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.

Littlewood–Paley and Lusin functions on stratified groups

Zhao, Jiman

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https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2007.09.007.pdf

Littlewood–Paley and Lusin functions on stratified groups

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