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')">

\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

References

[1] BarlettaE, LanducciM. 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, 162–393-106

[2] BerndtssonB. 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, 312209-218

[3] ChenL, McNealJD. 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, 3761–2407-430

[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] DiederichK, FischerB, FornæssJE. Hölder estimates on convex domains of finite type. Math. Z., 1999, 232143-61

[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] FeffermanCL, KohnJJ. Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds. Adv. in Math., 1988, 692223-303

[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] FornæssJE. 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, 1232335-345

[13] FornæssJE, LeeL, ZhangY. 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, 213495-512

[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] GrauertH, LiebI. 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, 56229-50

[16] GrundmeierD, SimonL, StensønesB. 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, 182531-571

[17] HenkinGM. Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications. Mat. Sb. (N.S.), 1969, 784611-632

[18] HenkinGM. 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, 822300-308

[19] HenkinGM . 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, 263211-212

[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] JinM, YuanY. 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, 3585523-528

[22] KerzmanN. 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] LanducciM. 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] LiSY . 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, 37796725-6750

[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] RangeRM. 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, 288163-74

[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] RangeRM, SiuY-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] SibonyN. 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, 622235-242

[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)

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.