Uniform Estimates of the Cauchy–Riemann Equation and Kerzman’s Question

Abstract

We answer a question raised by Kerzman in 1971. More precisely, we show that the canonical solution of the

\overline{\partial }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}

-equation satisfies the

L^p

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}

estimate on the polydisc for

p\in [1,\infty ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \in [1, \infty ]$$\end{document}

. Moreover, the

L^p

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}

estimates for

p\in [1,\infty ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \in [1, \infty ]$$\end{document}

of

\overline{\partial }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}

can also be obtained on the product of bounded

C^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}

planar domains by an observation based on the method developed in [Dong et al. arXiv:2006.14484].

\overline{\partial }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}

equation / Polydisc / Product domains / Uniform estimates / 32W05 / 32A25

References

[1] Barletta E, Landucci M. Optimal L∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^\infty $$\end{document} estimates for the canonical solution of the CR-equation in the nonsmooth case. Complex Variables Theory Appl., 1991, 16(2–3): 93-106

[2] Berndtsson B. A smooth pseudoconvex domain in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document} for which L∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^\infty $$\end{document}-estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} do not hold. Ark. Mat., 1993, 31(2): 209-218

[3] Chen L, McNeal JD. A solution operator for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on the Hartogs triangle and Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} estimates. Math. Ann., 2020, 376(1–2): 407-430

[4] Chen, L., McNeal, J.D.: Product domains, Multi-Cauchy transforms, and the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} equation. Adv. Math. 360, 106930, 42 pp. (2020)

[5] Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables. AMS/IP Studies in Advanced Mathematics, vol. 19. American Mathematical Society, Providence, RI; International Press, Boston, MA (2001)

[6] Diederich K, Fischer B, Fornæss JE. Hölder estimates on convex domains of finite type. Math. Z., 1999, 232(1): 43-61

[7] Dong, X., Pan, Y., Zhang, Y.: Uniform estimates for the canonical solution to the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}-equation on product domains. arXiv:2006.14484 (2020)

[8] Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (2010)

[9] Fassina, M., Pan, Y.: Supnorm estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on product domains in Cn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^n$$\end{document}. Acta Math. Sin. (Engl. Ser) 40(10), 2307–2323 (2024)

[10] Fefferman CL, Kohn JJ. Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds. Adv. in Math., 1988, 69(2): 223-303

[11] Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex, Annals of Mathematics Studies, No. 75. Princeton University Press, Princeton, NJ.; University of Tokyo Press, Tokyo (1972)

[12] Fornæss JE. Sup-norm estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}. Ann. of Math. (2), 1986, 123(2): 335-345

[13] Fornæss JE, Lee L, Zhang Y. On supnorm estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on infinite type convex domains in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}. J. Geom. Anal., 2011, 21(3): 495-512

[14] Fornæss, J.E., Sibony, N.: Smooth pseudoconvex domains in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document} for which the corona theorem and Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} fail. In: Complex Analysis and Geometry, pp. 209–222, Univ. Ser. Math., Plenum, New York (1993)

[15] Grauert H, Lieb I. Das Ramirezsche Integral und die Lösung der Gleichung ∂¯f=α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }} f=\alpha $$\end{document} im Bereich der beschränkten Formen. Rice Univ. Stud., 1970, 56(2): 29-50

[16] Grundmeier D, Simon L, Stensønes B. Sup-norm estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}. Pure Appl. Math. Q., 2022, 18(2): 531-571

[17] Henkin GM. Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications. Mat. Sb. (N.S.), 1969, 78(4): 611-632

[18] Henkin GM. Integral representation of functions in strictly pseudoconvex domains and applications to the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}-problem. Mat. Sb. (N.S.), 1970, 82(2): 300-308

[19] Henkin GM . A uniform estimate for the solution of the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}-problem in a Weil region. Uspehi Mat. Nauk, 1971, 26(3): 211-212

[20] Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North-Holland Publishing Co., Amsterdam (1990)

[21] Jin M, Yuan Y. On the canonical solution of ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on polydisks. C. R. Math. Acad. Sci. Paris, 2020, 358(5): 523-528

[22] Kerzman N. Hölder and Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} estimates for solutions of ∂¯u=f\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }} u=f$$\end{document} in strongly pseudoconvex domains. Comm. Pure Appl. Math., 1971, 24: 301-379

[23] Landucci M. On the projection of L2(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2(D)$$\end{document} into H(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(D)$$\end{document}. Duke Math. J., 1975, 42: 231-237

[24] Li SY . Solving the Kerzman’s problem on the sup-norm estimate for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on product domains. Trans. Amer. Math. Soc., 2024, 377(9): 6725-6750

[25] Ohsawa, T.: L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} Approaches in Several Complex Variables—Towards the Oka–Cartan Theory with Precise Bounds, 2nd edn. Springer Monographs in Mathematics. Springer, Tokyo (2018)

[26] Range RM. Integral kernels and Hölder estimates for ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document} on pseudoconvex domains of finite type in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}. Math. Ann., 1990, 288(1): 63-74

[27] Range, R.M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, vol. 108. Springer-Verlag, New York (1986)

[28] Range RM, Siu Y-T. Uniform estimates for the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}-equation on domains with piecewise smooth strictly pseudoconvex boundaries. Math. Ann., 1973, 206: 325-354

[29] Sibony N. Un exemple de domaine pseudoconvexe regulier où l’équation ∂¯u=f\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }} u = f$$\end{document} n’admet pas de solution bornée pour f bornée. Invent. Math., 1980, 62(2): 235-242

[30] Straube, E.J.: Lectures on the L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}-Sobolev Theory of the ∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bar{\partial }}$$\end{document}-Neumann Problem, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich (2010)

[31] Yuan, Y.: Uniform estimates of the Cauchy–Riemann equation on product domains, arXiv:2207.02592 (2022)

Rights & permissions

The Author(s)