Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

References

[1] W. W. Adams, C. A. Berenstein, P. Loustaunau, I. Sabadini and D. C. Struppa, Regular functions of several quaternionic variables and the Cauchy–Fueter complex, J. Geom. Anal. 9 (1999), no. 1, 1–15. 10.1007/BF02923085Search in Google Scholar

[2] J. Ahrens, M. G. Cowling, A. Martini and D. Müller, Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres, Math. Z. 294 (2020), no. 3–4, 1659–1686. 10.1007/s00209-019-02313-wSearch in Google Scholar

[3] S. Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, Bull. Sci. Math. 127 (2003), no. 1, 1–35. 10.1016/S0007-4497(02)00004-0Search in Google Scholar

[4] S. Alesker, Quaternionic Monge–Ampère equations, J. Geom. Anal. 13 (2003), no. 2, 205–238. 10.1007/BF02930695Search in Google Scholar

[5] S. Alesker, Valuations on convex sets, non-commutative determinants, and pluripotential theory, Adv. Math. 195 (2005), no. 2, 561–595. 10.1016/j.aim.2004.08.009Search in Google Scholar

[6] S. Alesker and E. Shelukhin, A uniform estimate for general quaternionic Calabi problem (with an appendix by Daniel Barlet), Adv. Math. 316 (2017), 1–52. 10.1016/j.aim.2017.05.023Search in Google Scholar

[7] S. Alesker and M. Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, J. Geom. Anal. 16 (2006), no. 3, 375–399. 10.1007/BF02922058Search in Google Scholar

[8] D. Alpay, F. Colombo, T. Qian and I. Sabadini, The H ∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators, J. Funct. Anal. 271 (2016), no. 6, 1544–1584. 10.1016/j.jfa.2016.06.009Search in Google Scholar

[9] D. Alpay, F. Colombo and I. Sabadini, On slice hyperholomorphic fractional Hardy spaces, Math. Nachr. 290 (2017), no. 17–18, 2725–2739. 10.1002/mana.201700011Search in Google Scholar

[10] D. Alpay, F. Colombo and I. Sabadini, Quaternionic de Branges Spaces and Characteristic Operator Function, Springer Briefs Math., Springer, Cham 2020. 10.1007/978-3-030-38312-1Search in Google Scholar

[11] R. J. Baston, Quaternionic complexes, J. Geom. Phys. 8 (1992), no. 1–4, 29–52. 10.1016/0393-0440(92)90042-YSearch in Google Scholar

[12] A. Bonfiglioli, Taylor formula for homogeneous groups and applications, Math. Z. 262 (2009), no. 2, 255–279. 10.1007/s00209-008-0372-zSearch in Google Scholar

[13] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their sub-Laplacians, Springer Monogr. Math., Springer, Berlin, 2007. Search in Google Scholar

[14] J. Bureš, A. Damiano and I. Sabadini, Explicit resolutions for the complex of several Fueter operators, J. Geom. Phys. 57 (2007), no. 3, 765–775. 10.1016/j.geomphys.2006.06.001Search in Google Scholar

[15] J. Bureš and V. Souček, Complexes of invariant operators in several quaternionic variables, Complex Var. Elliptic Equ. 51 (2006), no. 5–6, 463–485. 10.1080/17476930500482473Search in Google Scholar

[16] A.-P. Calderón and A. Zygmund, On higher gradients of harmonic functions, Studia Math. 24 (1964), 211–226. 10.4064/sm-24-2-211-226Search in Google Scholar

[17] D.-C. Chang, X. T. Duong, J. Li, W. Wang and Q. Wu, An explicit formula of Cauchy–Szegő kernel for quaternionic Siegel upper half space and applications, Indiana Univ. Math. J. 70 (2021), no. 6, 2451–2477. 10.1512/iumj.2021.70.8732Search in Google Scholar

[18] D.-C. Chang, I. Markina and W. Wang, On the Cauchy–Szegö kernel for quaternion Siegel upper half-space, Complex Anal. Oper. Theory 7 (2013), no. 5, 1623–1654. 10.1007/s11785-012-0282-2Search in Google Scholar

[19] F. Colombo and J. Gantner, Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, Oper. Theory Adv. Appl. 274, Birkhäuser/Springer, Cham, 2019. 10.1007/978-3-030-16409-6Search in Google Scholar

[20] F. Colombo, G. Gentili, I. Sabadini and D. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math. 222 (2009), no. 5, 1793–1808. 10.1016/j.aim.2009.06.015Search in Google Scholar

[21] F. Colombo, V. Souček and D. C. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys. 56 (2006), no. 7, 1175–1191. 10.1016/j.geomphys.2005.06.009Search in Google Scholar

[22] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994. Search in Google Scholar

[23] J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), no. 1, 69–70. 10.1090/S0002-9939-1981-0619983-8Search in Google Scholar

[24] G. Dafni, Hardy spaces on some pseudoconvex domains, J. Geom. Anal. 4 (1994), no. 3, 273–316. 10.1007/BF02921583Search in Google Scholar

[25] X. Dou, G. Ren, I. Sabadini and T. Yang, Weak slice regular functions on the n-dimensional quadratic cone of octonions, J. Geom. Anal. 31 (2021), no. 11, 11312–11337. 10.1007/s12220-021-00682-5Search in Google Scholar

[26] Z. Fan, M. Lacey and J. Li, Schatten classes and commutators of Riesz transform on Heisenberg group and applications, J. Fourier Anal. Appl. 29 (2023), no. 2, Paper No. 17. 10.1007/s00041-023-09995-1Search in Google Scholar

[27] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. 10.1007/BF01406845Search in Google Scholar

[28] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), no. 3–4, 137–193. 10.1007/BF02392215Search in Google Scholar

[29] M. Feldman and R. Rochberg, Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group, Analysis and Partial Differential Equations, Lecture Notes Pure Appl. Math. 122, Dekker, New York (1990), 121–159. Search in Google Scholar

[30] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University, Princeton, 1982. 10.1515/9780691222455Search in Google Scholar

[31] J. B. Garnett and R. H. Latter, The atomic decomposition for Hardy spaces in several complex variables, Duke Math. J. 45 (1978), no. 4, 815–845. 10.1215/S0012-7094-78-04539-8Search in Google Scholar

[32] D. Geller, Some results in H p theory for the Heisenberg group, Duke Math. J. 47 (1980), no. 2, 365–390. 10.1215/S0012-7094-80-04722-5Search in Google Scholar

[33] G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), no. 1, 279–301. 10.1016/j.aim.2007.05.010Search in Google Scholar

[34] C. R. Graham, The Dirichlet problem for the Bergman Laplacian. I, Comm. Partial Differential Equations 8 (1983), no. 5, 433–476. 10.1080/03605308308820275Search in Google Scholar

[35] S. Janson and T. H. Wolff, Schatten classes and commutators of singular integral operators, Ark. Mat. 20 (1982), no. 2, 301–310. 10.1007/BF02390515Search in Google Scholar

[36] J.-L. Journé, Calderón–Zygmund operators on product spaces, Rev. Mat. Iberoam- 1 (1985), no. 3, 55–91. 10.4171/rmi/15Search in Google Scholar

[37] S. Kołodziej and M. Sroka, Regularity of solutions to the quaternionic Monge–Ampère equation, J. Geom. Anal. 30 (2020), no. 3, 2852–2864. 10.1007/s12220-020-00394-2Search in Google Scholar

[38] A. Korányi, A Poisson integral for homogeneous wedge domains, J. Anal. Math. 14 (1965), 275–284. 10.1007/BF02806394Search in Google Scholar

[39] S. G. Krantz and S.-Y. Li, On decomposition theorems for Hardy spaces on domains in 𝐂 n and applications, J. Fourier Anal. Appl. 2 (1995), no. 1, 65–107. 10.1007/s00041-001-4023-6Search in Google Scholar

[40] L. Lanzani and E. M. Stein, The Cauchy–Szegő projection for domains in ℂ n with minimal smoothness, Duke Math. J. 166 (2017), no. 1, 125–176. 10.1215/00127094-3714757Search in Google Scholar

[41] S. Lord, E. McDonald, F. Sukochev and D. Zanin, Quantum differentiability of essentially bounded functions on Euclidean space, J. Funct. Anal. 273 (2017), no. 7, 2353–2387. 10.1016/j.jfa.2017.06.020Search in Google Scholar

[42] A. Nagel, J.-P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szegő kernels in 𝐂 2 , Ann. of Math. (2) 129 (1989), no. 1, 113–149. 10.2307/1971487Search in Google Scholar

[43] V. V. Peller, Nuclearity of Hankel operators, Mat. Sbornik 113 (1980), 538–581. Search in Google Scholar

[44] V. V. Peller, Hankel Operators and Their Applications, Springer Monogr. Math., Springer, New York, 2003. 10.1007/978-0-387-21681-2Search in Google Scholar

[45] T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann. 310 (1998), no. 4, 601–630. 10.1007/s002080050162Search in Google Scholar

[46] R. Rochberg and S. Semmes, A decomposition theorem for BMO and applications, J. Funct. Anal. 67 (1986), no. 2, 228–263. 10.1016/0022-1236(86)90038-8Search in Google Scholar

[47] R. Rochberg and S. Semmes, Nearly weakly orthonormal sequences, singular value estimates, and Calderon–Zygmund operators, J. Funct. Anal. 86 (1989), no. 2, 237–306. 10.1016/0022-1236(89)90054-2Search in Google Scholar

[48] G. Sarfatti, Quaternionic Hankel operators and approximation by slice regular functions, Indiana Univ. Math. J. 65 (2016), no. 5, 1735–1757. 10.1512/iumj.2016.65.5896Search in Google Scholar

[49] Y. Shi and W. Wang, The tangential k-Cauchy–Fueter complexes and Hartogs’ phenomenon over the right quaternionic Heisenberg group, Ann. Mat. Pura Appl. (4) 199 (2020), no. 2, 651–680. 10.1007/s10231-019-00895-0Search in Google Scholar

[50] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[51] E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Math. Notes 11, Princeton University, Princeton, 1972. Search in Google Scholar

[52] E. M. Stein and G. Weiss, Generalization of the Cauchy–Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163–196. 10.2307/2373431Search in Google Scholar

[53] R. S. Strichartz, Self-similarity on nilpotent Lie groups, Generalized Convex Bodies and Generalized Envelopes, Contemp. Math. 140, American Mathematical Society, Providence (1992), 123–157. 10.1090/conm/140/1197594Search in Google Scholar

[54] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge University, Cambridge, 1999. Search in Google Scholar

[55] D. Wan and W. Wang, On the quaternionic Monge–Ampère operator, closed positive currents and Lelong–Jensen type formula on the quaternionic space, Bull. Sci. Math. 141 (2017), no. 4, 267–311. 10.1016/j.bulsci.2015.03.001Search in Google Scholar

[56] W. Wang, The k-Cauchy–Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic k-regular functions, J. Geom. Phys. 60 (2010), no. 3, 513–530. 10.1016/j.geomphys.2009.11.011Search in Google Scholar

[57] W. Wang, On the linear algebra in the quaternionic pluripotential theory, Linear Algebra Appl. 562 (2019), 223–241. 10.1016/j.laa.2018.10.018Search in Google Scholar

[58] W. Wang, The Neumann problem for the k-Cauchy–Fueter complex over k-pseudoconvex domains in ℝ 4 and the L 2 estimate, J. Geom. Anal. 29 (2019), no. 2, 1233–1258. 10.1007/s12220-018-0037-zSearch in Google Scholar

[59] W. Wang and Q. Wu, Atomic decomposition theorem for Hardy spaces on products of Siegel upper half spaces and bi-parameter Hardy spaces, J. Geom. Anal. 33 (2023), Article No. 351. 10.1007/s12220-023-01384-wSearch in Google Scholar