We show that the $L^2(w)$ operator norm of the composition $M\!\circ
T_{\Omega}$, where $M$ is the maximal operator and $T_{\Omega}$ is a rough
homogeneous singular integral with angular part $\Omega\in
L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence
is sharp

Lerner, Andrei K.

English

arXiv

Andrei K. Lerner

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 356 (2018) 77–80Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comHarmonic analysisA note on weighted bounds for rough singular integralsUne note sur les bornes pondérées pour les intégrales singulières rugueusesAndrei K. Lerner 1Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israela r t i c l e i n f o a b s t r a c tArticle history:Received 27 September 2017Accepted after revision 24 November 2017Available online 8 December 2017Presented by the Editorial BoardWe show that the L2(w) operator norm of the composition M ◦ T , where M is the maximal operator and T is a rough homogeneous singular integral with angular part  ∈ L∞(Sn−1), depends quadratically on [w]A2 , and that this dependence is sharp.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons que la norme d’opérateur L2(w) du composé M ◦ T , où M est l’opérateur maximal et T est une intégrale singulière homogène rugueuse de partie angulaire  ∈L∞(Sn−1), dépend de manière quadratique de [w]A2 et que cette dépendance est précise.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionConsider a class of rough homogeneous singular integrals defined byT f (x) = p.v.∫Rnf (x − y)(y/|y|)|y|n dy,with  ∈ L∞(Sn−1) and having zero average over the sphere.In [7], Hytönen, Roncal and Tapiola proved that‖T‖L2(w)→L2(w) ≤ Cn‖‖L∞[w]2A2 , (1.1)where [w]A2 = supQ∫Q w∫Q w−1|Q |2 . Different proofs of this result, via a sparse domination, were given by Conde-Alonso, Culiuc, Di Plinio, and Ou [3], and by the author [8]. Recently, (1.1) was extended to maximal singular integrals by Di Plinio, Hytönen, and Li [4].E-mail address: lernera@math.biu.ac.il.1 The author was supported by ISF grant No. 447/16 and ERC Starting Grant No. 713927.https://doi.org/10.1016/j.crma.2017.11.0161631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.78 A.K. Lerner / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 77–80It was conjectured in [7] that the quadratic dependence on [w]A2 in (1.1) can be improved to the linear one. In this note, we obtain a strengthening of (1.1), which, to a certain extent, supports this conjecture.Theorem 1.1. For every w ∈ A2 , we have‖M◦T‖L2(w)→L2(w) ≤ Cn‖‖L∞[w]2A2 , (1.2)and this bound is optimal, in general.Here M denotes the standard Hardy–Littlewood maximal operator. Notice that ‖M‖L2(w)→L2(w)  [w]A2 , and this bound is sharp [1]. Therefore, (1.2) cannot be obtained via a simple combination of the sharp linear bound for M with (1.1). The proof of (1.2) is based essentially on the technique introduced in [8].2. PreliminariesRecall that a family of cubes S is called sparse if there exists 0 < α < 1 such that for every Q ∈ S , one can find a measurable set E Q ⊂ Q with |E Q | ≥ α|Q |, and the sets {E Q }Q ∈S are pairwise disjoint.Given a sublinear operator T , define the maximal operator Mp,T byMp,T f (x) = supQ x⎛⎜⎝ 1|Q |∫Q|T ( f χRn\3Q )|pdy⎞⎟⎠1/p.Denote 〈 f 〉p,Q =(1|Q |∫Q | f |p)1/p.Proposition 2.1. Assume that T and Mp,T are of weak type (1, 1) and, moreover, ‖Mp,T ‖L1→L1,∞ ≤ Kp for all p ≥ 2. Then‖T ‖L2(w)→L2(w) ≤ Cn(‖T ‖L1→L1,∞ + K )[w]2A2 . (2.1)Proof. This is just a combination of several known facts. By [8, Cor. 3.2], for every suitable f , g , there exists a sparse family S such that|〈T f , g〉| ≤ Cn(‖T ‖L1→L1,∞ + Kp′)∑Q ∈S〈 f 〉1,Q 〈g〉p,Q |Q | (p > 1).But it was shown in [3] (see the proof of Corollary A1 there) that this sparse bound implies (2.1). In particular, T with  ∈ L∞ satisfies the hypothesis of Proposition 2.1, namely, it was proved in [8] that‖Mp,T f ‖L1,∞ ≤ Cn‖‖L∞ p‖ f ‖L1 (p ≥ 1). (2.2)3. Proof of Theorem 1.1First, by a general extrapolation argument found in [9], the sharpness of (1.2) follows from ‖M ◦ T‖Lp→Lp ≥ c(p−1)2 as p → 1. The latter relation holds for a subclass of T with kernels satisfying the standard nondegeneracy assumptions. In particular, it can be easily checked for the Hilbert transform.Turn to the proof of (1.2). By homogeneity, one can assume that ‖‖L∞ = 1. The proof is based on two pointwise estimates:M(T f )(x)  MM f (x) + M1,T f (x) (3.1)andMp,(M1,T) f (x)  M f (x) + Mp,T f (x) (p ≥ 2) (3.2)(we use the usual notation A  B if A ≤ Cn B).Let us show first how to complete the proof using these estimates. By (2.2), M1,T is of weak type (1, 1). Applying (2.2)again along with (3.2) yields ‖Mp,(M1,T )‖L1→L1,∞  p. Therefore, by Proposition 2.1,‖M1,T‖L2(w)→L2(w)  [w]2A2 .This estimate, combined with (3.1) and Buckley’s linear bound for M [1], implies (1.2).A.K. Lerner / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 77–80 79It remains to prove (3.1) and (3.2). We start with (3.1). This estimate follows from the definition of M1,T and the standard fact that, for every cube Q containing the point x,1|Q |∫Q|T( f χ3Q )|  MM f (x). (3.3)For the sake of completeness, we outline the proof of (3.3). Combining the weak type (1, 1) and the L2 boundedness of T (see [2,10]) with interpolation and Yano’s extrapolation [6, p. 43], we obtain1|Q |∫Q|T( f χ3Q )|  ‖ f ‖L log L,3Q .By Stein’s L log L result [11],‖ f ‖L log L,Q  1|Q |∫QM f ,which, along with the previous estimate, implies (3.3).Turn to the proof of (3.2). Let R be an arbitrary cube containing the point x. Let y ∈ R and let Q be an arbitrary cube containing y.Assume that Q ≤ 12 R . Then Q ⊂ 2R and 3Q ⊂ 3R . Hence,1|Q |∫Q|T( f χRn\(3R∪3Q ))| ≤ M2R(T( f χRn\3R))(y), (3.4)where M2R denotes the maximal operator restricted to the cube 2R .Suppose now that R < 2Q . Then R ⊂ 5Q and 3R ⊂ 9Q . We obtain|T( f χ15Q \(3R∪3Q ))χQ (z)| 1|Q |∫15Q| f |  M f (x).Also,1|Q |∫Q|T( f χRn\15Q )|  1|5Q |∫5Q|T( f χRn\15Q )| M1,T f (x),and therefore,1|Q |∫Q|T( f χRn\(3R∪3Q ))|dz  M f (x) + M1,T f (x).This estimate, combined with (3.4), impliesM1,T( f χRn\3R)(y) = supQ y1|Q |∫Q|T( f χRn\(3R∪3Q ))| M2R(T( f χRn\3R))(y) + M f (x) + M1,T f (x).Therefore, by the Lp-boundedness of M ,⎛⎝ 1|R|∫RM1,T( f χRn\3R)pdy⎞⎠1/p⎛⎝ 1|R|∫2R|T( f χRn\3R)|pdy⎞⎠1/p+ M f (x) + M1,T f (x).Combining this estimate with|T( f χRn\3R)χ2R(y)|  M f (x) + |T( f χRn\6R)χ2R(y)|and using also that, by Hölder’s inequality, M1,T f ≤ Mp,T f , we obtain⎛⎝ 1|R|∫RM1,T( f χRn\3R)pdy⎞⎠1/p M f (x) + Mp,T f (x),which proves (3.2), and therefore, Theorem 1.1 is completely proved.80 A.K. Lerner / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 77–80Remark 3.1. Define the maximal singular integral byT  f (x) = supε>0∣∣∣∣∣∣∣∫|y|>εf (x − y)(y/|y|)|y|n dy∣∣∣∣∣∣∣.As we have mentioned in the introduction, it was shown in [4] that‖T ‖L2(w)→L2(w) ≤ Cn‖‖L∞[w]2A2 . (3.5)We outline an alternative proof of this result based on Theorem 1.1.Assume that ‖‖L∞ = 1. DenoteT j f (x) =∫2 j<|y|≤2 j+1f (x − y)(y/|y|)|y|n dy.Let  be a Schwartz function supported in the unit ball and such that ∫ = 1. Set k(x) = 2−kn(2−kx). It was proved in [5] thatT  f (x)  M f (x) + M(T f)(x) +∞∑j=0A j f (x), (3.6)whereA j f (x) = supk∈Z|T j+k f (x) − k ∗ T j+k f (x)|.The standard Fourier transform estimates (which again can be found in [5]) show that ‖A j‖L2→L2  2−α j with some fixed 0 < α < 1. Also, the standard convolution estimates imply A j f (x)  M f (x). Applying the interpolation with a change of measures along with the sharp reverse Hölder estimates (as it was done in [7]), we obtain∥∥∥∞∑j=0A j f∥∥∥L2(w) [w]2A2‖ f ‖L2(w),which, along with (3.6) and Theorem 1.1, proves (3.5).AcknowledgementI would like to thank the referee for useful comments that improved the presentation.References[1] S.M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1) (1993) 253–272.[2] A.P. Calderón, A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956) 289–309.[3] J.M. Conde-Alonso, A. Culiuc, F. Di Plinio, Y. Ou, A sparse domination principle for rough singular integrals, Anal. PDE 10 (5) (2017) 1255–1284.[4] F. Di Plinio, T.P. Hytönen, K. Li, Sparse bounds for maximal rough singular integrals via the Fourier transform, preprint, available at https://arxiv.org/abs/1706.09064.[5] J. Duoandikoetxea, J.L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (3) (1986) 541–561.[6] L. Grafakos, Classical Fourier Analysis, second edition, Graduate Texts in Mathematics, vol. 249, Springer, New York, 2009.[7] T.P. Hytönen, L. Roncal, O. Tapiola, Quantitative weighted estimates for rough homogeneous singular integrals, Isr. J. Math. 218 (1) (2017) 133–164.[8] A.K. Lerner, A weak type estimate for rough singular integrals, Rev. Mat. Iberoam. (2017), in press, available at https://arxiv.org/abs/1705.07397.[9] T. Luque, C. Pérez, E. Rela, Optimal exponents in weighted estimates without examples, Math. Res. Lett. 22 (1) (2015) 183–201.[10] A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1) (1996) 95–105.[11] E.M. Stein, Note on the class L log L, Stud. Math. 32 (1969) 305–310.

A note on weighted bounds for rough singular integrals

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