Solving the Kerzman's problem on the sup-norm estimate for $\dbar$ on product domains

In this paper, the author solves the long term open problem of Kerzman on sup-norm estimate for Cauchy-Riemann equation on polydisc in $n$-dimensional complex space. The problem has been open since 1971. He also extends and solves the problem on a bounded product domain $\Omega^n$, where $\Omega$ either is simply connected with $C^{1,\alpha}$ boundary or satisfies a uniform exterior ball condition with piecewise $C^1$ boundary.

[1]  R. X. Dong,et al.  Sharp pointwise and uniform estimates for ∂ , 2019, Analysis & PDE.

[2]  Ziming Shi.  Weighted Sobolev $$L^{p}$$ Estimates for Homotopy Operators on Strictly Pseudoconvex Domains with $$C^{2}$$ boundary , 2019, The Journal of Geometric Analysis.

[3]  J. McNeal,et al.  Product domains, multi-Cauchy transforms, and the ∂¯ equation , 2019, Advances in Mathematics.

[4]  Renjie Chen,et al.  Graham type theorem on classical bounded symmetric domains , 2019, Calculus of Variations and Partial Differential Equations.

[5]  Xianghong Gong.  Hölder estimates for homotopy operators on strictly pseudoconvex domains with $$C^2$$C2 boundary , 2017, 1702.08872.

[6]  Xianghong Gong.  H\"{o}lder estimates for homotopy operators on strictly pseudoconvex domains with $C^2$ boundary , 2017 .

[7]  E. Stein,et al.  The Bergman projection in $L^p$ for domains with minimal smoothness , 2012, 1201.4148.

[8]  D. Chakrabarti,et al.  The Cauchy–Riemann equations on product domains , 2009, 0911.0103.

[9]  S. Bell.  The Green's function and the Ahlfors map , 2007, 0708.3025.

[10]  S. Krantz,et al.  On the existence of smooth plurisubharmonic solutions for certain degenerate monge—ampère equations , 2000 .

[11]  R. Range.  Holomorphic Functions and Integral Representations in Several Complex Variables , 1998 .

[12]  Mei-Chi Shaw.  Optimal Hoelder and L p Estimates for ∂ b on the Boundaries of Real Ellipsoids in C n , 1991 .

[13]  Michael Grüter,et al.  The Green function for uniformly elliptic equations , 1982 .

[14]  M. Landucci.  On the projection of $L\sp{2}(D)$ into $H(D)$ , 1975 .

[15]  C. Fefferman.  The Bergman kernel and biholomorphic mappings of pseudoconvex domains , 1974 .

[16]  N. Kerzman.  Hölder and Lp estimates for solutions of ∂u = f in strongly pseudoconvex domains , 1971 .

[17]  P. Garabedian.  A partial differential equation arising in conformal mapping , 1951 .

[18]  Loredana Lanzani,et al.  Regularity of a @ -Solution Operator for Strongly C-Linearly Convex Domains with Minimal Smoothness , 2019 .

[19]  Hemant Kumar Pathak,et al.  Function Theory of Several Complex Variables , 2019, Complex Analysis and Applications.

[20]  J. Erik.  Sup-norm estimates for 8 in C2 , 2016 .

[21]  Y. Zhang.  On supnorm estimates for d on infinite type convex domains in C2 , 2012 .

[22]  M. Mitrea,et al.  On the Well-posedness of the Dirichlet Problem in Certain Classes of Nontangentially Accessible Domains , 2009 .

[23]  UNIFORM ESTIMATES FOR THE ^-EQUATION ON INTERSECTIONS OF STRICTLY PSEUDOCONVEX DOMAINS , 2007 .

[24]  N. Levenberg,et al.  Function theory in several complex variables , 2001 .

[25]  So-chin Chen,et al.  Partial differential equations in several complex variables , 2001 .

[26]  J. Michel,et al.  THE ∂ PROBLEM ON DOMAINS WITH PIECEWISE SMOOTH BOUNDARIES WITH APPLICATIONS , 1999 .

[27]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[28]  J. Michel,et al.  Local regularity for the tangential Cauchy-Riemann complex. , 1993 .

[29]  J. Leiterer,et al.  Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges , 1993 .

[30]  N. Sibony,et al.  Smooth Pseudoconvex Domains in ℂ 2 For Which the Corona Theorem and L p Estimates for ∂̄ Fail , 1993 .

[31]  S. Krantz.  Optimal lipschitz and Lp estimates for the equation ∂u = f on strongly pseudo-convex domains , 1974 .

[32]  Melanie Shaw.  Transactions of the American Mathematical Society on the Boundaries of Real Ellipsoids in En O. Introduction , 2022 .