Singular integrals on regular curves in the Heisenberg group. (English. French summary) Zbl 1483.43009
Summary: Let \(\mathbb{H}\) be the first Heisenberg group, and let \(k\in C^\infty(\mathbb{H}\smallsetminus \{0\})\) be a kernel which is either odd or horizontally odd, and satisfies
\[
|\nabla_{\mathbb{H}}^nk(p)|\leq C_n\|p\|^{-1-n},\quad p\in\mathbb{H} \smallsetminus \{0 \},\, n\geq 0.
\]
The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel \(k(p)=\nabla_{\mathbb{H}}\log \|p\|\). We prove that convolution with \(k\), as above, yields an \(L^2\)-bounded operator on regular curves in \(\mathbb{H}\). This extends a theorem of G. David to the Heisenberg group.
As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield \(L^2\) bounded operators on Lipschitz flags in \(\mathbb{H}\). This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by V. Chousionis and S. Li [Anal. PDE 10, No. 6, 1407–1428 (2017; Zbl 1369.28004)].
As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield \(L^2\) bounded operators on Lipschitz flags in \(\mathbb{H}\). This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by V. Chousionis and S. Li [Anal. PDE 10, No. 6, 1407–1428 (2017; Zbl 1369.28004)].
MSC:
| 43A80 | Analysis on other specific Lie groups |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 28A75 | Length, area, volume, other geometric measure theory |
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
Citations:
Zbl 1369.28004References:
| [1] | Bellaïche, André, The tangent space in sub-Riemannian geometry, (Sub-Riemannian Geometry. Sub-Riemannian Geometry, Progr. Math., vol. 144 (1996), Birkhäuser: Birkhäuser Basel), 1-78 · Zbl 0862.53031 |
| [2] | Burago, Dmitri; Burago, Yuri; Ivanov, Sergei, A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33 (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0981.51016 |
| [3] | Calderón, A.-P., Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA, 74, 4, 1324-1327 (1977) · Zbl 0373.44003 |
| [4] | Chousionis, Vasileios; Fässler, Katrin; Orponen, Tuomas, Boundedness of singular integrals on \(C^{1 , \alpha}\) intrinsic graphs in the Heisenberg group, Adv. Math., 354, Article 106745 pp. (2019) · Zbl 1427.42017 |
| [5] | Chousionis, Vasileios; Fässler, Katrin; Orponen, Tuomas, Intrinsic Lipschitz graphs and vertical β-numbers in the Heisenberg group, Am. J. Math., 141, 4, 1087-1147 (2019) · Zbl 1426.43003 |
| [6] | Chousionis, Vasileios; Li, Sean, Nonnegative kernels and 1-rectifiability in the Heisenberg group, Anal. PDE, 10, 6, 1407-1428 (2017) · Zbl 1369.28004 |
| [7] | Chousionis, Vasileios; Li, Sean; Zimmerman, Scott, Singular integrals on \(C^{1 , \alpha}\) regular curves in Carnot groups, J. Anal. Math. (2021), to appear |
| [8] | Chousionis, Vasileios; Mattila, Pertti, Singular integrals on Ahlfors-David regular subsets of the Heisenberg group, J. Geom. Anal., 21, 1, 56-77 (2011) · Zbl 1218.42006 |
| [9] | Chousionis, Vasileios; Mattila, Pertti, Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups, J. Reine Angew. Math., 691, 29-60 (2014) · Zbl 1297.43005 |
| [10] | Chousionis, Vasilis; Li, Sean; Zimmerman, Scott, The traveling salesman theorem in Carnot groups, Calc. Var. Partial Differ. Equ., 58, 1, Article 35 pp. (2019) · Zbl 1405.28003 |
| [11] | Christ, M., A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61, 2, 601-628 (1990) · Zbl 0758.42009 |
| [12] | Christ, Michael, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics, vol. 77 (1990), Published for the Conference Board of the Mathematical Sciences: Published for the Conference Board of the Mathematical Sciences Washington, DC, by the American Mathematical Society, Providence, RI · Zbl 0745.42008 |
| [13] | Citti, G.; Manfredini, M.; Pinamonti, A.; Serra Cassano, F., Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group, Calc. Var. Partial Differ. Equ., 49, 3-4, 1279-1308 (2014) · Zbl 1291.22011 |
| [14] | Coifman, R. R.; David, G.; Meyer, Y., La solution des conjecture de Calderón, Adv. Math., 48, 2, 144-148 (1983) · Zbl 0518.42024 |
| [15] | Coifman, R. R.; McIntosh, A.; Meyer, Y., L’intégrale de Cauchy définit un opérateur borné sur \(L^2\) pour les courbes lipschitziennes, Ann. Math. (2), 116, 2, 361-387 (1982) · Zbl 0497.42012 |
| [16] | David, G.; Semmes, S., Singular integrals and rectifiable sets in \(\text{R}^n\): Au-delà des graphes lipschitziens, Astérisque, 193, 152 (1991) · Zbl 0743.49018 |
| [17] | David, G.; Semmes, S., Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38 (1993), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0832.42008 |
| [18] | David, Guy, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Éc. Norm. Supér. (4), 17, 1, 157-189 (1984) · Zbl 0537.42016 |
| [19] | David, Guy, Opérateurs d’intégrale singulière sur les surfaces régulières, Ann. Sci. Éc. Norm. Supér. (4), 21, 2, 225-258 (1988) · Zbl 0655.42013 |
| [20] | David, Guy, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Mathematics, vol. 1465 (1991), Springer-Verlag: Springer-Verlag Berlin |
| [21] | Guy, David; Journé, Jean-Lin, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math. (2), 120, 2, 371-397 (1984) · Zbl 0567.47025 |
| [22] | Guy, David; Mattila, Pertti, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoam., 16, 1, 137-215 (2000) · Zbl 0976.30016 |
| [23] | David, Guy; Semmes, Stephen, Quantitative rectifiability and Lipschitz mappings, Trans. Am. Math. Soc., 337, 2, 855-889 (1993) · Zbl 0792.49029 |
| [24] | Di Donato, Daniela; Fässler, Katrin; Orponen, Tuomas, Metric rectifiability of \(\mathbb{H} \)-regular surfaces with Hölder continuous horizontal normal (2020), arXiv e-prints |
| [25] | Di Donato, Daniela; Fässler, Katrin, Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups, Ann. Mat. Pura Appl. (2021) |
| [26] | Fabes, E. B.; Jodeit, M.; Rivière, N. M., Potential techniques for boundary value problems on \(C^1\)-domains, Acta Math., 141, 3-4, 165-186 (1978) · Zbl 0402.31009 |
| [27] | Fabes, E. B.; Rivière, N. M., Dirichlet and Neumann problems for the heat equation in \(C^1\)-cylinders, (Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2. Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 179-196 · Zbl 0436.35039 |
| [28] | Fässler, K.; Orponen, T.; Rigot, S., Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group, Trans. Am. Math. Soc., 373, 8, 5957-5996 (2020) · Zbl 1446.28005 |
| [29] | Fässler, Katrin; Orponen, Tuomas, Riesz transform and vertical oscillation in the Heisenberg group, Anal. PDE (2021), to appear · Zbl 1457.46046 |
| [30] | Ferrari, Fausto; Franchi, Bruno; Pajot, Hervé, The geometric traveling salesman problem in the Heisenberg group, Rev. Mat. Iberoam., 23, 2, 437-480 (2007) · Zbl 1142.28004 |
| [31] | Franchi, Bruno; Serapioni, Raul; Serra Cassano, Francesco, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321, 3, 479-531 (2001) · Zbl 1057.49032 |
| [32] | Franchi, Bruno; Serapioni, Raul; Serra Cassano, Francesco, Intrinsic Lipschitz graphs in Heisenberg groups, J. Nonlinear Convex Anal., 7, 3, 423-441 (2006) · Zbl 1151.58005 |
| [33] | Franchi, Bruno; Serapioni, Raul; Serra Cassano, Francesco, Differentiability of intrinsic Lipschitz functions within Heisenberg groups, J. Geom. Anal., 21, 4, 1044-1084 (2011) · Zbl 1234.22002 |
| [34] | Garnett, John B.; Marshall, Donald E., Harmonic Measure, New Mathematical Monographs, vol. 2 (2008), Cambridge University Press: Cambridge University Press Cambridge, Reprint of the 2005 original · Zbl 1139.31001 |
| [35] | Grafakos, Loukas, Classical Fourier Analysis, Graduate Texts in Mathematics, vol. 249 (2014), Springer: Springer New York · Zbl 1304.42001 |
| [36] | Grafakos, Loukas, Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250 (2014), Springer: Springer New York · Zbl 1304.42002 |
| [37] | Hajłasz, Piotr; Zimmerman, Scott, Geodesics in the Heisenberg group, Anal. Geom. Metric Spaces, 3, 1, 325-337 (2015) · Zbl 1328.53039 |
| [38] | Hofmann, Steve, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J., 90, 2, 209-259 (1997) · Zbl 0941.42006 |
| [39] | Hofmann, Steve; Lewis, John L., \( L^2\) solvability and representation by caloric layer potentials in time-varying domains, Ann. Math. (2), 144, 2, 349-420 (1996) · Zbl 0867.35037 |
| [40] | Jerison, David S., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I, J. Funct. Anal., 43, 1, 97-142 (1981) · Zbl 0493.58021 |
| [41] | Jerison, David S., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II, J. Funct. Anal., 43, 2, 224-257 (1981) · Zbl 0493.58022 |
| [42] | Jones, Peter W., Lipschitz and bi-Lipschitz functions, Rev. Mat. Iberoam., 4, 1, 115-121 (1988) · Zbl 0782.26007 |
| [43] | Jones, Peter W., Square functions, Cauchy integrals, analytic capacity, and harmonic measure, (Harmonic Analysis and Partial Differential Equations. Harmonic Analysis and Partial Differential Equations, El Escorial, 1987. Harmonic Analysis and Partial Differential Equations. Harmonic Analysis and Partial Differential Equations, El Escorial, 1987, Lecture Notes in Math., vol. 1384 (1989), Springer: Springer Berlin), 24-68 · Zbl 0675.30029 |
| [44] | Jones, Peter W., Rectifiable sets and the traveling salesman problem, Invent. Math., 102, 1, 1-15 (1990) · Zbl 0731.30018 |
| [45] | Journé, Jean-Lin, Calderón-Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón, Lecture Notes in Mathematics, vol. 994 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0508.42021 |
| [46] | Juillet, Nicolas, A counterexample for the geometric traveling salesman problem in the Heisenberg group, Rev. Mat. Iberoam., 26, 3, 1035-1056 (2010) · Zbl 1206.28003 |
| [47] | Lewis, John L.; Murray, Margaret A. M., The method of layer potentials for the heat equation in time-varying domains, Mem. Am. Math. Soc., 114, 545 (1995), viii+157 · Zbl 0826.35041 |
| [48] | Li, Sean, Stratified β-numbers and traveling salesman in Carnot groups (February 2019), arXiv e-prints, page |
| [49] | Li, Sean; Schul, Raanan, The traveling salesman problem in the Heisenberg group: upper bounding curvature, Trans. Am. Math. Soc., 368, 7, 4585-4620 (2016) · Zbl 1350.53044 |
| [50] | Li, Sean; Schul, Raanan, An upper bound for the length of a traveling salesman path in the Heisenberg group, Rev. Mat. Iberoam., 32, 2, 391-417 (2016) · Zbl 1355.28005 |
| [51] | Mattila, Pertti; Serapioni, Raul; Serra Cassano, Francesco, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 9, 4, 687-723 (2010) · Zbl 1229.28004 |
| [52] | Naor, Assaf; Young, Robert, Vertical perimeter versus horizontal perimeter, Ann. Math. (2), 188, 1, 171-279 (2018) · Zbl 1397.46020 |
| [53] | Nazarov, Fedor; Tolsa, Xavier; Volberg, Alexander, On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1, Acta Math., 213, 2, 237-321 (2014) · Zbl 1311.28004 |
| [54] | Nazarov, Fedor; Tolsa, Xavier; Volberg, Alexander, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat., 58, 2, 517-532 (2014) · Zbl 1312.44005 |
| [55] | Nazarov, Fedor; Treil, Sergei; Volberg, Alexander, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Not., 9, 463-487 (1998) · Zbl 0918.42009 |
| [56] | Okikiolu, Kate, Characterization of subsets of rectifiable curves in \(\text{R}^n\), J. Lond. Math. Soc. (2), 46, 2, 336-348 (1992) · Zbl 0758.57020 |
| [57] | Orponen, Tuomas; Villa, Michele, Sub-elliptic boundary value problems in flag domains (June 2020), arXiv e-prints, page |
| [58] | Rigot, Séverine, Quantitative notions of rectifiability in the Heisenberg groups (Apr 2019), arXiv e-prints, page |
| [59] | Rudin, Walter, Functional Analysis, International Series in Pure and Applied Mathematics (1991), McGraw-Hill, Inc.: McGraw-Hill, Inc. New York · Zbl 0867.46001 |
| [60] | Semmes, Stephen, Analysis vs. geometry on a class of rectifiable hypersurfaces in \(\text{R}^n\), Indiana Univ. Math. J., 39, 4, 1005-1035 (1990) · Zbl 0796.42014 |
| [61] | Semmes, Stephen W., A criterion for the boundedness of singular integrals on hypersurfaces, Trans. Am. Math. Soc., 311, 2, 501-513 (1989) · Zbl 0675.42015 |
| [62] | Stein, Elias M.; Murphy, Timothy S., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Monographs in Harmonic Analysis (1993), Princeton University Press · Zbl 0821.42001 |
| [63] | Tolsa, Xavier, Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc. (3), 98, 2, 393-426 (2009) · Zbl 1194.28005 |
| [64] | Tolsa, Xavier, Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón-Zygmund Theory, Progress in Mathematics, vol. 307 (2014), Birkhäuser/Springer: Birkhäuser/Springer Cham · Zbl 1290.42002 |
| [65] | Verchota, Gregory, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59, 3, 572-611 (1984) · Zbl 0589.31005 |
| [66] | Zimmerman, Scott, Singular integrals on \(C_{w^\ast}^{1 , \alpha}\) regular curves in Banach duals (December 2020), arXiv e-prints, page |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
