Abstract
This article develops a novel approach to the representation of singular integral operators of Calderón–Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón–Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of T(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the \(A_2\) theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative \(A_p\) estimates which are best known, and sharp in the range \(\max \{p,p'\}\ge 3\) for the fully cancellative case.
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Alpert, B.K.: A class of bases in \(L^{2}\) for the sparse representation of integral operators. SIAM J. Math. Anal. 24(1), 246–262 (1993)
Astala, K., Iwaniec, T., Saksman, E.: Beltrami operators in the plane. Duke Math. J. 107(1), 27–56 (2001)
Barron, A., Conde-Alonso, J.M., Ou, Y., Rey, G.: Sparse domination and the strong maximal function. Adv. Math. 345, 1–26 (2019)
Barron, A., Pipher, J.: Sparse domination for bi-parameter operators using square functions (2017). arXiv:1709.05009
Benea, C., Bernicot, F.: Conservation de certaines propriétés à travers un contrôle épars d’un opérateur et applications au projecteur de Leray-Hopf. Ann. Inst. Fourier (Grenoble) 68(6), 2329–2379 (2018)
Bényi, Á., Torres, R.H.: The discrete Calderón reproducing formula of Frazier and Jawerth. Functional analysis, harmonic analysis, and image processing: a collection of papers in honor of Bjorn Jawerth, pp. 79–107 (2017)
Brocchi, G., A sparse quadratic T1 theorem, New York. J. Math. 26, 1232–1272 (2020).
Chang, S.-Y.A., Fefferman, R.: A continuous version of duality of \(H^{1}\) with BMO on the bidisc. Ann. Math. (2) 112(1), 179–201 (1980)
Conde-Alonso, J.M., Culiuc, A., Di Plinio, F., Ou, Y.: A sparse domination principle for rough singular integrals. Anal. PDE 10(5), 1255–1284 (2017)
Cruz, V., Mateu, J., Orobitg, J.: Beltrami equation with coefficient in Sobolev and Besov spaces. Can. J. Math. 65(6), 1217–1235 (2013)
Cruz, V., Tolsa, X.: Smoothness of the Beurling transform in Lipschitz domains. J. Funct. Anal. 262(10), 4423–4457 (2012)
Cruz-Uribe, D.V., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer Basel AG, Basel (2011)
Culiuc, A., Di Plinio, F., Ou, Y.: Uniform sparse domination of singular integrals via dyadic shifts. Math. Res. Lett. 25(1), 21–42 (2018)
David, G., Journé, J.-L.: A boundedness criterion for generalized Calderón–Zygmund operators. Ann. Math. (2) 120(2), 371–397 (1984)
Dragičević, O., Volberg, A.: Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms. Mich. Math. J. 51(2), 415–435 (2003)
Duoandikoetxea, J.: Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260(6), 1886–1901 (2011)
Fefferman, R.: \(A^{p}\) weights and singular integrals. Am. J. Math. 110(5), 975–987 (1988)
Fefferman, R., Pipher, J.: Multiparameter operators and sharp weighted inequalities. Am. J. Math. 119(2), 337–369 (1997)
Fefferman, R.: Harmonic analysis on product spaces. Ann. Math. (2) 126(1), 109–130 (1987)
Ferguson, S.H., Lacey, M.T.: A characterization of product BMO by commutators. Acta Math. 189(2), 143–160 (2002)
Ferguson, S.H., Sadosky, C.: Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures. J. Anal. Math. 81, 239–267 (2000)
Figiel, T.: Singular Integral Operators: A Martingale Approach. Geometry of Banach Spaces (Strobl, 1989), pp. 95–110 (1990)
Frazier, M., Torres, R., Weiss, G.: The boundedness of Calderón–Zygmund operators on the spaces \(\dot{F}^{\alpha ,q}_p\). Rev. Mat. Iberoam. 4(1), 41–72 (1988)
Frazier, M., Jawerth, B., Weiss, G.: Littlewood–Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, vol. 79. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (1991)
Hagelstein, P., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. 367(11), 7999–8032 (2015)
Hart, J., Oliveira, L.: Hardy space estimates for limited ranges of Muckenhoupt weights. Adv. Math. 313, 803–838 (2017)
Holmes, I., Petermichl, S., Wick, B.D.: Weighted little bmo and two-weight inequalities for Journé commutators. Anal. PDE 11(7), 1693–1740 (2018)
Hytönen, T., Lappas, S.: Dyadic representation theorem using smooth wavelets with compact support (2020). arXiv:2003.04019
Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175(3), 1473–1506 (2012)
Journé, J.-L.: Calderón–Zygmund operators on product spaces. Rev. Mat. Iberoam. 1(3), 55–91 (1985)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)
Lacey, M.T.: An elementary proof of the \(A_{2}\) bound. Isr. J. Math. 217(1), 181–195 (2017)
Lacey, M.T., Petermichl, S., Reguera, M.C.: Sharp \(A_{2}\) inequality for Haar shift operators. Math. Ann. 348(1), 127–141 (2010)
Lacey, M.T., Sawyer, E.T., Uriarte-Tuero, I.: Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane. Acta Math. 204(2), 273–292 (2010)
Lerner, A.K.: Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals. Adv. Math. 226(5), 3912–3926 (2011)
Lerner, A.K.: A simple proof of the \(A_{2}\) conjecture. Int. Math. Res. Not. IMRN 14, 3159–3170 (2013)
Lerner, A.K.: On pointwise estimates involving sparse operators. N. Y. Math. 22, 341–349 (2016)
Lerner, A.K.: Quantitative weighted estimates for the Littlewood-Paley square function and Marcinkiewicz multipliers. Math. Res. Lett. 26(2), 537–556 (2019)
Li, K., Martikainen, H., Ou, Y., Vuorinen, E.: Bilinear representation theorem. Trans. Am. Math. Soc. 371(6), 4193–4214 (2019)
Li, K., Martikainen, H., Vuorinen, E.: Bilinear Calderón–Zygmund theory on product spaces. Journal de Mathématiques Pures et Appliquées 138, 356–412 (2020)
Martikainen, H.: Representation of bi-parameter singular integrals by dyadic operators. Adv. Math. 229(3), 1734–1761 (2012)
Meyer, Y.: Continuité sur les espaces de Hölder et de Sobolev des opérateurs définis par des intégrales singulières. Recent progress in Fourier analysis (El Escorial, 1983), pp. 145–172 (1985)
Meyer, Y., Coifman, R.: Wavelets, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge (1997) (Calderón-Zygmund and multilinear operators, Translated from the 1990 and 1991 French originals by David Salinger)
Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I. Invent. Math. 119(2), 199–233 (1995)
Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II. Math. Z. 221(2), 267–291 (1996)
Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Bi-parameter paraproducts. Acta Math. 193(2), 269–296 (2004)
Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Multi-parameter paraproducts. Rev. Mat. Iberoam. 22(3), 963–976 (2006)
Nazarov, F., Treil, S., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12(4), 909–928 (1999)
Ou, Y.: Multi-parameter singular integral operators and representation theorem. Rev. Mat. Iberoam. 33(1), 325–350 (2017)
Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_{p}\) characteristic. Am. J. Math. 129(5), 1355–1375 (2007)
Petermichl, S., Treil, S., Volberg, A.: Why the Riesz transforms are averages of the dyadic shifts? In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), pp. 209–228 (2002)
Petermichl, S., Volberg, A.: Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J. 112(2), 281–305 (2002)
Prats, M.: Sobolev regularity of the Beurling transform on planar domains. Publ. Mat. 61(2), 291–336 (2017)
Prats, M.: Sobolev regularity of quasiconformal mappings on domains. J. Anal. Math. 138(2), 513–562 (2019)
Prats, M., Tolsa, X.: A T (P) theorem for Sobolev spaces on domains. J. Funct. Anal. 268(10), 2946–2989 (2015)
Rahm, R., Sawyer E.T., Wick, B.D., Weighted Alpert Wavelets. J. Fourier Anal. Appl. 27(1), 1 (2021)
Tao, T.: Lecture notes for 247b (2007). https://math.ucla.edu/~tao/247b.1.07w/notes7.pdf
Torres, R.H.: Boundedness results for operators with singular kernels on distribution spaces. Mem. Am. Math. Soc. 90(442), viii+172 (1991)
Vagharshakyan, A.: Recovering singular integrals from Haar shifts. Proc. Am. Math. Soc. 138(12), 4303–4309 (2010)
Wang, K.: The generalization of paraproducts and the full T1 theorem for Sobolev and Triebel–Lizorkin spaces. J. Math. Anal. Appl. 209(2), 317–340 (1997)
Wilson, J.M.: Weighted norm inequalities for the continuous square function. Trans. Am. Math. Soc. 314(2), 661–692 (1989)
Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)
Acknowledgements
The authors are deeply thankful to Alexander Barron, Henri Martikainen and Yumeng Ou for illuminating conversations on bi-parameter T(1) theorems and weighted norm inequalities. The authors gratefully acknowledge Walton Green for his insightful reading and suggestions which led to significant improvements to the clarity of the statements and exposition.
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F. Di Plinio was partially supported by the National Science Foundation under the grants NSF-DMS-2000510, NSF-DMS-2054863. B. D. Wick’s research partially supported in part by NSF grant NSF-DMS-1800057 as well as ARC DP190100970. Data sharing not applicable to this article as no datasets were generated or analyzed.
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Di Plinio, F., Wick, B.D. & Williams, T. Wavelet representation of singular integral operators. Math. Ann. 386, 1829–1889 (2023). https://doi.org/10.1007/s00208-022-02443-3
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DOI: https://doi.org/10.1007/s00208-022-02443-3