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Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

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Abstract

This article is concerned with the geometry of germs of real analytic surfaces in \(({\mathbb {C}}^2,0)\) having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in \(({\mathbb {C}}^2,0)\) having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions \(\{\tau _1,\tau _2\}\) at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to \(\{z_1z_2=const\}\) (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

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Notes

  1. Through the paper, for any \({{\mathcal {S}}}\subset {{\mathbb {R}}}\), \(|{{\mathcal {S}}}|\) denotes its Lebesgue measure.

References

  1. Arnold, V.I.: Proof of a theorem by A.N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18(5), 9–36 (1963)

  2. Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften. Springer (1988)

  3. Avila, A., Fayad, B., Krikorian, R.: A KAM scheme for \({{\rm SL}}(2,{\mathbb{R}})\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21(5), 1001–1019 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings, Princeton Math. Ser., vol. 47. Princeton University Press, Princeton (1999)

  5. Berti, M., Biasco, L., Procesi, M.: KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. Éc. Norm. Supér. (4) 46(2), 301–373 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bost, J.-B.: Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolmogorov, Arnol’d, Moser, Rüssmann, Zehnder, Herman, Pöschel,...). In Séminaire Bourbaki, vols. 133–134 of Astérisque, pp. 113–157. Société Mathématiques de France, exposé 639 (1986)

  8. Broer, H.W., Huitema, G.W., Sevryuk, M.B.: Quasi-periodic motions in families of dynamical systems, Lecture Notes in Mathematics, vol. 1645. Springer (1996)

  9. Bruno, A.D.: Analytical form of differential equations. Trans. Mosc. Math. Soc. 25, 131–288 (1971)

    Google Scholar 

  10. Bruno, A.D.: Analytical form of differential equations. Trans. Mosc. Math. Soc. 26, 199–239 (1972)

    Google Scholar 

  11. Cartan, E.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 4(1), 333–354 (1932)

  12. Chern, S.S., Moser, J.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chierchia, L., Pinzari, G.: The planetary \(N\)-body problem: symplectic foliation, reductions and invariant tori. Invent. Math. 186, 1–77 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coffman, A.: Analytic stability of the CR cross-cap. Pac. J. Math. 226(2), 221–258 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Coffman, A.: CR singularities of real fourfolds in \({\mathbb{C}}^3\). Ill. J. Math. 53(3), 939–981 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Eliasson, L.H.: Floquet solutions for the \(1\)-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992)

    Article  MATH  Google Scholar 

  17. Eliasson, L.H.: Absolutely convergent series expansions for quasi periodic motions. Math. Phys. Electron. J. 2, Paper 4 (1996) (electronic)

  18. Féjoz, J.: Démonstration du ‘théorème d’Arnold’ sur la stabilité du système planétaire (d’après Herman). Ergod. Theory Dyn. Syst. 24(5), 1521–1582 (2004)

    Article  MATH  Google Scholar 

  19. Forstnerič, F., Stout, E.L.: A new class of polynomially convex sets. Ark. Mat. 29(1), 51–62 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gong, X.: On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents. Comment. Math. Helv. 69(4), 549–574 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gong, X.: Fixed points of elliptic reversible transformations with integrals. Ergod. Theory Dyn. Syst. 16(4), 683–702 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gong, X.: Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics. Indiana Univ. Math. J. 53(1), 83–96 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gong, X., Stolovitch, L.: Real submanifolds of maximum complex tangent space at a CR singular point I. Invent. Math. 206, 293–377 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gong, X., Stolovitch, L.: Real submanifolds of maximum complex tangent space at a CR singular point II. J. Differ. Geom. 112, 121–198 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190(1), 209–260 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, X., Yin, W.: A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric. Int. Math. Res. Not. IMRN 15, 2789–2828 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Huang, X., Yin, W.: A Bishop surface with a vanishing Bishop invariant. Invent. Math. 176(3), 461–520 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Huang, X., Yin, W.: Flattening of CR singular points and analyticity of the local hull of holomorphy I. Math. Ann. 365(1–2), 381–399 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang, X., Yin, W.: Flattening of CR singular points and analyticity of the local hull of holomorphy II. Adv. Math. 308, 1009–1073 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Huang, X., Fang, H.: Flattening a non-degenerate CR singular point of real codimension two. Geom. Funct. Anal. 28(2), 289–333 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations. Graduate Studies in Mathematics, vol. 86. American Mathematical Society, Providence (2008)

  33. Klingenberg, W., Jr.: Asymptotic curves on real analytic surfaces in \({{\bf C}}^2\). Math. Ann. 273(1), 149–162 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kolmogorov, A.N.: On the preservation of conditionally periodic motions under small variations of the Hamilton function. Dokl. Akad. Nauk SSSR 98(4), 527–530 (1954). English translation in “Selected Works”, Kluwer

  35. Kossovskiy, I., Zaitsev, D.: Convergent normal form for real hypersurfaces at a generic Levi-degeneracy. J. Reine Angew. Math. 749, 201–225 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lamel, B., Mir, N.: Convergence and divergence of formal CR mappings. Acta Math. 220(2), 367–406 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Marmi, S., Moussa, P., Yoccoz, J.-C.: Linearization of generalized interval exchange maps. Ann. Math. 176(3), 1583–1646 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1, 1–20 (1962)

  39. Moser, J.: Analytic surfaces in \({\bf C}^2\) and their local hull of holomorphy. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 397–410 (1985)

  40. Moser, J., Webster, S.M.: Normal forms for real surfaces in \({\bf C}^{2}\) near complex tangents and hyperbolic surface transformations. Acta Math. 150(3–4), 255–296 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pyartli, A.S.: Diophantine approximation on submanifolds of Euclidean space. Funct. Anal. Appl. 3, 303–306 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rüssmann, H.: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6(2), 119–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sevryuk, M.B.: Reversible systems. Lecture Notes in Mathematics, vol. 1211. Springer, Berlin (1986)

  44. Sevryuk, M.B.: Mathematical Reviews MR1406428 (97f:58103) of [21] Mathematical Reviews. American Mathematical Society, Providence (1997)

  45. Sevryuk, M.B.: The reversible context 2 in KAM theory: the first steps. Regul. Chaotic Dyn. 16(1–2), 24–38 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sevryuk, M.B.: KAM theory for lower dimensional tori within the reversible context 2. Mosc. Math. J. 12(2), 435–455, 462 (2012)

  47. Stolovitch, L.: Singular complete integrability. Publ. Math. Inst. Hautes Études Sci. 91, 133–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stolovitch, L.: A KAM phenomenon for singular holomorphic vector fields. Publ. Math. Inst. Hautes Études Sci. 102, 99–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems. I. Commun. Pure Appl. Math. 28(1), 91–140 (1975)

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Acknowledgements

The authors thank X. Gong for his interest, stimulating discussions that help them to substantially improve the exposition of the main results. The authors would also like to thank L. Lempert and M. Procesi for their interests and sharp comments which help them to improve the text. The authors thank also the anonymous referees for the careful review and helpful suggestions.

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Correspondence to Laurent Stolovitch.

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Communicated by Ngaiming Mok.

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This work has been supported by the French government through the ANR grant ANR-15-CE40-0001-03 for the project Bekam and through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.

Appendix A: Proof of Lemma 6.11

Appendix A: Proof of Lemma 6.11

Let \(\varsigma :=\max \{\Vert f_1-f_2\Vert _{{{\mathcal {O}}},\beta '',r''},\ \Vert g_1-g_2\Vert _{{{\mathcal {O}}},\beta '',r''}\}\), which is, by (98), smaller than \(\frac{\beta '^2}{16}\). Let \(\omega \in {{{\mathcal {O}}}}(r'', \beta '')\). In order to estimate its norm, let us first decompose the following expression:

$$\begin{aligned}&h(e^{{\mathrm{i}}b\alpha (\xi \eta )}\xi +f_1,e^{-{\mathrm{i}}b\alpha (\xi \eta )}\eta +g_1)-h(e^{{\mathrm{i}}b\alpha (\xi \eta )}\xi +f_2,e^{-{\mathrm{i}}b\alpha (\xi \eta )}\eta +g_2) \nonumber \\&\quad = \sum _{l\ge 0}\left( h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\nonumber \right. \\&\left. \qquad -h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) \end{aligned}$$
(204)
$$\begin{aligned}&\ \ \ \ \ \ \cdot \, \left( e^{{\mathrm{i}}b \alpha }\xi +f_2\right) ^l\nonumber \\&\qquad +\, \sum _{l\ge 1} h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1) \, \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^l- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^l \right) \nonumber \\ \end{aligned}$$
(205)
$$\begin{aligned}&\qquad + \, \text {similar expressions involving }h_{0,j}\text { instead of } h_{l,0}. \end{aligned}$$
(206)
  • Terms in (204)

Expanding \(h_{l,0}\) around \(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2\) in (204), we obtain:

$$\begin{aligned}&h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1 g_1)\nonumber \\&\qquad -h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2) \nonumber \\&\quad = \sum _{k\ge 1}\frac{1}{k!}h^{(k)}_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\nonumber \\&\qquad \cdot \left( (e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1 g_1)-(e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\right) ^k.\nonumber \\ \end{aligned}$$
(207)

Combining Corollary 6.5 together with Remark 6.6 and (98), we obtain, for \((\xi ,\eta )\in {{\mathcal {C}}}^{r''}_{\omega ,\beta ''}\) and \(m=1,2\):

$$\begin{aligned}&|\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_m+e^{{\mathrm{i}}b\alpha }\xi g_m+f_m g_m-\omega |\\&\quad \le |\xi \eta -\omega |+|e^{-{\mathrm{i}}b\alpha }\eta f_m+e^{{\mathrm{i}}b\alpha }\xi g_m+f_m g_m|\\&\quad<\beta ''+2e^{\frac{9}{8}{{\tilde{\beta }}}}r''\cdot \frac{\beta '^2}{16}+\frac{\beta '^4}{256}\\&\quad<\frac{\beta '}{2}+\frac{101}{200}\cdot \frac{\beta '}{16}+\frac{\beta '}{256}<\beta '. \end{aligned}$$

By Cauchy’s inequality, we have, for all \( (\xi ,\eta )\in {{\mathcal {C}}}_{\omega ,\beta ''}\) and \({{\tilde{k}}}\ge 1\),

$$\begin{aligned} \frac{1}{{{\tilde{k}}}!}\left| h^{({{\tilde{k}}})}_{l,0}(\xi \eta )\right| \le \sup _{|z-\xi \eta |=\frac{\beta '}{2}}|h_{l,0}(z)|\left( \frac{2}{\beta '}\right) ^{\tilde{k}} \le \left( \frac{2}{\beta '}\right) ^{\tilde{k}}|h_{l,0}|_{\omega ,\beta '}. \end{aligned}$$
(208)

We recall that, for \(|z|<1\),

$$\begin{aligned} \sum _{{{\tilde{k}}}\ge k}C_{{{\tilde{k}}}}^k z^{{{\tilde{k}}}-k} =\frac{1}{k!}\sum _{{{\tilde{k}}}\ge 0} \frac{d^k}{dz^k}\left( z^{\tilde{k}}\right) =\frac{1}{k!}\frac{d^k}{dz^k}\left( \frac{1}{1-z}\right) =(1-z)^{-(k+1)}. \end{aligned}$$

Hence, developing \(h^{(k)}_{l,0}\) around \(\xi \eta \), we have, for \(k\ge 1\),

$$\begin{aligned}&\frac{1}{k!}\left\| h^{(k)}_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\right\| _{\omega ,\beta '',r''}\nonumber \\&\quad \le \sum _{{{\tilde{k}}}\ge k} \frac{\left| h^{({{\tilde{k}}})}_{l,0}\right| _{\omega ,\beta ''}}{k!({{\tilde{k}}}-k)!} \Vert e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2\Vert ^{{{\tilde{k}}}-k}_{\omega ,\beta '',r''}\nonumber \\&\quad = \sum _{{{\tilde{k}}}\ge k} \frac{{{\tilde{k}}}!}{({{\tilde{k}}}-k)!\cdot k!}\cdot \frac{\left| h^{(\tilde{k})}_{l,0}\right| _{\omega ,\beta ''}}{{{\tilde{k}}}!} \Vert e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2\Vert ^{\tilde{k}-k}_{\omega ,\beta '',r''}\nonumber \\&\quad \le |h_{l,0}|_{\omega ,\beta '}\sum _{{{\tilde{k}}}\ge k}C_{{{\tilde{k}}}}^k \left( \frac{2}{\beta '}\right) ^{{{\tilde{k}}}} \left( \frac{\beta '^2}{16}\right) ^{{{\tilde{k}}}-k}\nonumber \\&\quad = \left( \frac{2}{\beta '}\right) ^{k} |h_{l,0}|_{\omega ,\beta '}\sum _{{{\tilde{k}}}\ge k}C_{{{\tilde{k}}}}^k \left( \frac{\beta '}{8}\right) ^{{{\tilde{k}}}-k}\nonumber \\&\quad =\left( \frac{2}{\beta '}\right) ^{k} |h_{l,0}|_{\omega ,\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-(k+1)}. \end{aligned}$$
(209)

Recalling that \(|\omega |<r''^2-\beta ''\), by Lemma 3.3, we have, for \(-1\le b\le 1\),

$$\begin{aligned}&\left\| (e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)-(e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right\| _{\omega ,\beta '',r''} \nonumber \\&\quad \le \left\| e^{-{\mathrm{i}}b\alpha }\eta (f_1-f_2)\right\| _{\omega ,\beta '',r''}+ \left\| e^{{\mathrm{i}}b\alpha }\xi (g_1-g_2)\right\| _{\omega ,\beta '',r''} \nonumber \\&\qquad + \Vert (f_1-f_2)g_1-f_2(g_1-g_2)\Vert _{\omega ,\beta '',r''} \nonumber \\&\quad< e^{\frac{9}{8}{{\tilde{\beta }}}}r''\left( \Vert f_1-f_2\Vert _{\omega ,\beta '',r''}+\Vert g_1-g_2\Vert _{\omega ,\beta '',r''}\right) \nonumber \\&\qquad + \, \Vert f_1-f_2\Vert _{\omega ,\beta '',r''}\Vert g_1\Vert _{\omega ,\beta '',r''}+\Vert f_2\Vert _{\omega ,\beta '',r''}\Vert g_1-g_2\Vert _{\omega ,\beta '',r''} \nonumber \\&\quad< \frac{101}{50} r''\varsigma +\frac{\beta '^2}{8}\varsigma <\frac{13}{25}\varsigma . \end{aligned}$$
(210)

Since \(\frac{\varsigma }{\beta '}<\frac{\beta '}{8}\) and according to (85), we have \(1-\frac{\beta '}{8}>\frac{99}{100}\) and \(1-\frac{26\varsigma }{25\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-1}>\frac{99}{100}\), so that \(\frac{26}{25}\left( 1-\frac{\beta '}{8}\right) ^{-2}\left( 1-\frac{26\varsigma }{25\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-1}\right) ^{-1}<\frac{27}{25}.\) Combining (209) and (210), together with (19), we obtain

$$\begin{aligned}&\left\| h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1 g_1)-h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\right\| _{\omega ,\beta '',r''}\nonumber \\&\quad \le \sum _{k\ge 1}\frac{1}{k!}\Vert h^{(k)}_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\Vert _{\omega ,\beta '',r''}\nonumber \\&\qquad \, \cdot \left\| (e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)-(e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right\| _{\omega ,\beta '',r''}^k \nonumber \\&\quad< \, \Vert h\Vert _{\omega ,\beta ',r'} r'^{-l} \cdot \sum _{k\ge 1}\left( \frac{26\varsigma }{25\beta '}\right) ^k \left( 1-\frac{\beta '}{8}\right) ^{-(k+1)} \nonumber \\&\quad = \, \Vert h\Vert _{\omega ,\beta ',r'} r'^{-l}\frac{26\varsigma }{25\beta '} \left( 1-\frac{\beta '}{8}\right) ^{-2}\left( 1-\frac{26\varsigma }{25\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-1}\right) ^{-1} \nonumber \\&\quad < \, \frac{27\varsigma }{25\beta '}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} r'^{-l}. \end{aligned}$$
(211)

Hence, according to Lemma 3.3, for \(l\ge 0\),

$$\begin{aligned}&\left\| \left( h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1 g_1)-h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2)\right) \right. \\&\qquad \cdot \, \left. (e^{{\mathrm{i}}b\alpha } \xi +f_2)^l\right\| _{\omega ,\beta '',r''}\\&\quad <\frac{27\varsigma }{25\beta '}\Vert h\Vert _{\omega ,\beta ',r'} r'^{-l}\left( e^{\frac{9}{8} {{\tilde{\beta }}}} r''+\frac{\beta '^2}{16}\right) ^{l}. \end{aligned}$$

On the other hand, (95) implies \(2r''{{\tilde{\beta }}} <\frac{r'-r''}{8}\). Indeed, since \(0<r''<r'<\frac{1}{4}\) and \(0<\beta <1\), then \(8\beta ^{\frac{1}{2}}<(r'-r'')r''\) implies

$$\begin{aligned} 2r''{{\tilde{\beta }}}=2r''\cdot 16\beta ^{\frac{5}{4}}<8\beta ^{\frac{5}{4}}<8\beta<8\beta ^{\frac{1}{2}}\cdot \frac{(r'-r'')r''}{8}<\frac{r'-r''}{8}. \end{aligned}$$
(212)

Therefore, according to (86), we have

$$\begin{aligned} r'-e^{\frac{9}{8}{{\tilde{\beta }}}} r''-\frac{\beta '^2}{16}>r'-r''-2{{\tilde{\beta }}} r''>r'-r''-\frac{r'-r''}{8}=\frac{7}{8}(r'-r''). \end{aligned}$$

As a consequence, we have

$$\begin{aligned}\sum _{k\ge 0}r'^{-k}\left( e^{\frac{9}{8} {{\tilde{\beta }}}} r''+\frac{\beta '^2}{16}\right) ^{k} =\frac{r'}{r'-e^{\frac{9}{8}{{\tilde{\beta }}}} r''-\frac{\beta '^2}{16}}<\frac{8r'}{7(r'-r'')}.\end{aligned}$$

Thus, under (95), the \(\Vert \cdot \Vert _{{{\mathcal {O}}},\beta '',r''}\)-norm of (204) is bounded by

$$\begin{aligned} \frac{27\varsigma }{25\beta '}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}\sum _{l\ge 0}r'^{-l}\left( e^{\frac{9}{8} {{\tilde{\beta }}}} r''+\frac{\beta '^2}{16}\right) ^{l}< & {} \frac{27}{25}\cdot \frac{8r'}{7(r'-r'')}\frac{\varsigma }{\beta '}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}\nonumber \\< & {} \frac{5r'}{4(r'- r'')}\frac{\varsigma }{\beta '}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}. \end{aligned}$$
(213)

To show (204)\(\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \) provided that \(f_1,f_2,g_1,g_2\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \), it remains to verify the analyticity on \({{{\mathcal {O}}}}(r'',\beta '') \) for the coefficients (204)\(_{l,j}\), \(l,j\ge 0\), \(lj=0\). According to (207), (209) and (210), for \({{\tilde{l}}}\ge 0\),

$$\begin{aligned}&h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)-h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2\\&\quad +e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2) \in {{{\mathcal {A}}}}_{\beta '',r''}\left( {{{\mathcal {O}}}}\right) ,\end{aligned}$$

and, by (23) and (211), for \(\tilde{l}\ge 0\), for \(l,j\ge 0\) with \(lj=0\),

$$\begin{aligned}&\left| \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \right. \\&\quad \quad \left. \left. \ \ \ - \, h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2) \right) _{l,j}\right| _{{{{\mathcal {O}}}}(r'',\beta '')} \le \frac{27\varsigma }{25\beta '}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{ r'^{{{\tilde{l}}}}r''^{l+j}}. \end{aligned}$$

Note that, for \(l\ge 0\),

$$\begin{aligned} (204)_{l,0}= \,&\sum _{{{\tilde{l}}}\ge 0}\sum _{0\le k \le l}\left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{\tilde{l}}\right) _{k,0}\\&\cdot \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \\&\left. -h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{l-k,0} \\&+ \, \sum _{{{\tilde{l}}}\ge 0} \sum _{k \ge 1}\left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{l+k,0} (\xi \eta )^k\\&\cdot \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \\&\left. -h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{0,k}, \end{aligned}$$

and for \(j\ge 1\),

$$\begin{aligned} (204)_{0,j}=&\sum _{{{\tilde{l}}}\ge 0}\sum _{0\le k \le j}\left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,k}\\&\cdot \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \\&\left. -h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{0,j-k}\\&+ \, \sum _{{{\tilde{l}}}\ge 0} \sum _{k \ge 1}\left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,j+k} (\xi \eta )^k\\&\cdot \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \\&\left. -h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{k,0}, \end{aligned}$$

where, by Lemma 6.5 and by (23), for \(l,j\ge 0\) with \(lj=0\),

$$\begin{aligned}&\left| \left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{l,j}\right| _{{{{\mathcal {O}}}}(r'',\beta '')} \le \left( e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\frac{\beta '^2}{16}\right) ^{{{\tilde{l}}}}r''^{-(l+j)}. \end{aligned}$$

Then we see that, for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '') \),

$$\begin{aligned}&\sum _{{{\tilde{l}}}\ge 0}\sum _{0\le k \le l}\left| \left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{k,0}(\omega )\right. \\&\qquad \cdot \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \\&\left. \left. \qquad -h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{l-k,0}(\omega )\right| \\&\quad \le \frac{27(l+1)\varsigma \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{25\beta ' r''^{l} }\sum _{{{\tilde{l}}}\ge 0}\left( e^{\frac{9}{8}{{\tilde{\beta }}}}\frac{r''}{r'}+\frac{\beta '^2}{16r'}\right) ^{\tilde{l}}, \\&\sum _{{{\tilde{l}}}\ge 0} \sum _{k \ge 1}\left| \left( (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{l+k,0}(\omega )\cdot \omega ^k\right. \\&\qquad \cdot \left. \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right. \right. \\&\left. \left. \qquad -h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2g_2)\right) _{0,k}(\omega )\right| \\&\quad \le \sum _{{{\tilde{l}}}\ge 0}\sum _{k\ge 1} \left( e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\frac{\beta '^2}{16}\right) ^{\tilde{l}}r''^{-(l+k)}(r''^2-\beta '')^k\cdot \frac{27\varsigma }{25\beta '}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{ r'^{\tilde{l}}r''^{k}}\\&\quad \le \frac{27\varsigma \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{25\beta ' r''^{l} } \sum _{k\ge 1}\left( 1-\frac{\beta ''}{r''^2} \right) ^k \sum _{{{\tilde{l}}}\ge 0}\left( e^{\frac{9}{8}{{\tilde{\beta }}}}\frac{r''}{r'}+\frac{\beta '^2}{16r'}\right) ^{\tilde{l}}, \end{aligned}$$

which implies the analyticity of (204)\(_{l,0}\) under (95), and it is similar for that of (204)\(_{0,j}\). Hence, (204)\(\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \).

  • Terms in (205)

Note that (19) and (208) imply that, for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '')\), for \(l\ge 0\),

$$\begin{aligned}&\Vert h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1) \Vert _{\omega ,\beta '',r''}\nonumber \\&\quad \le \sum _{k\ge 0}\frac{|h^{(k)}_{l,0}|_{\omega ,\beta ''}}{k!}\Vert e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1 \Vert ^k_{\omega ,\beta '',r''}\nonumber \\&\quad \le |h_{l,0}|_{\omega ,\beta '}\sum _{k\ge 0}\left( \frac{2}{\beta '}\right) ^{k}\left( 2e^{\frac{9}{8}{{\tilde{\beta }}}}r'' \frac{\beta '^2}{8}+\frac{\beta '^4}{256}\right) ^k\le \frac{101}{100}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{r'^l}. \end{aligned}$$
(214)

In (205), we have, for \(l\ge 1\),

$$\begin{aligned} \Vert (e^{{\mathrm{i}}b\alpha }\xi +f_1)^l- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^l\Vert _{\omega ,\beta '', r''}\le & {} \sum _{k=1}^l C_{l}^k \Vert e^{{\mathrm{i}}b\alpha }\xi +f_1\Vert _{\omega ,\beta '', r''}^{l-k} \Vert f_2-f_1\Vert _{\omega ,\beta '', r''}^k\nonumber \\< & {} \sum _{k=1}^l C_{l}^k \left( e^{\frac{9}{8}{{\tilde{\beta }}}} r''+\frac{\beta '^2}{16}\right) ^{l-k} \varsigma ^k\nonumber \\= & {} \left( e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\frac{\beta '^2}{16}+\varsigma \right) ^l- \left( e^{\frac{9}{8}{{\tilde{\beta }}}} r''+\frac{\beta '^2}{16}\right) ^{l}\nonumber \\\le & {} l \left( e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma \right) ^{l-1} \varsigma . \end{aligned}$$
(215)

Furthermore, by (95), we have \(\beta ^{\frac{5}{4}}<\beta ^{\frac{1}{2}}<\frac{r'-r''}{32}\). Recalling that \(\varsigma <\frac{\beta '^2}{8}\le \frac{(16\beta ^\frac{5}{4})^2}{8}\) and using (212), we have

$$\begin{aligned} 1-\frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}> 1-\frac{(1+2{{\tilde{\beta }}}) r''}{r'} > \frac{r'-r''}{r'}-\frac{r'-r''}{8r'} =\frac{7(r'-r'')}{8r'}. \end{aligned}$$
(216)

Therefore, the \(\Vert \cdot \Vert _{{{\mathcal {O}}},\beta '', r''}\)-norm of (205) is bounded by

$$\begin{aligned}&\sum _{l\ge 1} \Vert h_{l,0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\Vert _{\omega ,\beta '', r''}\nonumber \\&\qquad \cdot \, \Vert (e^{{\mathrm{i}}b\alpha }\xi +f_1)^l- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^l \Vert _{{{\mathcal {O}}},\beta '', r''}\nonumber \\&\quad< \frac{101\varsigma }{100r'}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}\sum _{l\ge 1}l \left( \frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}\right) ^{l-1}\nonumber \\&\quad =\frac{101\varsigma }{100r'}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} \left( 1-\frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}\right) ^{-2} \nonumber \\&\quad< \frac{101\varsigma }{100r'}\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}\cdot \frac{8^2 r'^2}{7^2(r'-r'')^2} <\frac{7r'\varsigma \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{5(r'-r'')^2}. \end{aligned}$$
(217)

For \(l\ge 0\), we have that (205)\(_{l,0}\) equals to

$$\begin{aligned}&\sum _{{{\tilde{l}}}\ge 1}\sum _{0\le k \le l} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{l-k,0}\\&\quad \quad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{k,0}\\&\quad \quad + \, \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{l+k,0} \\&\qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,k}(\xi \eta )^k\\&\quad \quad + \, \sum _{{{\tilde{l}}}\ge 1} \sum _{k \ge 1} \left( h_{{{\tilde{l}}},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{0,k} \\&\qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{l+k,0}(\xi \eta )^k, \end{aligned}$$

and for \(j\ge 1\), (205)\(_{0,j}\) equals to

$$\begin{aligned}&\sum _{{{\tilde{l}}}\ge 1}\sum _{0\le k \le j} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{0,j-k} \\&\qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,k}\\&\qquad + \, \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{0,j+k} \\&\qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{k,0} (\xi \eta )^k\\&\qquad + \, \sum _{{{\tilde{l}}}\ge 1} \sum _{k \ge 1} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{k,0} \\&\qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,j+k} (\xi \eta )^k. \end{aligned}$$

If \(f_1,f_2,g_1,g_2\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \), then by (214)–(216) and (19), we see the analyticity of (205)\(_{l,0}\), since for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '') \),

$$\begin{aligned}&\left| \sum _{{{\tilde{l}}}\ge 1}\sum _{0\le k \le l} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{l-k,0}(\omega ) \right. \\&\left. \quad \quad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{k,0}(\omega )\right| \\&\quad \le \sum _{{{\tilde{l}}}\ge 1}\sum _{0\le k \le l}\frac{101}{100}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{r'^{\tilde{l}}r''^{l-k}}\cdot \frac{{{{\tilde{l}}}} (e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma )^{{{{\tilde{l}}}}-1} \varsigma }{r''^{k}}\\&\quad =\frac{101}{100}\frac{ (l+1) \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} \varsigma }{r' r''^{l}} \sum _{{{\tilde{l}}}\ge 1} {{{\tilde{l}}}} \left( \frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}\right) ^{\tilde{l}-1}, \\&\left| \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{l+k,0} (\omega )\right. \\&\left. \qquad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{0,k}(\omega )\cdot \omega ^k\right| \\&\quad \le \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \frac{101}{100}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{r'^{\tilde{l}}r''^{l+k}}\cdot \frac{{{{\tilde{l}}}} (e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma )^{{{{\tilde{l}}}}-1} \varsigma }{r''^{k}}\cdot (r''^2-\beta '')^k\\&\quad =\frac{101}{100}\frac{ \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} \varsigma }{r' r''^{l}} \sum _{{{\tilde{l}}}\ge 1} {{{\tilde{l}}}} \left( \frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}\right) ^{\tilde{l}-1}\sum _{k \ge 1} \frac{(r''^2-\beta '')^k}{r''^{2k}}, \\&\left| \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \left( h_{\tilde{l},0}(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_1+e^{{\mathrm{i}}b\alpha }\xi g_1+f_1g_1)\right) _{0,k} (\omega ) \right. \\&\left. \quad \quad \times \left( (e^{{\mathrm{i}}b\alpha }\xi +f_1)^{{{\tilde{l}}}}- (e^{{\mathrm{i}}b\alpha }\xi +f_2)^{{{\tilde{l}}}}\right) _{l+k,0}(\omega )\cdot \omega ^k\right| \\&\quad \le \sum _{{{\tilde{l}}}\ge 1}\sum _{k \ge 1} \frac{101}{100}\frac{\Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}}{r'^{\tilde{l}}r''^{k}}\cdot \frac{{{{\tilde{l}}}} (e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma )^{{{{\tilde{l}}}}-1} \varsigma }{r''^{l+k}}\cdot (r''^2-\beta '')^k\\&\quad =\frac{101}{100}\frac{ \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} \varsigma }{r' r''^{l}} \sum _{{{\tilde{l}}}\ge 1} {{{\tilde{l}}}} \left( \frac{e^{\frac{9}{8}{{\tilde{\beta }}}}r''+\beta '^2+\varsigma }{r'}\right) ^{\tilde{l}-1}\sum _{k \ge 1} \frac{(r''^2-\beta '')^k}{r''^{2k}}. \end{aligned}$$

The proof for (205)\(_{0,j}\) is similar, hence (205)\(\in {{{\mathcal {A}}}}_{\beta '',r''}\left( {{{\mathcal {O}}}}\right) \).

Combining (213), (217) and similar estimates obtained for expressions (206), we obtain

$$\begin{aligned}&\Vert h(e^{{\mathrm{i}}b\alpha }\xi +f_1,e^{-{\mathrm{i}}b\alpha }\eta +g_1)-h(e^{{\mathrm{i}}b\alpha }\xi +f_2,e^{-{\mathrm{i}}b\alpha }\eta +g_2)\Vert _{{{\mathcal {O}}},\beta '', r''}\\&\quad<\frac{1}{r'- r''} \left( \frac{14r'}{5(r'- r'')}+\frac{5r'}{2\beta '}\right) \varsigma \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'} < \, \frac{3r'}{(r'-r'')\beta '}\varsigma \Vert h\Vert _{{{\mathcal {O}}},\beta ',r'}, \end{aligned}$$

since (95) implies that \(\beta '\le {{\tilde{\beta }}}<\frac{r'-r''}{64}\). This finishes the proof of Lemma 6.11. \(\square \)

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Stolovitch, L., Zhao, Z. Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions. Math. Ann. 386, 587–672 (2023). https://doi.org/10.1007/s00208-022-02408-6

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